Grade 6 Math Curriculum Notebook

Statistics and Probability 6SP5a
Core Content
Cluster Title: Summarize and describe distributions.
Standard: 5. Summarize numerical data sets in relation to their context, such as by: a. Reporting the number of observations.
MASTERY Patterns of Reasoning:
Conceptual:
• Understand what an observation is (e.g. sample size, n size) and how it relates to numerical data sets.
• Understand and explain why the number of observations is important to summarizing numerical data sets
Procedural:
• Find and report the total number of observations given a plot. Representational:
• Show where the number of observations is or can be represented in a data display (line plot, histogram, box plot) and explain why that representation is efficient or not efficient.
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand how to collect data.
• Understand how to find data sets in media such as newspapers and webpages Procedural:
• Be able to count the number of observations Representational:
• Represent data in line plots, histograms, and box plots Academic Vocabulary
Data set, n - size, Observation, Sample size
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Instructional Strategies Used
Show examples of plots and ask students to find the total observations. Be sure to clarify the difference between intervals and observations.
Assessment Tasks Used
Skill-based Task
Students will find total observations given different plots.
Resources Used
http://www.ade.az.gov/standards/math/2010MathStandards/Gradelevel/MathGr6.pdf
Problem Task
Have students use data found in newspaper or other media to interpret total number of observations in that data set. Have them explain why the number of observations is important for that set of data.
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Statistics and Probability 6SP5b
Core Content
Cluster Title: Summarize and describe distributions.
Standard: 5. Summarize numerical data sets in relation to their context, such as by:
b. Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.
MASTERY Patterns of Reasoning:
Conceptual:
• Understand how the data was gathered.
• Justify the appropriateness of the process used for data collection.
• Understand the importance of the units used in the data sets.
Procedural:
• Identify and describe the attribute being measured.
• Describe how the data was gathered. Representational:
• Interpret labels given on the plot including horizontal and vertical axes, the number line, title, and legend.
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Experience with horizontal and vertical axes
• Experience with units of measurement. Procedural:
• Set up a plot including the axes, the number line, the title, and the legend Representational:
• Represent intervals within units of measure Academic Vocabulary
Abbreviations for common measurements, Attribute, characteristic, investigation
Instructional Strategies Used
Resources Used
Bring in samples of plots from media and have students identify attributes. Assessment Tasks Used
Skill-based Task:
Students will identify attributes and unit of measurement of a given data set
http://www.ade.az.gov/standards/math/2010MathStandards/Gradelevel/MathGr6.pdf
Problem Task:
Students will gather data, and defend measurement technique and choice of unit.
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Statistics and Probability 6SP5c
Core Content
Cluster Title: Summarize and describe distributions.
Standard: 5. Summarize numerical data sets in relation to their context, such as by:
c. Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
MASTERY Patterns of Reasoning:
Conceptual:
• Interpret a set of numerical data beyond measures of center and variability by noticing and describing patterns and deviations.
• Understand the representation of a set of numerical data in the context of the data.
• Understand mean absolute deviation Procedural:
• Determine variability such as interquartile range (use mean absolute deviation as an extension).
• Describe the overall pattern of data.
• Describe any striking deviations from the overall pattern (outliers). Representational:
• Create and use data plots to interpret a set of data Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand measures of center and how they are used to interpret a set of data.
• Understand measures of variability and how they are used to interpret a set of data.
• Understand absolute value.
Procedural:
• Calculate mean, median, and range of a data set. Representational:
• Represent data with plots
Academic Vocabulary and Notation
1st Quartile (Q1), 2nd Quartile (Q2), 3rd Quartile (Q3), 4th Quartile (Q4), Box plot, Distribution, Dot plot, Histogram, Interquartile range, Upper quartile, Lower quartile, Median, Upper endpoint, Lower endpoint, Deviation
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Instructional Strategies Used
Resources Used
Have each student choose two to three books from the library and have them record the number of pages in each. Graph the data as a class on a line plot, then interpret the data and create a box plot. Use the box plot to show variation of data, and then interpret deviation from mean using specific data from the pages recorded
http://www.deltastate.edu/docs/math/Mitchell3.pdf
http://www.ade.az.gov/standards/math/2010MathStandards/Gradelevel/MathGr6.pdf
pages 41-43
Assessment Tasks Used
Skill-based Task:
Students will gather and summarize data using a box plot
Problem Task:
They will also interpret differences in given values compared to the mean (deviation from the mean), noting patterns or deviations in the data display.
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Statistics and Probability 6SP5d
Core Content
Cluster Title: Summarize and describe distributions.
Standard: 5. Summarize numerical data sets in relation to their context, such as by:
d. Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.
MASTERY Patterns of Reasoning:
Conceptual:
• Understand how the context of the data gathered may determine the measures of center and variability chosen to interpret the data.
• Understand how the shape of the data distribution may determine the choice of measure of center and variability.
• Understand that measures of center emphasize different attributes for the data set. (e.g., mean income v. median income)
Procedural:
• Choose the most appropriate measure of center (mean or median) and variability (range).
Representational:
• Represent a set of numerical data in order to show the shape of the data distribution (e.g. in a dot plot)
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Know definitions of mean, median, and range. Procedural:
• Accurately plot a set of numerical data in a dot plot. Representational:
• Represent a set of numerical data in a dot plot (line plot). Academic Vocabulary
Context, data distribution
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Instructional Strategies Resources Used Used
http://www.ohiorc.org/pm/math/richproblemmath.aspx?pmrid=62
Expose students to data This is a rich mathematical problem from the Ohio Resource Center from a company. Students which addresses all four subparts of standard 6.SP.5
must choose the best
measure of center for
different situations (such as choosing between mean and median when showing profit or loss).
After showing students two graphs using the same data, but different scales and/or intervals have students describe the difference in shape.
Assessment Tasks Used
Skill-based Task:
Given a data plot graph with background
information, students will identify the contexthow the context of the data gathered could have
of the data and the measure of center used and will explain why that measure of center was chosen.
influenced the shape of the data (how more music students were on the high end of the honors and awards and grades).
“According to the National Education Longitudinal Study of 1988, music students received more academic honors and awards than non-music students. A higher percentage of music participants received As, As/Bs, and Bs than non-music participants.” (Source: NELS:88 First Follow-up, 1990, National Center for Education Statistics, Washington D.C.)
Problem Task:
Interpret the following statement in terms of
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Pacing Guide
Pacing Guide for Mathematics Essentials
Term 1
Fluently divide multi-digit numbers using the standard algorithm. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
6NS2 6NS3
Write, read, and evaluate expressions in which letters stand for numbers.
6EE2
Term 2
Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions. Understand that positive and negative numbers are used together to describe quantities having opposite directions or values; use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.
6NS1 6NS5
Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
6EE7
Term 3
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
6RP1
6RP2 6RP3c
Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
6NS6c
Term 4
Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
6G1
Summarize numerical data sets in relation to their context, such as by:
a. Reporting the number of observations.
b. Describing the nature of the attribute under investigation, including how it was measured and its units of measurement
c. Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered (only measures of center).
6SP5
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Units
Planning Guide: Jay McTighe, an expert in unit planning and author of Understanding by Design, has written four point to consider when planning units. They are presented below.
UbD Design Standards Stage 1 – To what extent does the design:
1. focus on the “Big ideas” of targeted content? Consider: are . . .
– the targeted understandings enduring, based on transferable, big ideas at the heart of the
discipline and in need of “uncoverage”?
– the targeted understandings framed as specific generalizations?
– the “big ideas” framed by questions that spark meaningful connections, provoke genuine
inquiry and deep thought, and encourage transfer?
– appropriate goals (e.g., content standards, benchmarks, curriculum objectives) identified? – valid and unit-relevant knowledge and skills identified?
Stage 2 – To what extent do the assessments provide:
2. fair, valid, reliable and sufficient measures of the desired results? Consider: are . . .
– students asked to exhibit their understanding through “authentic” performance tasks? – appropriate criterion-based scoring tools used to evaluate student products and
performances?
– a variety of appropriate assessment formats provide additional evidence of learning? Stage 3 – To what extent is the learning plan:
3. effective and engaging? Consider: will students . . .
– know where they’re going (the learning goals), why (reason for learning the content), and
what is required of them (performance requirements and evaluative criteria)?
– be hooked – engaged in digging into the big ideas (e.g., through inquiry, research, problem- solving, experimentation)?
– have adequate opportunities to explore/experience big ideas and receive instruction to equip them for the required performance(s)?
– have sufficient opportunities to rethink, rehearse, revise, and/or refine their work based upon timely feedback?
– have an opportunity to self-evaluate their work, reflect on their learning and set future goals? Consider: the extent to which the learning plan is:
– tailored and flexible to address the interests and learning styles of all students?
– organized and sequenced to maximize engagement and effectiveness?
Overall Design – to what extent is the entire unit:
4. coherent, with the elements of all 3 stages aligned?
Grant Wiggins and Jay McTighe 2005
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Assessment Standards
Q1
Q2
Q3
Q4
Choice of SAGE interim test or grade level common assessment using District LA, school/teacher created, SAGE formative, UTIPS, or commercial test
SAGE Interim, ELA only, no Writing
Grade Level Common Assessment using District LA, school/teacher created, SAGE formative, UTIPS, or commercial test
SAGE summative Window for 2016-17 is Mar 27 – May 19
SAGE Blueprints
Grade 6
50 Operational Items
Domain
Min
Max
Ratios and Proportional Relationships (RP)
28%
32%
The Number System (NS)
18%
22%
Expressions and equations (EE)
28%
34%
Geometry/Statistics and Probability (G/SP)
16%
20%
DOK1
18%
32%
DOK2
46%
62%
DOK3
8%
20%
Disclosure: Depth of Knowledge (DOK) and Elements of Rigor are essential components of the Utah Mathematics Core Standards. As such, DOK and Elements of Rigor are integrated into the Student Assessment of Growth and Excellence (SAGE) assessment items. All students will see a variety of DOK and Elements of Rigor on the SAGE summative assessment. For more information about DOK and Elements of Rigor please see: http//www.schools.utah.gov/assessment/Criterion-Referenced-Tests/Math.aspx
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Purpose of Testing (from USBE testing ethics training)
The purpose of statewide assessment is for accountability.
When administered properly, standardized assessments allow students to demonstrate their abilities, knowledge, aptitude, or skills (see R277 – 404). Valid and reliable results from uniform assessments provide information used by:
Students, to determine how well they have learned the skills and curriculum they are expected to know;
Parents, to know whether their student is gaining the skills and competencies needed to be competitive and successful;
Teachers, to gauge their students’ understanding and identify potential areas of improvement in their teaching;
LEAs (districts or charter schools), to evaluate programs and provide additional support;
State, for school accountability; and
Public, to evaluate schools and districts.
As educators, we are obligated to provide students with an opportunity to demonstrate their knowledge and skills fairly and accurately.
Educators involved with the state – wide assessment of students must conduct testing in a fair and ethical manner (see Utah Code 53A-1-608; R277-404).
The best test preparation a teacher can provide is good instruction throughout the year that covers the breadth and depth of the standards for a course, using varied instructional and assessment activities tailored to individual students.
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Ethical Assessment Practices (USBE ethics training) Licensed Utah Educators should:
• Ensure students are enrolled in appropriate courses and receive appropriate instruction
• Provide instruction to the intended depth and breadth of the course curriculum
• Provide accommodations throughout instruction to eligible students as identified by an
ELL, IEP, or 504 team.
• Use a variety of assessments methods to inform instructional practices
• Introduce students to various test-taking strategies throughout the year
• Provide students with opportunities to engage with available training test to ensure that
they can successfully navigate online testing systems, and to ensure that local
technology configurations can successfully support testing.
• Use formative assessments throughout the year using high-quality, non-secure test
questions aligned to Utah Standards.
Licensed Utah Educators shall ensure that:
• An appropriate environment reflective of an instructional setting is set for testing to limit distractions from surroundings or unnecessary personnel.
• All students who are eligible for testing are tested.
• A student is not discouraged from participating in state assessments, but upon a
parent’s opt-out request (follow LEA procedures), the student is provided with a
meaningful educational activity.
• Tests are administered in-person and testing procedures meet all test administration
requirements.
• Active test proctoring occurs: walking around the room to make sure that each
student has or is logged into the correct test; has appropriate testing materials
available to them; and are progressing at an appropriate pace.
• No person is left alone in a test setting with student tests left on screen or open.
• The importance of the test, test participation, and the good faith efforts of all
students are not undermined.
• All information in the Test Administration Manual (TAM) for each test administered
is reviewed and strictly followed (see 53A-1-608; R277-404).
• Accommodations are provided for eligible students, as identified by an ELL, IEP, or
504 team. These accommodations should be consistent with accommodations
provided during instruction throughout the instructional year.
• Any electronic devices that can be used to access non-test content or to
record/distribute test content or materials shall be inaccessible by students (e.g., cell phones, recording devices, inter-capable devices). Electronic security of tests and student information must not be compromised.
• Test materials are secure before, during and after testing. When not in use, all materials shall be protected, where students, parents cannot gain access.
No one may enter a student’s computer-based test to examine content or alter a student’s response in any way either on the computer or a paper answer document for any reason.
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Unethical Assessment Practices (USBE ethics training)
It is unethical for educators to jeopardize the integrity of an assessment or the validity of student responses.
Unethical practices include:
• Providing students with questions from the test to review before taking the test.
• Changing instruction or reviewing specific concepts because those concepts appear on
the test.
• Rewording or clarifying questions, or using inflection or gestures to help students
answer.
• Allowing students to use unauthorized resources to find answers, including dictionaries,
thesauruses, mathematics tables, online references, etc.
• Displaying materials on walls or other high visibility surfaces that provide answer to
specific test items (e.g., posters, word walls, formula charts, etc.).
• Reclassifying students to alter subgroup reports.
• Allowing parent volunteers to assist with the proctoring of a test their child is taking or
using students to supervise other students taking a test.
• Allowing the public to view secure items or observe testing sessions.
• Reviewing a student’s response and instructing the student to, or suggesting that the
student should, rethink his/her answers.
• Reproducing, or distributing, in whole or in part, secure test content (e.g., taking
pictures, copying, writing, posting in a classroom, posting publically, emailing).
• Explicitly or implicitly encouraging students to not answer questions, or to engage in
dishonest testing behavior.
• Administering tests outside of the prescribed testing window for each assessment.
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Intervention Standards
PCSD MTSS/RTI Model
Provo City School District's Academic MTSS (Multi-Tiered Systems of Support) details the system for providing Tier 1, 2, and 3 instruction; interventions; and assessment to help each student receive appropriate support. It is detailed below.
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Unpacking the Complexity of MTSS Decision Making
Successful MTSS implementation is a highly complex process that involves the following tasks:
• Gathering accurate and reliable data
• Correctly interpreting and validating data
• Using data to make meaningful instructional changes for students
• Establishing and managing increasingly intensive tiers of support
• Evaluating the process at all tiers to ensure the system is working
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Utah’s Multi-Tiered System of Supports USBE website:
http://www.schools.utah.gov/umtss/UMTSS-Model.aspx
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Supplemental Resources
Provo City School District’s Instructional Model
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• Student focus
• Educator credibility
• Meeting norms
• Professional Learning Communities (PLC)/Collaboration
• Civility policy
• Appearance and interactions
• Continual Leaning
• Testing ethics
• Research orientation
• Policy adherence
• Culture
• Safety–emotional and physical
• Physical classroom space
• Relationships
• Family connections
• Procedures
• Classroom management
• Student artifacts
• Student focus
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• Formative evaluation
• Summative evaluation
• Feedback:
• Performance of understanding
• Self-reported grades
• Student self-evaluation
• Testing ethics
• Differentiation
• Data analysis
• Response to interventions (RTI)/Multi-tiered system of success (MTSS)
• Lesson design
• Teacher clarity: share LT, share SC, share PoU
• Evidence-based instructional strategies
• Based on data
• Student engagement
• DOK – Depth of Knowledge
• Differentiation
• Student ownership of learning
• Curriculum notebook
• RTI/MTSS
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• State standards
• Curriculum map/pacing guide
• Units
• Objectives
• Curriculum Notebooks
• Course essentials
• Current
• Planning
Professional Association
The National Council of Teachers of Mathematics NCTM is the largest professional association for mathematics teachers.
Their website is at: https:// http://www.nctm.org/
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Evidence-based Pedagogical Practices
Hattie's Visible Learning
John Hattie, creator of Visible Learning, is a leading education researcher who has analyzed meta analyses in order to rank education practices (and factors) from most effective to least effective.
Hattie's list of highest ranking factors can be found at: https://visible-learning.org/hattie-ranking-influences-effect-sizes-learning-achievement/
or
https://visible-learning.org/nvd3/visualize/hattie-ranking-interactive-2009-2011-2015.html
Hattie's original book on the topic can be found at:
https://www.amazon.com/Visible-Learning-Synthesis-Meta-Analyses- Achievement/dp/0415476186
Definitions of Hattie's factors can be found at:
https://www.amazon.com/Visible-Learning-Synthesis-Meta-Analyses- Achievement/dp/0415476186
National Reading Panel Research
The federal government commissioned a National Reading Panel to review and compile the best evidence of effective practices for reading instruction.
The full report and executive summary can be accessed at:
https://lincs.ed.gov/communications/NRP
Learning Targets
Provo City School District employs the use of learning targets, success criteria, formative assessment, and feedback. A basis of study on these topics is the book, Learning Targets, by Connie Moss and Susan Brookhart, can be found at: https://www.amazon.com/Learning-Targets-Helping-Students-Understanding- ebook/dp/B008FOKP5S.
The district has produced four videos that demonstrate elements of learning target instruction and can be found at:
http://provo.edu/teachingandlearning/learning-targets-videos/
Teacher Resource Guide
Provo City School District's Teacher Resource Guide helps teachers meet the Utah Effective Teaching Standards and includes effective teaching practices. It can be found at: http://provo.edu/teachingandlearning/wp-content/uploads/sites/4/2016/01/11182016-TRG- fixed.pdf
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Glossary
Assessment Standards
College and Career Readiness
Curriculum Resources
ELA
Essential Learning Standards
Evidence-based Pedagogical Practices
Intervention Standards
Language Standards
A set of criteria to guide the assessment of student learning in a course that is based on Standards/Essentials of the course; this might include formative assessment practices, summative assessments/practices, common assessment plans, feedback practices, and a schedule for testing.
The College and Career Readiness (CCR) anchor standards
and grade-specific standards are necessary complements—the former providing broad standards, the latter providing additional specificity—that together define the skills and understandings that all students must demonstrate.
The materials teachers use to plan, prepare, and deliver instruction, including materials students use to learn about the subject. Such materials include texts, textbooks, tasks, tools, and media. Sometimes organized into a comprehensive program format, they often provide the standards, units, pacing guides, assessments, supplemental resources, interventions, and student materials for a course.
English Language Arts, includes components of Reading, Writing, Speaking and Listening, and Language.
These are also known as power standards. They are particular standards/objectives/indicators that a school/district defines as critical for student learning. In fact, they are so critical that students will receive intervention if they are not learned. Essentials are chosen because they: 1. have endurance, 2. have leverage, and 3. are important for future learning.
A list of teaching strategies that are supported by adequate, empirical research as being highly effective.
A set of criteria to guide teachers to provide additional instruction to students who did not master the content in Tier 1 instruction. This might include: commercial intervention programs, teacher- developed intervention materials, diagnostic testing, RTI/MTSS processes, and a list of essential knowledge/skills that will prompt intervention if the student does not demonstrate mastery.
(L) A component of ELA Standards that focus on conventions of standard English grammar, usage, and mechanics as well as learning other ways to use language to convey meaning effectively.
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Math Content Standards
Mathematical Practice Standards
MTSS
Pacing Guide
Pathways of Progress
Performance of Understanding.
Provo Way Instructional Model
Reading Standards: Foundational Skills
SAGE
(MC) Math Content Standards identify the knowledge of concepts and the skills students need for college and career readiness.
The 8 Mathematical Practice Standards describe varieties of expertise that mathematics educators at all levels seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. All 8 mathematical practice standards are essential standards.
Multi-Tiered Systems of Support is an approach to academic and behavioral intervention. It is part of the intervention standards.
The order and timeline of the instruction of standards, objectives, indicators, and Essentials over the span of a course (semester or year).
(POP) An evaluation of individual student growth or improvement over time compared to other students with the same level of initial skills. It empowers educators to set goals that are meaningful, ambitious, and attainable.
(PoU). Student results that provide compelling evidence that the student has acquired the learning target. (Brookhart, 2012).
The five areas of expectations for successful instruction identified by Provo City School District.
(RF) A component of ELA Standards that focus on helping students gain a foundation where curriculum is intentionally and coherently structured to develop rich content knowledge within and across grades as well as acquiring the habits of reading independently and closely, which are essential to their future success.
Student Assessment of Growth and Excellence. This is the state end of level test for ELA and Math grades 3 – 8, and Science grades 4 – 8.
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