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Mathematics Content FrameworkVERSION 1.0 - SY12-13
CITY OF CHICAGORahm Emanuel
Mayor
CITY BOARD OF EDUCATIONDavid J. Vitale
President
Jesse H. RuizVice President
Members:Henry S. Bienen
Dr. Mahalia A. HinesPenny Pritzker
Rodrigo A. SierraAndrea L. Zopp
CHICAGO PUBLIC SCHOOLSJean-Claude Brizard
Chief Executive Officer
Jennifer Cheatham, Ed.D.Chief of Instruction
Chicago Public SchoolsPage 2
June, 2012
Dear CPS Educators,
We’re pleased to share the first version of the CPS Mathematics Content Framework. This Framework provides a clear path toward implementation of the Common Core State Standards for Mathematics (CCSS-M), a more rigorous set of expectations to help our students develop the critical analysis and problem-solving skills necessary for college and career success.
The development of this Framework was greatly informed by the Partnership for Assessment of Readiness for College and Careers (PARCC) Model Mathematics Content Framework as well as other mathematics education organizations. The PARCC Framework outlines the critical focus areas and progression of skill development necessary to meet the expectations of the CCSS-M. Teachers from schools across the District, including our Early Adopter pilot schools, collaborated with the Department of Mathematics and Science to build this Framework, and critical feedback was provided by Network teams, as well as local and national mathematics experts. We cannot thank them enough for their thoughtful feedback.
The Framework provides clear expectations for implementation in the form of Planning Guides that outline the scope of learning for the 2012-2013 school year. These Planning Guides provide guidance to help teachers build deep conceptual understanding of mathematics, procedural fluency, and critical-thinking and problem-solving skills. The Planning Guides identify sample tasks that are designed to measure the conceptual understanding and mathematical thinking expected of the CCSS-M. Many of these tasks are built from the Mathematics Assessment Resource Service (MARS) tasks. CPS is taking a gradual transition approach to the CCSS-M, and as such it is important to note that the Planning Guides were based on a three-year Bridge Plan. The Bridge Plan outlines the transition across all grades to full implementation of all CCSS-M standards by 2014-15.
The Framework also references a Toolset that provides a sample lesson plan template, sample grade-specific tasks, a tool for examining and modifying lessons, and a professional resource list. Included in this document are sample tasks for grade 8 and Algebra I. Complete Toolsets for grades 6-8, as well as Algebra I and Geometry, will be available on our Knowledge Management site: https://ocs.cps.k12.il.us/sites/IKMC, and on the Department of Mathematics and Science website at http://cmsi.cps.k12.il.us/.
Before diving into this document, it’s important to first become familiar with the standards. Earlier this year, every CPS teacher received a personal copy of the standards. Please read through the standards, take note of the learning progressions and the marriage between the standards for mathematical content and practice, and reflect on what shifts will occur in your classroom as a result.
As always, if you have feedback, ideas for resources or have questions about the new standards, do not hesitate to contact us at [email protected].
We look forward to continuing to work with mathematics educators to further refine our strategy and continue to provide support and resources for implementation. This journey together will help ensure that all students reach a level of achievement that puts them on the path to success in college and career. Thank you for all you do every day for our students.
Sincerely,
Jean-Claude Brizard Jennifer Cheatham, Ed.D.
COMMON CORE CPS
CPS Mathematics Content Framework_Version 1.0 Page 3
Table of ContentsCPS Mathematics Content Framework - Version 1.0
IntroductionAcknowledgements ..................................................................................................................................................................5
Overview ..................................................................................................................................................................................7
An Introduction: The Common Core State Standards for Mathematics ...................................................................................8
• K-8 Content Domains .....................................................................................................................................................8
• 9-12 Content Standards ................................................................................................................................................9
• Common Core Standards for Mathematical Practice: K-12 .........................................................................................9
CPS Instructional Shifts in Mathematics .................................................................................................................................11
• Shift 1: Focus on the critical areas to develop deep conceptual understanding and procedural fluency ..................11
• Shift 2: Integrate mathematical practices standards throughout instruction .............................................................12
• Shift 3: Maintain coherence and continuity to link learning within and across grade levels ......................................13
• Shifts in Action: Comparing the Standards in the Middle Grades...............................................................................14
• Shifts in Action: Comparing the Standards in High School .........................................................................................15
The CPS Mathematics Content Framework ............................................................................................................................16
• District-wide Expectations ...........................................................................................................................................16
- CPS Bridge Plan for Mathematics ..........................................................................................................................16
- Grades K – 5 ..........................................................................................................................................................17
- Grades 6 – 8 ..........................................................................................................................................................17
- Algebra I and Geometry ........................................................................................................................................17
- CPS Mathematics Planning Guides ........................................................................................................................18
- Grade 6-8 Planning Guides ....................................................................................................................................19
- High School Course Planning Guides .....................................................................................................................20
- District-wide Benchmark Performance Assessments ............................................................................................21
• CPS Mathematics Toolsets ...........................................................................................................................................22
Getting Started .......................................................................................................................................................................23
Implementation of CCSS: Planning with ALL Learners in Mind ..............................................................................................27
Works Cited ............................................................................................................................................................................29
Planning GuidesGrade 6 Mathematics Planning Guide - SY12-13 Introduction ...............................................................................................33
Grade 6 Mathematics Planning Guide - SY12-13 ....................................................................................................................35
Grade 7 Mathematics Planning Guide - SY12-13 Introduction ...............................................................................................39
Grade 7 Mathematics Planning Guide - SY12-13 ....................................................................................................................41
Grade 8 Mathematics Planning Guide - SY12-13 Introduction ...............................................................................................44
Grade 8 Mathematics Planning Guide - SY12-13 ....................................................................................................................46
TOC continued on next page
Chicago Public SchoolsPage 4
Algebra I Planning Guide - SY12-13 Introduction ...................................................................................................................49
Algebra I Planning Guide - SY12-13 ........................................................................................................................................51
Geometry Planning Guide - SY12-13 Introduction .................................................................................................................68
Geometry Planning Guide - SY12-13 ......................................................................................................................................70
Sample TasksGrade 8 Sample Task ..............................................................................................................................................................88
Algebra I Sample Task .............................................................................................................................................................90
Appendix: Resources to Support the Learning Expected by the CCSS-M ...................................................................93
CPS Mathematics Content Framework_Version 1.0 Page 5
Acknowledgements
The work represented in this document and all associated pieces represents a collaborative effort on the part of many. Teachers and Instructional Support Leaders (ISLs) throughout CPS contributed countless hours to share their best thinking and extensive experience in planning, developing, reviewing, and revising these materials. While these materials represent our best understanding about how to address the challenge of implementing the Common Core State Standards for Mathematics in every CPS classroom, we know there is much to be learned. As we begin our transition to meeting this challenge, we welcome feedback about ways to make the materials more useful.
THANKS to the following for their contribution:
Contributors to 6-8 Materials
Anne AgostinelliTeacherJohn C. Haines School
Angela DumasInstructional Support LeaderO’Hare Network
Alanna MertensMathematics SpecialistDept. of Mathematics & Science
Rosalind AliTeacherMorton School of Excellence
Tom EllewInstructional Support LeaderAustin-North Lawndale Network
Bill PassTeacherEli Whitney School
Jennifer BoyceInstructional Support LeaderSkyway Network
Ruby EverageInstructional Support LeaderGarfield-Humboldt Park Network
Faylesha L. PorterMathematics SpecialistDept. of Mathematics & Science
Carolyn Chatman-WallsInstructional Support LeaderAustin-North Lawndale Network
Carla Ann GurgoneTeacherEvergreen Academy Middle School
Jennifer RathInstructional Support LeaderFulton Network
Patrick ClancyTeacherJames G. Blaine School
Teresa HugginsInstructional Support LeaderRock Island Network
Allayna RatliffInstructional Support LeaderGarfield-Humboldt Park Network
Laura CoppTeacherIrene C. Hernandez Middle School
Katie LaBombardTeacherLittle Village Academy
Aisha StrongInstructional Support LeaderFullerton Network
Gavin CreadenTeacherAgustin Lara Academy
Brenda LisenbyTeacherJames R. Doolittle, Jr. School
Leslie Swain-StoreInstructional Support LeaderFullerton Network
Barb CrumMathematics SpecialistDept. of Mathematics & Science
Matt McLeodMathematics SpecialistDept. of Mathematics & Science
Donna ThigpenInstructional Support LeaderEnglewood-Gresham Network
Antoinette MayoInstructional Support LeaderMidway Network
Rachel WilliamsTeacherMatthew Gallistel Language Academy
Chicago Public SchoolsPage 6
Contributors to High School Materials
Amanda BabbTeacherDavid G. Farragut Career Academy
David GoodrichTeacherLincoln Park High School
John NewmanInstructional Support LeaderSouthwest Side Network
Yvette BarclayTeacherDaniel Hale Williams Prep School of Medicine
Lorraine V. HarrisTeacherMichelle Clark Academic Prep Magnet High School
Therese PlunkettTeacherRogers C. Sullivan High School
Keith CoxTeacherRoberto Clemente Community Academy High School
Therese HodgettsTeacherMarie Sklodowska Curie Metropolitan High School
Caycee SledgeInstructional Support LeaderAlternative Schools
Dorian “Boo” DruryInstructional Support LeaderNorth/Northwest Side Network
Hannah LangTeacherDavid G. Farragut Career Academy
Terri Townes-KellyInstructional Support LeaderFar South Side Network
Thomas FrayneInstructional Support LeaderSouth Side Network
Rickey MurffMathematics SpecialistDept. of Mathematics & Science
Earl N. WilliamsTeacherManley Career Community Academy High School
Carrie WillisTeacherDaniel Hale Williams Prep School of Medicine
Special thanks to our partners in the Chicago STEM Education Consortium (C-STEMEC) who give so generously of their time and expertise in support of high-quality mathematics and science education in CPS:
• DePaul University, STEM Center
• Loyola University of Chicago, Center for Science and Math Education
• University of Chicago, Center for Elementary Mathematics and Science Education
• University of Illinois at Chicago, Learning Sciences Research Institute
CPS Mathematics Content Framework_Version 1.0 Page 7
Intr
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The Common Core State Standards for Mathematics (CCSS-M), initiated by the Council of Chief State School Officers (CCSSO) and the National Governors Association (NGA), articulate the skills and understandings that K-12 students must demonstrate in order to be college- and career-ready in mathematics by the end of high school. The CCSS-M are unprecedented in their unified vision of what students are expected to achieve, and the standards are more cohesive and challenging than what has typically existed before. As of spring of 2012, these standards have been adopted by 46 states.
While the Common Core provide the expected results for students’ achievement, there is no mandate for how teachers are to instruct. “Teachers are thus free to provide students with whatever tools and knowledge their professional judgment and experience identify as most helpful for meeting the goals set out in the Standards” (CCSS, 2010, p. 4). Therefore, the CPS Department of Mathematics and Science has been collaborating with teachers to develop the CPS Mathematics Content Framework—a tool that will guide teachers as they implement the CCSS-M.
Making the transition to these new higher standards will not be easy. All educators, at every level of the system, must work together to create the structures and supports that will help teachers provide rigorous learning experiences in their classrooms every day, experiences that are aligned with the new standards, so that every student is prepared for success after high school.
This document includes:
� An introduction to the Common Core State Standards for Mathematics
� The instructional shifts needed to implement the CCSS-M and how these shifts are evident in the CPS Mathematics Content Framework
• Focus on critical areas to develop deep conceptual understanding and procedural fluency
• Integrate the mathematical practice standards throughout instruction
• Maintain coherence and continuity to link learning within and across grades
� Components of the CPS Mathematics Content Framework
• CPS Mathematics Bridge Plan
• CPS Mathematics Planning Guides
• How Performance Assessments support the CPS Mathematics Content Framework
• Mathematics Toolsets: Sample Tasks (for each planning guide), Lesson Plan Template, Tool for Examining and Modifying Lessons/Tasks, Sample of Modified Lessons/Tasks, and Recommended Professional Resource List
� A suggested process for designing and implementing a Standards-based curriculum
Chicago Public SchoolsPage 8
An IntroductionThe Common Core State Standards for Mathematics
The Common Core State Standards for Mathematics, CCSS-M, define what students should understand and be able to do in mathematics. They set grade-specific standards that follow coherent learning progressions, progressions that reflect what research tells us about how a student’s mathematical knowledge, skill, and understanding develop over time. The standards were developed to address the critical need to develop all students so they can succeed in college and careers and be competitive in the global workforce. They raise the bar dramatically for what we expect students to know and be able to do.
K-8 Content DomainsIn K-8, the emphasis of the CCSS-M is all about focus on mastery of the critical skills at each grade. No longer will curricula be “a mile wide but an inch deep.” This focus is different from prior standards, and will allow more time to develop fluency in key skills as well as deep conceptual understanding.
The CCSS-M define, by each grade level, what students should understand and be able to do. The content standards are structured in a hierarchy: clusters are groups of related standards; domains are larger groups of related standards.
For example, in a Grade 7 standard, the hierarchy looks like this:
Common Core State Standard for Mathematics 7th Grade Example
Expressions and Equations
Solve real-life and mathematical problems using numerical and algebraic expressions and equations
7.EE.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
Content Cluster
Content Domain
Content Standard
Grade 7 Example
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There are eleven content domains across grades K-8. The Expressions and Equations domain is highlighted below to illustrate the focus of Year 1, as described below, on page 17.
Content Domains, K-8
K 1 2 3 4 5 6 7 8Counting & CardinalityOperations and Algebraic ThinkingNumber and Operations in Base 10
FractionsMeasurement and DataGeometry
Ratios and Proportional RelationshipsThe Number SystemStatistics and Probability Expressions and Equations
Functions
K 1 2 3 4 5 6 7 8
The sequencing of skill development represented by these content domains is different than the sequence found in the former Illinois State Learning Standards and in all instructional materials. The work of transitioning to the CCSS-M sequence of skill development for CPS, K-8, is described in the CPS Bridge Plan for Mathematics and in grade-level Planning Guides, described later in this document.
9-12 Content StandardsAt the high school level, the focus of the CCSS-M is the complex application of what students have learned, K-8. This is captured by the fact that modeling is listed as its own conceptual category. No conceptual category is isolated or addressed by a single high school mathematics course, and every high school course includes content standards from more than one conceptual category. The conceptual categories are:
� Number and Quantity
� Algebra
� Functions
� Modeling
� Geometry
� Statistics and Probability
A student’s work in each conceptual category will cross the boundaries of a number of traditional high school mathematics courses. The work of integrating the content standards associated with these conceptual categories across the traditional course structure found in most CPS high schools is addressed in the CPS Mathematics Planning Guides, described later in this document.
Common Core Standards for Mathematical Practice: K-12The mathematical practices describe how we expect students to engage with the content and represent varieties of expertise that educators should seek to develop in their students, expertise that students should demonstrate in solving mathematics problems. When compared to the former Illinois state standards, the CCSS-M practices represent a new challenge: the practices need to become an integral part of mathematics instruction and must be incorporated into the lesson along with, not apart from, the content standards.
Chicago Public SchoolsPage 10
The CCSS-M Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Although the Standards for Mathematical Practice are enumerated individually, they are integrated into the content standards. This means that in the transition to CCSS-M described below, the practices are explicitly connected to the content standards, because the practices are not meant to be –in fact cannot be – taught independently.
Teaching and learning at every grade should incorporate all Standards for Mathematical Practice. Mathematics teachers, K-12, should provide in every lesson the opportunity to develop and demonstrate mastery of these practices.
CPS Mathematics Content Framework_Version 1.0 Page 11
Intr
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The creation of the CPS instructional shifts in mathematics, below, was informed by (1) carefully analyzing the CCSS-M and (2) synthesizing other documents that describe instructional shifts, documents that were created by other states school districts and educational partners. By simplifying and consolidating the shifts, we hope CPS mathematics educators will be able to more easily understand and adopt them.
Teachers are likely to be in different stages in practicing these shifts; however, focusing on them will help us build a common understanding of what is needed in mathematics instruction as we move forward towards full implementation.
Instructional Shifts in Mathematics
Focus on critical areas to develop deep conceptual understanding and procedural fluency.
Integrate the mathematical practice standards throughout instruction.
Maintain coherence and continuity to link learning within and across grades.
Shift 1: Focus on the critical areas to develop deep conceptual understanding and procedural fluencyWhat does it mean to focus on critical areas to develop deep conceptual understanding and procedural fluency?
The CCSS-M ask us to teach more deeply to a more focused set of standards. According to Student Achievement Partners, “As a first step in implementing the Common Core State Standards for Mathematics focus strongly where the standards focus.”
In classrooms, this means that teachers understand how the CCSS-M prioritize standards within learning progressions, so that instructional time and energy are focused on critical concepts in a given grade. In this way, students develop strong foundational knowledge and deep conceptual understanding and are able to transfer mathematical skills and understanding across concepts and grades.
As the Standards state, “Conceptual understanding and procedural skill are equally important” (CCSS, 2010, p. 4). In mathematics, students need procedural fluency that requires them to not only know algorithms, but also how and when to use them appropriately. Students need to be able to compute flexibly, accurately and efficiently. This will help them access more complex concepts in later mathematics.
Why is this shift so important?
This shift is important because, as educators, we are often asked to cover a wide range of concepts in each grade level or course. Teaching fewer concepts means educators will have time to be more intentional about both building deeper conceptual understanding in their students and developing a variety of algorithms that students can use with dexterity as they approach any mathematical situation or problem. When our students develop deeper understandings and greater procedural fluency, they are better able to apply those understandings and skills to other contexts, both mathematical and real-world.
Chicago Public SchoolsPage 12
How should teachers focus on critical areas to develop deep conceptual understanding and procedural fluency?
“Rather than racing to cover everything in today’s mile-wide, inch-deep curriculum, educators are encouraged to use the power of the eraser and significantly narrow and deepen the way time and energy is spent in the math classroom. Focus deeply on only those concepts that are emphasized in the standards so that students can gain strong foundational conceptual understanding, a high degree of procedural skill and fluency, and the ability to apply the math they know to solve problems inside and outside the math classroom.“(http://www.achievethecore.org/steal-these-tools/focus-in-math)
Teachers should focus on the critical areas through careful planning of what to teach and when. They should consider carefully whether or not students should achieve mastery of the standard or concept, not presuming it will be taught next year. Teaching to fewer standards gives teachers more time to engage students in rigorous mathematical tasks, providing them the opportunity to both explore the concepts and construct the requisite depth of understanding. This time also allows for students to work in small groups, engage in hands-on exploration, identify patterns and analyze data, while thinking, speaking, and writing critically about their learning. Through these rigorous lessons, students will gain deep conceptual understandings and procedural fluency.
How is this shift represented in the CPS Mathematics Content Framework-Version 1.0?
This shift is represented through the CPS Content Framework-Version 1.0 Planning Guides, where we lay out a carefully selected set of standards that are the focus of instruction.
In grades 6-8, this focus is based on the Expressions and Equations learning progression. In high school, some CCSS-M standards that address specific course content are deferred in Year 1 to provide teachers with more time to make the transition to new content. More information about this transition is provided in the CPS Bridge Plan for Mathematics (described on page 16) and in the Planning Guides.
Shift 2: Integrate mathematical practice standards throughout instructionWhat does it mean to integrate mathematical practice standards throughout instruction?
Teachers need to explicitly incorporate the Standards for Mathematical Practice in everyday instruction. These practice standards work in concert with the content standards, as neither of them stands alone. This begins with teachers regularly using rigorous mathematical tasks (tasks with a high level of cognitive complexity), and encouraging students to grapple with these rich mathematical tasks in a way that fosters facility with the Standards for Mathematical Practice. This combination of practice and content through rich tasks and student engagement is the means by which we allow students to develop their own understanding of the mathematical content – and ultimately leads to optimal student learning gains.
Why is this shift so important?
The shift to integrate the Standards for Mathematical Practice is important because the CCSS-M ask so much more from our students than prior standards. The Standards for Mathematical Practice are actually both a means to achieve the content standards and also an expectation of student performance on their own. When we are successful in implementing the Standards for Mathematical Practice, we equip students with the ability to take their mathematics learning with them into their lives and future careers. We may not know what careers are waiting for them in their future, but we do know that in our technologically advancing world we need to prepare students to think mathematically. That is, they need to be able to make mathematical sense of any given situation or problem. By using mathematical practices to master mathematical content, they will have command of a comprehensive “bag of tricks,” mathematical tools and skills that will help them determine the most efficient and accurate solution, after considering a range of possibilities. In other words, mathematical practices in combination with the content standards are essential to prepare students for success.
How should teachers integrate mathematical practices throughout instruction?
In mathematics classrooms that integrate the mathematical practices with content instruction, teachers implement rich
CPS Mathematics Content Framework_Version 1.0 Page 13
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mathematical tasks and typically act as facilitators, using probing questions to stimulate students and intentionally providing students with multiple opportunities to interact directly with the content through regular mathematical discourse (either individually or in groups of students). In such a role, the teacher guides the instruction in an inquiry-based approach, in stark contrast to many direct-instruction programs in which the teacher is primarily a lecturer.
In these classrooms, the way students use practices to approach mathematical problems and arrive at an answer is just as important as arriving at the correct answer. Small-group cooperative learning may be used extensively in the mathematics classroom as an instructional strategy. The teacher, circulating among and interacting with the groups, encourages them to find ways to solve problems without giving them an example to be emulated, as in traditional classrooms. Students may also demonstrate facility by working individually (through journal writing, etc.).
With every lesson, teachers are deliberate in providing students with opportunities to develop and demonstrate mathematical practices. Through these experiences, teachers enable and encourage students to make sense of the content on their own, understanding how their previous learning informs their current tasks, and how their current tasks will build further competencies.
How is this shift represented in the CPS Mathematics Content Framework – Version 1.0?
In the CPS Mathematics Content Framework’s Planning Guides, specific Standards for Mathematical Content are associated with a list of applicable Standards for Mathematical Practice. For each practice standard listed, we describe how it looks in the context of the content being taught. Additionally, the possible tasks referenced in the Planning Guides are examples of tasks that integrate and foster students’ use of the mathematical practices.
Shift 3: Maintain coherence and continuity to link learning within and across grade levelsWhat does it mean to maintain coherence and continuity of learning within and across grade levels?
“The Common Core State Standards in mathematics were built on progressions: narrative documents describing the progression of a topic across a number of grade levels, informed both by research on children’s cognitive development and by the logical structure of mathematics.” (http://math.arizona.edu/~ime/progressions/) Mathematics teachers who maintain coherence and continuity of the learning progressions both (1) understand the foundation of the mathematics that led to what they are currently teaching and (2) inform their lessons with an understanding of the mathematics their students will encounter next.
Why is this shift so important?
It is extremely important for mathematics educators to carefully connect the learning within and across grades so that students can build new understandings on solid foundations. With a clear understanding of the connections between what comes before and after a particular point in the progression, teachers can address any missing prerequisite understanding or skills (revealed by assessment) and determine the next steps to move students forward from that point.
How should teachers address learning progressions in their instruction?
In 2014-2015, after the CPS transition to the CCSS-M is complete (per the CPS Bridge Plan for Mathematics, page 16), learning progressions will fully guide and inform instructional planning and delivery decisions. Teachers will be very familiar with the progressions, so they will know what mathematical content their students have learned in preceding years, and will orient instruction to that starting point. During instruction, teachers will focus on the grade-level content with an eye on future content expectations for any set of standards, or within a particular learning progression.
How is this shift represented in the CPS Mathematics Content Framework – Version 1.0?
The learning progressions serve as the basis for the content standards chosen for the Planning Guides. For example, the CPS Mathematics Content Framework-Version 1.0 takes the learning progression of Expressions and Equations as the focus for learning in grades 6-8. This progression provides much of the foundational learning for students to prepare for the Algebra I content standards. The learning progressions and associated content standards are also reflected in the Prior Knowledge and in the sample tasks described in the Planning Guides.
Chicago Public SchoolsPage 14
Shifts in Action: Comparing the Standards, Middle GradesThe CCSS-M guide the way for students to become capable and confident in mathematics. In raising the bar, they expose gaps in our current approaches to teaching and learning mathematics.
For example, consider how skills and practices differ in the two mathematics standards below.
The differences between these two standards are stark. In the former Illinois Learning Standard (ILS), the students were expected to solve a linear equation provided to them. In the CCSS-M, students are expected to define a variable, construct their own equation or inequality, and solve the problem (either in a context or not) all the while reasoning about the relationship between the quantities, the components of the equation or inequality, solving the problem and then deciding whether or not the solution is reasonable. (Mathematical Practices 1, 2 and 4)
In other words, to demonstrate mastery of the CCSS-M standard, students must apply mathematical thinking while demonstrating content knowledge. (SHIFT 2: Integrate the mathematical practice standards throughout instruction)
This particular standard also represents one of the clusters in the critical focus areas for grade 7 (Solve real-life and mathematical problems using numerical and algebraic expressions and equations). By expecting instruction to be focused in these critical areas, the standards build coherence from one grade to the next. (SHIFT 3: Maintain coherence and continuity to link learning within and across grade levels.)
By establishing these areas of focus, the standards eliminate some topics previously covered in the former ILS. This is the means by which the expected depth of understanding is achieved. (SHIFT 1: Focus on critical areas to develop deep conceptual understanding and procedural fluency).
K-8 Mathematics Performance Standards
Former Illinois State Learning Standard Common Core State Standard for Mathematics
IL STATE: (8.A.3b) Solve problems using linear expressions, equations, and inequalities.
Middle/Junior high schoolSTATE GOAL 8: Use algebraic and analytical methods to identify and describe patterns and relationships in data, solve problems and predict results.
A. Describe numerical relationships using variables and patterns.
CCSS-M: (7.EE.4) Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
Grade 7Expressions and EquationsSolve real-life and mathematical problems using numerical and algebraic expressions and equations
Standard = Content Standard = Content + Mathematical Practices
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9-12 Mathematics Performance Standards
College Readiness Standards Common Core State Standards for Mathematics
CRS: EEI Solve real world problems using first-degree equations.
College Readiness Standards Expressions, Equations, and Inequalities score range: 24-27
CCSS-M: (a-REI) Solve linear equations and inequalities in one variable with coefficients represented by letters.
Algebra Reasoning with equations and inequalities.Understanding solving equations as a process of reasoning and explaining reasoning.
Standard = Content Standard = Content + Mathematical Practices
Shifts in Action: Comparing the Standards in High SchoolThere are similar differences between standards at the high school level, as illustrated in the two mathematics standards below.
In the former College Readiness Standard (CRS) standard, students were expected to create and solve an equation for a real-world problem. In the new CCSS-M standard, the students are expected to identify the variables and quantities represented in a real-world problem and then determine the best model—linear equation, linear inequality, quadratic equation, quadratic inequality, rational equation, or exponential equation—to solve it. (SHIFT 2: Integrate the mathematical practice standards throughout instruction)
Students are then expected to write the equation or inequality that best models the problem and then solve the equation or inequality.
Finally, they are expected to interpret the solution in the context of the problem, per the domain. In explaining their reasoning, students demonstrate their deep understanding of the concepts and demonstrate the process for solving the mathematical task (SHIFT 1: Focus on critical areas to develop deep conceptual understanding and procedural fluency).
Chicago Public SchoolsPage 16
CPS Mathematics Content Framework
The CPS Mathematics Content Framework provides the comprehensive view of how we develop mathematically capable and confident students, as defined by the CCSS-M, given the mathematics shifts in instruction described above.
Our primary objective is to provide tools and structures that will support teachers in the design of strong, school-based mathematics instruction. In the following section, we first describe District-wide expectations for the implementation of the Mathematics Content Framework-Version 1.0. Next, we describe the components of the Mathematics Toolset(s) and how they are intended to help teachers implement the CCSS-M. See below for an overview of the following sections.
Framework components include:
� District-wide Expectations
• CPS Mathematics Bridge Plan
• CPS Mathematics Planning Guides
• District-wide Benchmark Performance Assessments
� Mathematics Toolsets
District-wide ExpectationsCPS Bridge Plan for Mathematics
In 2012-2013, networks and schools will begin implementing the CPS Bridge Plan for Mathematics, the three-year blueprint that will guide the full implementation of the CPS Mathematics Content Framework.
The CPS Bridge Plan for Mathematics is an aggressive strategy that is intentional and deliberate in its design. It defines how we phase in new content standards and build our capacity to make the requisite shifts in instruction. In the first year of transition, we have taken into consideration that (1) there is a vast difference between the CCSS-M and the former ILS and CRS, with new content at each grade level and major shifts in instruction and approach, (2) all of the mathematics instructional materials are currently aligned to the ILS and CRS, and (3) full implementation in Year 1 would create huge gaps in student learning (students would not have the expected prior knowledge to be successful). Therefore, the Bridge Plan and other Framework components strategically focus on grades 6-8, and High School Algebra I and Geometry in Year 1 (school year 2012-13).
As you can see, by year 2014-2015 the District will be ready to implement the complete CPS Mathematics Content Framework, K-12. The following chart illustrates the three-year transition:
Grade ‘12-’13Year 1
‘13-‘14Year 2
‘14-’15Year 3
K-5 ENCOURAGED: Integrate Standards for Mathematical Practice into instruction
Implement Mathematics Content Framework 1.0 (Grades K-5)
100% CCSS-M Content
6-8 Implement Mathematics Content Framework 1.0 (Grades 6-8)
Implement Mathematics Content Framework 2.0 (Grades 6-8)
100% CCSS-M Content
High School Implement Mathematics Content Framework 1.0 (Algebra I and Geometry)
Implement Mathematics Content Framework 2.0 (Algebra I, Geometry, and Algebra II)
100% CCSS-M Content
CPS Mathematics Content Framework_Version 1.0 Page 17
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Grades K – 5
In school year 2012-13, K-5 teachers will fully implement the Common Core State Standards for Literacy (CCSS-L). In order to respect the challenges of transitioning to the CCSS-L, we are deferring the implementation of the new Common Core Standards for Mathematical Content to the following school year (2013-14). Teachers in grades K-5 are encouraged to integrate Standards for Mathematical Practice into the content that is currently being taught. In other words, teachers may adjust how students engage with the content, but not change the content itself, as a way to prepare for the 2013-14 school year.
Grades 6 – 8
In Year 1 of the transition to CCSS-M, 2012-2013, grades 6-8 mathematics teachers will be teaching:
� to the former ILS before the ISAT in March, using their current mathematics instructional materials
� to the new CCSS-M standards, per the scope described by the Planning Guides, after the ISAT
A Look at Transition in Practice: Grades 6-8 � Through PD and “deep dives” in the Expressions
and Equations progression, teachers will build capacity around the CCSS-M and instructional shifts
� Prior to ISAT, they apply these CCSS-M instructional shifts as they approach “old” content
• Analyzing instructional tasks
• Enriching them to be more aligned with CCSS-M
• Embedding math practices
� After ISAT, teachers shift to a deeper focus on targeted CCSS-M, using guidance and resources in the Planning Guides
Specifically, after the March ISAT, grades 6-8 mathematics teachers will focus on the Expressions and Equations learning progression (as defined in the Progressions for the Common Core State Standards in Mathematics: 6-8, Expressions and Equations, 2011).
This progression was intentionally chosen to build the foundation for 2012-2013 8th graders to enter the following year’s high school Algebra I course ready for content that is tightly aligned with the CCSS-M. Specifically, the Grade 8 CCSS-M includes the “algebra of lines,” which was formerly taught in the first half of high school Algebra I courses.(“Algebra of lines” refers to equations, graphs of linear relationships, and systems of linear equations.)
Likewise, 2012-2013 6th and 7th graders will enter the following year’s mathematics classes well-prepared to succeed in classrooms where content is articulated along this critical progression. With each year in the Bridge Plan, the focus is on expanding students’ cognitive development in alignment with the logical structure of the CCSS-M.
Focus for Building CCSS-M Capacity, Grades 6-8
Both professional development and assessments in support of Year 1 CCSS-M implementation will be aligned with the Expressions and Equations content standards included in the Planning Guides.
Algebra I and Geometry
Because the majority of CPS high schools are teaching the traditional high school mathematics course sequence, the high school mathematics Planning Guides are designed for Algebra I and Geometry courses.
� In Year 1 of the transition to CCSS-M (2012-13), the Algebra I and Geometry Planning Guides integrate both (1) the CCSS-M content standards that students did not cover in the previous school year (2011-12) and (2) a carefully chosen subset of the CCSS-M content expected in these courses. In this way, the Algebra I and Geometry Planning Guides address CCSS-M expectations, taking into account the expected skills and proficiencies CPS students will bring to the classroom.
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� In Year 2 (2013-14), the CPS Mathematics Planning Guides will have a stronger emphasis on the CCSS-M, as students will be expected to have learned the content outlined in Year 1.
What’s in store for Algebra I, 2012-2013: This critical course extends the mathematics that students learn in middle school. Prior to CCSS-M, the “algebra of lines” has not been fully introduced in middle school mathematics courses. (“Algebra of lines” refers to equations, graphs of linear relationships, and systems of linear equations.) The Algebra I Planning Guide includes these concepts; it does not include the Statistics and Probability conceptual category in order to provide teachers and students with the time necessary to address the content standards associated with the algebra of lines.
What’s in store for Geometry, 2012-2013: This course formalizes and extends students’ geometric experiences from the middle grades. In the Geometry Planning Guide, student learning will focus on defining geometric shapes, points, lines, and planes (concepts found in CCSS-M Grades 7 and 8 ). Applications of Probability will not be included in Year 1, but will be included in Year 2 (2013-2014). This focus on geometric construction is new and therefore requires more time.
Focus for Building CCSS-M Capacity, Algebra I and Geometry
Both professional development and assessments in support of Year 1 CCSS-M implementation will be aligned with the following CCSS-M content standards associated with these Big Ideas in the Planning Guides (from Appendix A: Designing High School Mathematics Courses on the Common Core State Standards (2011):
� Algebra I: Quadratic Functions and Modeling; Linear and Exponential Relationships
� Geometry: Congruence, Proof, and Constructions; Circles With and Without Coordinates
Please note: Although the focus for building CCSS-M capacity in high school mathematics courses centers on content standards associated with these Big Ideas as noted above, instruction should also address other content standards outlined in the Planning Guides.
CPS Mathematics Planning Guides
The purpose of the CPS Mathematics Planning Guides is to provide teachers with a roadmap, to help them determine the appropriate approaches to address both the instructional shifts and the areas of instructional focus, as described in the Bridge Plan. The CPS Mathematics Planning Guides define the District’s expectations about what happens in classrooms by identifying the strategic CCSS-M standards in content and practice that CPS students will be expected to learn. Assessments, described in the next section of this document, will expect students have learned the material covered in the Planning Guides.
The Planning Guides are informed by (1) the Mathematics Model Content Frameworks developed by PARCC, the Partnership for Assessment of Readiness for College and Careers (2011), (2) the instructional shifts for mathematics, and (3) the transitional approach to implementation, as outlined in the CPS Bridge Plan for Mathematics.
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Grades 6-8 Planning Guides
Middle School Planning Guide: Grade 8 example
Structure and use: The purpose of the Planning Guides, grades 6-8, is to define the scope of the CCSS-M content standards to be taught post-ISAT, in the 2012-2013 school year. The Introduction in each Planning Guide provides essential directions for CCSS-M implementation. Within each grade-level guide, the Expressions and Equations learning progression, or “Big Idea,” is articulated by topics. Each topic provides instructional guidance, supported by key content standards and applicable Standards for Mathematical Practice, key terms and ideas, and sample tasks.
Timing and instructional materials: Schools and teachers are expected to continue using current instructional materials and to address the Illinois Assessment Frameworks from the beginning of the year until the ISAT is administered. In addition to this content, teachers will explicitly integrate the Standards for Mathematical Practice into their lessons on a regular basis. Following the ISAT, the grades 6-8 Planning Guides form the basis for instruction in the classroom. Teachers will continue to integrate the Standards for Mathematical Practice with the content standards, in order to achieve the depth of conceptual understanding and fluency that are expected by the CCSS-M.
Please note: For many CPS middle grades classrooms, some of the CCSS-M content standards in the applicable Planning Guide will be covered within the scope of a classroom’s instructional materials prior to ISAT. We are not asking teachers to change the order of these lessons. We are asking them to teach them with the level of rigor expected by the CCSS-M, integrating the Standards for Mathematic Practice.
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Big Idea Assessment: Expressions and EquationsSee the 8th Grade Toolset*: MARS task: Sorting Functions, Summative for this Big Idea
Topic: What is a Function?CCSS-M Content Standards8.F.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding
output.8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.8.F.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).8.F.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a
graph that exhibits the qualitative features of a function that has been described verbally.
Connections to Standards for Mathematical Practice
Standards for Mathematical Practice How it applies…
MP7- Look for and make use of structure.How different components of an equation affect or are represented in the graph of the equation; how components of the graph can lead to develop-ing an equation
MP8 - Look for and express regularity in repeated reasoning.
Through drawing the graphs of various equations, students will recognize the effect of different portions of the equation on the graphs; through repeating calculations, students express the generalized set of operations and create an equation.
Key Ideas and Terms for “What is a Function?”
Key Ideas Key Terms
• Definition and characteristics of different types of functions• Relationship between graph and equation• Development and consistent use of the language of functions
Function, input, output, independent variable, dependent variable, rate of change, Cartesian plane, slope, intercept, proportional
Terms should be deeply understood within the context of their use. Not to be considered standalone vocabulary exercises.
Prior Knowledge • Use of letters as variables rather than boxes, etc. as in earlier grades
• Familiarity with graphing in all four quadrants of the Cartesian plane
• Understanding of the meaning of operations, particularly with integers
Students will be able to: • Move between tables of value and function rules to represent functions.
• Graph functions using input-output values as ordered pairs and identify type of function through the shape of the graph.
• Match corresponding tables, graphs, equations.
• Draw a graph given verbal descriptions.
• Identify shapes of graphs of parent functions (linear, quadratic, cubic, exponential, etc.) and the effect of different components of the equation (leading coefficient, constant term, etc.).
• Graph proportional relationships.
• Compare different proportional relationships represented in different forms.
Possible task(s)* CME Algebra 1 text, Chapter 3, for shapes of graphs of parent functions
Grade 8 Mathematics Planning Guide – SY12-13Big Idea: Expressions and Equations
*The Grade 8 mathematics toolset includes this and other resources and can be found online at https://ocs.cps.k12.il.us/sites/IKMC/default.aspx and on the Department of Mathematics and Science website (cmsi.cps.k12.il.us).
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High School Course Planning Guides
High School Planning Guides are specific to each traditional high school mathematics course (e.g., Algebra I, Geometry).
High School Mathematics Planning Guide: Algebra I example
Structure and use: Each high school mathematics course Planning Guide describes the scope of CCSS-M content standards to be taught in the 2012-2013 school year. The Introduction in each Planning Guide provides essential directions for CCSS-M implementation. Each course is broken down into Big Ideas. A performance task, or problem, is referenced that illustrates what students will know and be able to do by the successful completion of the Big Idea. Within each Big Idea, topics provide more instructional guidance, supported by key content standards and associated mathematical practices.
Timing and instructional materials: Teachers are expected to cover the entire scope of content outlined in the high school mathematics course Planning Guides, in the order most appropriate to their students, instructional materials, and learning environments.
Using their current instructional materials, teachers will integrate the Standards for Mathematical Practice with the content standards, and enhance their instruction with rich mathematical tasks in order to achieve the depth of conceptual understanding and fluency that are expected by the CCSS-M. Tools to analyze tasks for rigor, sample rigorous mathematical tasks, and other resources to support lesson enhancement are included in the Planning Guide toolsets.
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Planning Guides
Big Idea Assessment: Foundations for AlgebraSee Algebra I Toolset*: Central Park
Topic: Operations on Rational NumbersCCSS-M Content Standards6-NS.7c Understandtheabsolutevalueofarationalnumberasitsdistancefrom0onthenumberline;interpretabsolutevalueasmagnitudeforapositiveoranegativequantityina
real-worldsituation.7-NS.1b Understand p + qasthenumberlocatedadistanceof|q|from p,inthepositiveornegativedirectiondependingonwhetherqispositiveornegative.Showthatanumberand
itsoppositehaveasumof0(areadditiveinverses).Interpretsumsofrationalnumbersbydescribingreal-worldcontexts.7-NS.1c Understandthatsubtractionofrationalnumbersasaddingtheadditiveinverse,p – q = p+(-q).Showthatthedistanceoftworationalnumbersonthenumberlineisthe
absolutevalueoftheirdifference,andapplythisprincipleinreal-worldcontexts.7-NS.1d Applypropertiesofoperationsasstrategiestoaddandsubtractrationalnumbers.7-NS.2a Understandthatmultiplicationisextendedfromfractionstorationalnumbersbyrequiringthatoperationscontinuetosatisfythepropertiesofoperations,particularlythe
distributiveproperty,leadingtoproductssuchas(-1)(-1)=1andtherulesformultiplyingsignednumbers.Interpretproductsofrationalnumbersbydescribingreal-worldcontexts.
ConnectionstoStandardsforMathematicalPractice
Standards of Mathematical Practice How it applies…
MP2-Reasonabstractlyandquantitatively. Manipulatethemathematicalrepresentationbyshowingtheprocess consideringthemeaningofthequantitiesinvolved.
MP3-Constructviableargumentsandcritiquethereasoningofothers. Justify(orallyandinwrittenform)theapproachused,includinghowitfitsinthecontextfromwhichthedataarose.
MP7-Lookforandmakeuseofstructure. Usepatternsorstructuretomakesenseofmathematicsandconnectpriorknowledgetosimilarsituationsandextendtonovelsituations.
PriorKnowledge • Thenumberline
• Rationalnumbers
• Additiveinverse
• Propertiesofoperations
• Propertiesofoperations,esp.thedistributiveproperty
Studentswillbeableto: • Findtheabsolutevalueofarationalnumber.
• Explainthemeaningofabsolutevalueasitsdistancefrom0onthenumberline.
• Addrationalnumbers.
• Usethenumberlineandthedefinitionofabsolutevaluetoexplainhowtoaddrationalnumbers.
• Explainwhythesumoftwooppositenumbersis0.
• Applytheskillofaddingrationalnumberstotherealworld.
• Subtractrationalnumbers.
• Usetheconceptsofabsolutevalueandadditiveinversetoexplainhowtosubtractrationalnumbers.
• Applytheskillofsubtractingrationalnumberstotherealworld.
• Usepropertiesofoperationswhenaddingandsubtractingrationalnumbers.
• Multiplyrationalnumbers.
• Interpretproductsofrationalnumbersinrealworldcontexts.
Possibletask(s)**TheAlgebraItoolsetincludesthisandotherresourcesandcanbefoundonlineathttps://ocs.cps.k12.il.us/sites/IKMC/default.aspxandontheDepartmentofMathematicsandSciencewebsite(cmsi.cps.k12.il.us).
Algebra I Planning Guide – SY12-13 Big Idea: Foundations for Algebra
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District-wide Benchmark Performance Assessments
Assessments are critical to a successful transition to the CCSS-M. In order to prepare our students for the rigor defined by the CCSS-M, our assessments must be designed to measure the conceptual understanding and the mathematical thinking expected of the CCSS-M. This is also the design of the upcoming Partnership for Assessment of Readiness of College and Career (PARCC) Assessment system for mathematics, which will be the accountability measure for CPS in 2014-15.PARCC tests will be designed to measure the knowledge, skills and understandings essential to achieving college and career readiness – as defined by the CCSS-M. To measure the full range of the standards, all assessments will include tasks that require students to connect mathematical content and mathematical practices. (PARCC Model Content Frameworks for Mathematics, October 2011)
The CPS Mathematics Performance Assessments will help students prepare for the PARCC tests by assessing material described in the Planning Guides. More importantly, these kinds of assessments mimic tasks that students will be expected to perform in college and/or career.
* The MARS (Mathematics Assessment Resource Service) tasks demand substantial chains of reasoning and non-routine problem solving. Most MARS tasks are designed to be accessible to most learners: the tasks begin with questions slightly below the level of difficulty of the grade level, and are meant to lead students into the concept and activate their prior knowledge. The tasks increase in rigor and difficulty such that teachers are able to determine student misconceptions.
** These performance tasks will also be used to measure student growth for teacher evaluation.
Performance Assessment for Grades K-5
BOY: These assessments will not be provided centrally by CPS.
During Year: OPTIONAL assessments based on MARS Tasks*, TBD
EOY: These assessments will not be provided centrally by CPS.
Performance Assessment for Grades 6-8
BOY: Pre-assessment of CCSS-M standards aligned with the Expressions and Equations learning progression**
During Year: Formative Assessments options based on MARS Tasks*, TBD
EOY: Post-assessment of CCSS-M standards aligned with the Expressions and Equations learning progression**
Performance Assessment Provided by District for High School Algebra I and Geometry
Will be aligned to the focus for building CCSS-M capacity for Algebra I and Geometry, Year 1:
• Algebra I: Quadratic Functions and Modeling; Linear and Exponential Relationships
• Geometry: Congruence, Proof, and Constructions; Circles With and Without Coordinates
BOY: Pre-assessment of CCSS-M standards aligned with the focus areas of learning for Algebra I and Geometry, respectively**
During Year: Formative Assessments options based on MARS Tasks*, TBD
EOY: Post-assessment of CCSS-M aligned with the focus areas of learning for Algebra I and Geometry, respectively**
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CPS Mathematics Toolsets
To support our transition to CCSS-M, we have developed tools for teachers to use as they plan and implement instruction aligned with CCSS-M expectations. These tools are described below and can be found at https://ocs.cps.k12.il.us/sites/IKMC/default.aspx and also at http://cmsi.cps.k12.il.us.
� Sample Lesson Plan Template
� Samples of Rigorous Tasks (for each Planning Guide, Grades 6-8, Algebra I, and Geometry)
� Tool for Examining and Modifying Lessons/Tasks
� Samples of Modified Lessons/Tasks
� Recommended Professional Resources
Sample Lesson Plan Template
The Lesson Plan Template provides a structure to guide teachers as they think through the components of a CCSS-M aligned lesson. These components include: the topic; standards (practice and content) and evidence of them; lesson flow and timing; potential facilitation questions; and anticipated potential misconceptions and how to address them.
Sample Rigorous Tasks at each grade level
Within each Planning Guide, rigorous performance tasks, or problems, are referenced at both the Big Idea and topic levels. These can be found at https://ocs.cps.k12.il.us/sites/IKMC/default.aspx and also at http://cmsi.cps.k12.il.us.
Lesson Analysis and Modification Tool
This tool is a rubric with an associated form with criteria to help teachers look at a lesson from their current materials and decide how well it is aligned to CCSS-M expectations (e.g., depth of conceptual understanding, integration of Standards for Mathematical Practice, focused on content standards, etc.). If a lesson is not well-aligned, teachers use the tool’s criteria to increase the rigor of the lesson.
Samples of Modified Lessons/Activities
These samples include (1) an original lesson, (2) the completed analysis tool (above), and (3) the final lesson as it was modified to address the shortcomings identified through the analysis.
Recommended Professional Resources
A list of vetted resources that will be helpful to teachers as they transition to CCSS-M in the classroom, which include: sample tasks and lessons, standards clarification and guidance, and pedagogical support. Many are free; others we recommend because we feel they are worth the price. The list will be updated as more tools and resources are developed, as more cities and states transition to CCSS-M.
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When teachers meet with their grade-level and/or course teams to develop a CCSS-M plan, how should they begin? The goal is for teacher teams to work together to determine how to implement the CCSS-M and instructional shifts, as described by this document – while considering the context of their school (teachers, students, instructional materials, etc.). The following planning outline will help all teams begin the transition to CCSS-M in their classrooms.
1. Become familiar with the CPS Mathematics Content Framework-Version 1.0 (this document), including the instructional shifts and the CPS Bridge Plan for Mathematics.
2. Use the CCSS-M materials to become familiar with Standards for Mathematical Practice. Teams are encouraged to work together to develop new instructional approaches that support these practice standards.
3. Become familiar with:
a. The kinds of high-cognitive demand tasks that are expected by the CCSS-M
b. Ways to analyze tasks in current instructional materials for rigor
c. Techniques to enhance the rigor of current instructional materials. In the appendix of this document, there are general resources and tools that support the kind of learning expected in the CCSS-M. Tools in the Planning Guides include grade- or course-specific resources, including sample MARS tasks. These tasks balance content and practice, for an integrated approach to instruction and performance assessment. In addition, general mathematics tools to support rigorous instruction are available online at https://ocs.cps.k12.il.us/sites/IKMC/default.aspx and on the Department of Mathematics and Science website (cmsi.cps.k12.il.us).
4. If you teach grades 6-8 mathematics, follow Getting Started - A
5. If you teach Algebra 1 or Geometry, follow Getting Started - B
6. If you teach mathematics in grades K-5, or any high school course other than Algebra 1 and Geometry— in other words, outside the scope of Year 1—follow Getting Started - C.
Getting Started
Things to Remember � The planning process is iterative. In other words, as you plan for instruction, an intentional effort should
be always be made to incorporate the instructional shifts in all lessons. Though we are gradually phasing in the Standards for Mathematical Content (in grades 6-8, Algebra I, and Geometry), the Standards for Mathematical Practice should be integrated in all mathematics instruction, per the Year 1 CPS Bridge Plan for Mathematics. Use the Planning Guides and Mathematics Toolsets to inform your planning and instruction.
� The most important parts of the process will be your instruction, collaboration with colleagues, and ongoing reflection about how your students are doing. As you implement your plans, continue to meet with your colleagues to study students’ work and revise instruction accordingly.
� Never underestimate your professional judgment. Your knowledge of your students and their needs should always be the forerunner in your planning.
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1. Consider the three instructional shifts that teachers must implement in order to support student success in meeting the CCSS-M. What instructional strategies are already being used that support these shifts? What adjustments to instruction are needed to address the instructional shifts? Integrating the Standards for Mathematical Practice into instruction may involve substantial shifts in instructional strategies.
2. Use the CCSS-M materials to become familiar with the content standards associated with the Expressions and Equations learning progression.
3. Become familiar with the Planning Guide and toolset for your grade level.
4. Before ISAT, plan to enhance your current instructional materials at the points where the Expressions and Equations content standards (from Planning Guide) are addressed, to provide the rigor expected by the CCSS-M. Use the resources referenced in appendix (in this document) and in the Mathematics toolset to analyze and modify your lessons and tasks.
5. After ISAT, use the Planning Guide to inform your planning and instruction. If your materials did not cover the expected Expressions and Equations content standards before ISAT, use the Planning Guide to plan instruction for remaining time.
6. Construct (or revise) your plans by addressing each component laid out in the Planning Guide.
a. What other standards need to be revisited or introduced to (1) support any missing skills/ knowledge? And (2) develop deep conceptual understandings and procedural fluency?
b. What tasks will help build the skills and knowledge your students need in order to master the key content standards, or be prepared for the next content standards in learning progression?
c. How can you supplement activities in your instructional materials to be more rigorous? High quality tasks will ignite student learning and provide a solid foundation upon which to build more complex mathematics.
d. How will you integrate the Standards for Mathematical Practice? Adjust your instructional strategies to enable and encourage students to make sense of the content on their own?
e. How can you use formative assessments and scoring tools to align with your Planning Guide and specific topics and tasks?
GETTING STARTED A: For grades 6-8 (Year 1 of Bridge Plan)
Guidance Notes
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1. Consider the three instructional shifts that teachers must implement in order to support student success in meeting the CCSS-M. What instructional strategies are already being used that support these shifts? What adjustments to instruction are needed to address the instructional shifts? Integrating the Standards for Mathematical Practice into instruction may involve substantial shifts in instructional strategies.
2. Use the CCSS-M materials (including Appendix A) to become familiar with the content standards associated with the focus areas for building CCSS-M capacity in Year 1.
a. Algebra I: Quadratic Functions and Modeling; Linear and Exponential Relationships
b. Geometry: Congruence, Proof, and Constructions; Circles With and Without Coordinates
3. Become familiar with the detailed expectations for learning in your course Planning Guide and the toolset to support your instruction. The scope described in the Planning Guide will lay the foundation for student success in Year 2 of our transition to CCSS-M.
4. Construct (or revise) your plans by addressing each component laid out in the Planning Guide.
a. Considering your students, instructional materials, and other factors, in what order will you present the Big Ideas? Against what timetable?
b. What other content standards need to be revisited or introduced to (1) support any missing skills/ knowledge? And (2) develop deep conceptual understandings and procedural fluency?
c. What tasks will help build the skills and knowledge your students need in order to master the key content standards, or be prepared for the next content standards outlined in the next Big Idea you will cover?
d. How can you supplement activities in your instructional materials to be more rigorous? High quality tasks will ignite student learning and provide a solid foundation upon which to build more complex mathematics.
e. How will you integrate the Standards for Mathematical Practice? Adjust your instructional strategies to enable and encourage students to make sense of the content on their own?
f. How can you use formative assessments and scoring tools to align with your Planning Guide and specific topics and tasks?
GETTING STARTED B: For Algebra 1 and Geometry (Year 1 of Bridge Plan)
Guidance Notes
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1. Use your current instructional materials as a starting point.
2. To the extent possible, integrate the Standards for Mathematical Practice into instruction. Every teacher is strongly encouraged to apply the Standards for Mathematical Practice into every CPS mathematics classroom and activity even though teachers may not be implementing the content standards this year. Integrating mathematical practices into instruction may involve substantial shifts in instructional strategies (such as the art of open-ended questioning, as referenced in the Shifts, page 11)
3. Begin to increase rigor of mathematical tasks. Use the resources in the Appendix to assess your lessons for rigor and supplement your current instructional materials, as appropriate.
4. Consider how to upgrade assessments to reflect the work you are doing with the Standards for Mathematical Practice. For example, choose problems with higher cognitive demand, such as open-ended or extended/constructed response.
GETTING STARTED C: For grades K-5, 11-12 mathematics classrooms (outside scope of Year 1 of Bridge Plan)
Guidance Notes
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Planning with ALL Learners in Mind
The Common Core State Standards were written for ALL K-12 students and, as such, effective implementation of the Standards rests on the intentional planning of instruction to provide access to learning for all students.
Student diversity is always present whether recognized or not; it is a given in every classroom. Within every group of students, teachers can anticipate that there will be a variety of skills, affinities, challenges, experiences, cultural lenses, aptitudes, interests, English language proficiency levels, (in the case of ELLs, native language proficiency levels), represented. As teachers engage in initial stages of curriculum planning (i.e. the clustering of standards, selection of texts and tasks, and the design of performance assessments) they must simultaneously consider the variety of learner profiles among their students. It is critical that teachers start curriculum planning with both the Standards and the Learners in mind.
Intentional planning for a diverse student group from the outset will maximize the likelihood that all students will be able to successfully access information, process concepts, and demonstrate their learning. Early in the year or course, data from various sources such as cumulative folders, screeners, pre-tests, Individualized Education Plans (IEPs), Individual Bilingual Instruction Plans (IBIPs), parent questionnaires, and getting-to-know-you activities, give teachers important preliminary information about every individual student that will influence their plans. As teachers better get to know individual students and their particular learning needs, over time they can continuously adjust curricular plans and personalize instructional strategies for more tailored differentiation.
Having initial plans that are universally-designed will position teachers to serve most students well, but in the process of personalizing the plan and as teachers would know, there will be certain elements that are crucial to include explicitly for particular groups of students. For example, while every child is unique and will therefore benefit from attention to their individual learner profile, a student who has been identified with a disability, by law and best practice, will require instructional supports based upon the IEP team’s best thinking relative to academic and functional need. Similarly, while every child is in the process of developing language and will therefore benefit from an educational experience that is designed for a range of social and academic English levels, a student who has been identified as an English Language Learner (ELL), by law and best practice, will have needs that must be addressed in particular ways. In both cases, it is important for teachers to specifically recognize and articulate in their curriculum plans and in their methods of instruction how they will tailor learning for these individuals.
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Article 14, Illinois School Code, Title 23, Administrative Code.226
Establishes the requirements for the treatment of children and the provision of special education and related services pursuant to the Individuals with Disabilities Education Improvement Act.
http://www.isbe.net/rules/archive/pdfs/226ark.pdf
Article 14C, Illinois School Code, Title 23 Illinois Administrative Code. 228
Establishes requirements for school districts’ provision of services to students in preschool through grade 12 who have been identified as limited English proficient.
http://www.isbe.net/rules/archive/pdfs/228ark.pdf
CAST CAST is a nonprofit research and development organization that works to expand learning opportunities for all individuals, especially those with disabilities, through Universal Design for Learning.
www.cast.org
Illinois Resource Center The IRC provides assistance to teachers serving linguistically and culturally diverse students, including English language learners (ELLs), in grades PK-12.
www.thecenterweb.org/irc
National Center on Universal Design for Learning
The National UDL Center supports the effective implementation of UDL by connecting stakeholders in the field and providing resources and information about: UDL Basics, Advocacy, Implementation Research, Community, Resources.
www.udlcenter.org
STAR NET Region II STAR NET Region II provides technical assistance, training and resources to professionals and families supporting the education of young children, with an emphasis on children with special needs.
www.thecenterweb.org/starnet
Understanding Language Understanding Language aims to heighten educator awareness of the critical role that language plays in the new Common Core State Standards and Next Generation Science Standards.
www.ell.stanford.edu
World-Class Instructional Design and Assessment
WIDA advances academic language development and academic achievement for linguistically diverse students through high quality standards, assessments, research, and professional development for educators.
www.wida.us
Resources for Addressing Universal Variability
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Intr
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tionWorks Cited
The Common Core Standards Writing Team. (2011) Progressions for the Common Core State Standards in Mathematics: 6-8, Expressions and Equations. Tucson, AZ: University of Arizona. Retrieved March 17, 2012. From http://math.arizona.edu/~ime/progressions/
PARCC Model Content Frameworks for Mathematics. (2011) Partnership for Assessment of Readiness for College and Careers.
National Governors Association and Council of Chief State School Officers. (2010) Common Core State Standards for Mathematics.
The Silicon Valley Mathematics Initiative. How Performance Tasks are Used to Inform Instruction. The Noyce Foundation. Retrieved 23 April, 2012. http://www.noycefdn.org/documents/math/howperformancetasks.pdf
Student Achievement Partners. (2012). Retrieved April 23, 2012, from http://www.achievethecore.org/steal-these-tools
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Planning GuidesThe following Planning Guides outline the scope of learning for grades 6-8 and Algebra I and Geometry for the 2012-2013 school year. These Planning Guides provide guidance to help teachers build deep conceptual understanding of mathematics, procedural fluency, and critical thinking and problem-solving skills. https://ocs.cps.k12.il.us/sites/IKMC and on the Department of Mathematics and Science website: http://cmsi.cps.k12.il.us/.
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Grade 6 Mathematics Planning Guide – SY12-13Introduction
The purpose of this Planning Guide is to define the scope of the Common Core State Standards for Mathematics (CCSS-M) Content Standards to be taught in the 2012-2013 school year. It has been designed by CPS teachers to be useful to CPS teachers during the three-year transition to full implementation of the CCSS-M.
In Year 1 of the transition to CCSS-M, 2012-2013, grades 6-8 mathematics teachers will be teaching:
� To the former ILS before the ISAT in March. The expectation is for schools and teachers to continue using their current mathematics instructional materials and to address the Illinois Assessment Frameworks from the beginning of the year until the ISAT is administered. In addition to this content, teachers will explicitly integrate the Standards for Mathematical Practice into their lessons on a regular basis.
� To the new CCSS-M standards, per the scope described by this Planning Guide, after the ISAT, using their instructional materials and other resources to integrate mathematical practices with rigorous mathematical tasks that support the scope of content standards. A toolset to support this instruction includes (1) samples of rigorous grade-specific tasks (including samples of formative assessment options based on MARS (Mathematics Assessment Resource Service) tasks) and (2) general tools that include a sample lesson planning template, a tool for analyzing and modifying lessons/tasks; samples of modified lessons/tasks; and a list of professional resources. These tools are available at https://ocs.cps.k12.il.us/sites/IKMC/default.aspx and on the Department of Mathematics and Science website (cmsi.cps.k12.il.us).
Specifically, after the March ISAT, grades 6-8 mathematics teachers will focus on the Expressions and Equations progression, as defined in the Progressions for the Common Core State Standards in Mathematics: 6-8, Expressions and Equations (2011). This progression was chosen because the Grade 8 CCSS-M includes the “algebra of lines,” which was formerly taught in the first half of high school Algebra I courses (“algebra of lines” refers to equations, graphs of linear relationships, and systems of linear equations). Concentrating on this particular progression allows grades 6-8 classrooms to hone in on the vertical articulation of this very important concept, focusing on students’ cognitive development and the logical structure of the CCSS-M.
Please note: If the content standards in this Planning Guide are covered within the scope of a classroom’s instructional materials prior to ISAT, teachers should teach to these with the level of rigor expected by the CCSS-M, integrating the Standards for Mathematic Practice with the content. Post-ISAT is an opportunity to reinforce content standards already addressed, and to focus more deeply on the Expressions and Equations content standards that were not covered prior to ISAT.
During the first 3 quarters, teachers will explicitly integrate the Standards for Mathematical Practice (below) into their lessons on a regular basis. In the 4th quarter, instruction that integrates Standards for Mathematical Practice with targeted content standards is supported by “how to” guidance in this Planning Guide. By teaching the mathematical practices alongside the indicated content standards, students will be more likely to achieve the depth of conceptual understanding and procedural fluency expected by the CCSS-M.
Chicago Public SchoolsPage 34
The CCSS-M Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Following the ISAT, this Planning Guide will form the basis for instruction in Grade 6 mathematics classrooms. This year’s focus of learning, Expressions and Equations, is broken into four topics: 1) Using Ratio Reasoning to Solve Problems, 2) Algebraic Expressions, 3) Single Variable Expressions, Equations, and Inequalities, and 4) Graphing in the Coordinate Plane. Topic components include a targeted set of content standards, target mathematical practices (and how they apply to the specific topic), key ideas for learning, key terms, and sample instructional tasks.
The guide assumes 9 weeks for instruction, including time for formative and summative assessments. The topics are sequenced in a way that we believe best develops and connects the mathematical content of the CCSS-M. However, teachers should review the topics and decide the order and time allocation appropriate for their classrooms, given their students, instructional materials, and other considerations. The order of the standards included in a topic does not imply a sequence of the content. Some standards may be revisited several times while addressing the topic, while others may be only partially addressed, depending on the mathematical focus of the topic.
Finally, this document reflects our current thinking about the transition to the CCSS-M. We welcome feedback about your experience with the document. Please share your thoughts with your network staff who will forward to the Department of Mathematics and Science.
CPS Mathematics Content Framework_Version 1.0 Page 35
Plan
ning
Gui
des
Gra
de 6
Mat
hem
atics
Pla
nnin
g G
uide
– S
Y12-
13Bi
g Id
ea: E
xpre
ssio
ns a
nd E
quati
ons
Big
Idea
Ass
essm
ent:
Expr
essi
ons a
nd E
quati
ons
See
6th G
rade
Too
lset
*: S
ampl
e 6th
Gra
de a
sses
smen
t (co
llecti
on o
f pro
blem
s fro
m Il
lust
rativ
e M
ath)
Topi
c: U
sing
Rati
o Re
ason
ing
to S
olve
Pro
blem
sCC
SS-M
Con
tent
Sta
ndar
ds6.
RP.1
U
nder
stan
d th
e co
ncep
t of a
ratio
and
use
ratio
lang
uage
to d
escr
ibe
a ra
tio re
latio
nshi
p be
twee
n tw
o qu
antiti
es.
6.RP
.2
Und
erst
and
the
conc
ept o
f a u
nit r
ate
a/b
asso
ciat
ed w
ith a
ratio
a:b
with
b≠0
, and
use
rate
lang
uage
in th
e co
ntex
t of a
ratio
rela
tions
hip.
6.
RP.3
U
se ra
tio a
nd ra
te re
ason
ing
to so
lve
real
-wor
ld a
nd m
athe
mati
cal p
robl
ems,
e.g
., by
reas
onin
g ab
out t
able
s of e
quiv
alen
t rati
os, t
ape
diag
ram
s, d
oubl
e nu
mbe
r lin
e di
agra
ms
or e
quati
ons.
a.
Mak
e ta
bles
of e
quiv
alen
t rati
os re
latin
g qu
antiti
es w
ith w
hole
-num
ber m
easu
rem
ents
, find
miss
ing
valu
es in
the
tabl
es, a
nd p
lot t
he p
airs
of v
alue
s on
the
coor
dina
te
plan
e. U
se ta
bles
to c
ompa
re ra
tios.
b.
So
lve
unit
rate
pro
blem
s inc
ludi
ng th
ose
invo
lvin
g un
it pr
icin
g an
d co
nsta
nt sp
eed.
c.
Fi
nd a
per
cent
of a
qua
ntity
as a
rate
per
100
; sol
ve p
robl
ems i
nvol
ving
find
ing
the
who
le, g
iven
a p
art a
nd th
e pe
rcen
t.d.
U
se ra
tio re
ason
ing
to c
onve
rt m
easu
rem
ent u
nits
; man
ipul
ate
and
tran
sfor
m u
nits
app
ropr
iate
ly w
hen
mul
tiply
ing
or d
ivid
ing
quan
tities
.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds f
or M
athe
mati
cal P
racti
ceHo
w it
app
lies…
MP2
– R
easo
n ab
stra
ctly
and
qua
ntita
tivel
y.Es
peci
ally
in u
nit c
onve
rsio
n (6
.RP.
3d) -
cre
ating
a sy
mbo
lic re
pres
enta
tion
of th
e pr
oble
m a
nd c
onte
xtua
lizin
g th
e re
sult
of th
e ca
lcul
ation
s bac
k to
the
cont
ext o
f the
pro
blem
.
MP4
– M
odel
ing
with
mat
hem
atics
.St
uden
ts a
re a
ble
to id
entif
y im
port
ant q
uanti
ties i
n a
prac
tical
situ
ation
an
d m
ap th
eir r
elati
onsh
ips u
sing
diag
ram
s, tw
o w
ay ta
bles
, gra
phs,
flow
-ch
arts
, etc
. and
use
this
info
to d
eriv
e a
form
ula.
Key
Idea
s and
Term
s fo
r “U
sing
Ratio
Rea
soni
ng
to S
olve
Pro
blem
s“
Key
Idea
sKe
y Te
rms
•U
nder
stan
d an
d us
e ra
tiona
l rea
soni
ng to
cal
cula
te a
uni
t rat
e•
Appl
y a
ratio
to c
alcu
late
the
com
positi
on o
f a se
t•
Use
ratio
reas
onin
g to
der
ive
a ge
nera
lized
des
crip
tion
of a
set
Ratio
, pro
porti
on, u
nit r
ate,
equ
ival
ent
Term
s sho
uld
be d
eepl
y un
ders
tood
with
in th
e co
ntex
t of t
heir
use.
Not
to
be c
onsid
ered
stan
dalo
ne v
ocab
ular
y ex
erci
ses.
Prio
r Kno
wle
dge
•U
se e
quiv
alen
t fra
ction
s as a
stra
tegy
to a
dd a
nd su
btra
ct fr
actio
ns
•M
ultip
ly a
nd d
ivid
e fr
actio
ns
•Si
mpl
ify fr
actio
ns in
to lo
wes
t ter
ms
Stud
ents
will
be
able
to:
•Cr
eate
and
des
crib
e a
ratio
rela
tions
hip
betw
een
two
quan
tities
.•
Expr
ess r
atios
in d
iffer
ent f
orm
s.•
Und
erst
and
and
deriv
e un
it ra
te.
•De
term
ine
equi
vale
nt ra
tios.
•Cr
eate
tabl
es o
f equ
ival
ent r
atios
.•
Use
ratio
rela
tions
hips
to d
eter
min
e pe
rcen
t, pa
rt, o
r who
le g
iven
the
othe
rs.
•Ap
ply
ratio
reas
onin
g w
hen
conv
ertin
g un
its o
f mea
sure
men
t.
Poss
ible
task
s*M
ARS
Task
: Can
dies
; Ele
vato
rs; B
oys a
nd G
irls T
ask;
Sug
ar C
ooki
es; S
hade
s of B
lue
*The
Gra
de 6
mat
hem
atics
tool
set i
nclu
des t
his a
nd o
ther
reso
urce
s and
can
be
foun
d on
line
at h
ttps:
//oc
s.cp
s.k1
2.il.
us/s
ites/
IKM
C/de
faul
t.asp
x an
d on
the
Depa
rtm
ent o
f Mat
hem
atics
and
Sci
ence
web
site
(cm
si.cp
s.k1
2.il.
us).
Chicago Public SchoolsPage 36
Topi
c: A
lgeb
raic
Exp
ress
ions
CCSS
-M C
onte
nt S
tand
ards
6.EE
.1
Writ
e an
d ev
alua
te n
umer
ical
exp
ress
ions
invo
lvin
g w
hole
-num
ber e
xpon
ents
6.EE
.2
Writ
e, re
ad, a
nd e
valu
ate
expr
essio
ns in
whi
ch le
tters
stan
d fo
r num
bers
6.EE
.3
Appl
y th
e pr
oper
ties o
f ope
ratio
ns to
gen
erat
e eq
uiva
lent
exp
ress
ions
. 6.
EE.4
Id
entif
y w
hen
two
expr
essio
ns a
re e
quiv
alen
t.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds f
or M
athe
mati
cal P
racti
ceHo
w it
app
lies…
MP2
- Re
ason
abs
trac
tly a
nd q
uanti
tativ
ely.
Whe
n ev
alua
ting
expr
essio
ns (3
x +
7 w
hen
x=4)
stud
ents
are
dec
onte
xtua
lizin
g th
e in
form
ation
but
then
mus
t int
erpr
et th
e re
sult
as it
app
lies t
o th
e co
ntex
t of
the
prob
lem
.
MP7
- Lo
ok fo
r and
mak
e us
e of
stru
ctur
e.W
hen
com
parin
g tw
o ex
pres
sions
, stu
dent
s mus
t und
erst
and
the
stru
ctur
e of
th
e tw
o (v
aria
bles
, coe
ffici
ents
, exp
onen
ts) i
n or
der t
o de
term
ine
equi
vale
nce.
MP8
– L
ook
for a
nd e
xpre
ss re
gula
rity
in re
peat
ed re
ason
ing.
This
is a
pow
erfu
l pra
ctice
whe
n m
ovin
g fr
om n
umer
ical
and
arit
hmeti
c re
a-so
ning
to a
lgeb
raic
repr
esen
tatio
n. B
y att
endi
ng to
cal
cula
tions
whe
n so
lvin
g sp
ecifi
c in
stan
ces o
f a p
robl
em, s
tude
nts w
ill b
e ab
le to
gen
eral
ize th
e pr
oces
s an
d w
rite
an a
lgeb
raic
exp
ress
ion.
Key
Idea
s and
Term
s fo
r “Al
gebr
aic
Ex
pres
sions
”
Key
Idea
sKe
y Te
rms
•Ap
ply
arith
meti
c op
erati
ons t
o va
riabl
e ex
pres
sions
•Bu
ild a
lgeb
raic
exp
ress
ions
from
a c
onte
xt
Expr
essio
n, e
valu
ate,
var
iabl
e, e
quiv
alen
t, co
effici
ent,
quoti
ent,
sum
, ter
m, p
rod-
uct,
fact
or, d
istrib
utive
pro
pert
y, sim
plifi
ed
Term
s sho
uld
be d
eepl
y un
ders
tood
with
in th
e co
ntex
t of t
heir
use.
Not
to b
e co
nsid
ered
stan
dalo
ne v
ocab
ular
y ex
erci
ses.
Prio
r Kno
wle
dge
•Fl
uenc
y w
ith a
rithm
etic
oper
ation
s and
thei
r con
necti
ons t
o re
al w
orld
con
text
s
•Ex
perie
nce
in si
mpl
ifyin
g nu
mer
ical
exp
ress
ions
follo
win
g th
e or
der o
f ope
ratio
ns•
Tran
slate
from
a v
erba
l des
crip
tion
of a
list
of o
pera
tions
to a
n ar
ithm
etic
repr
esen
tatio
n w
ith n
umbe
rs
•Fl
uenc
y w
ith m
athe
mati
cal t
erm
s rep
rese
nting
ope
ratio
ns
Stud
ents
will
be
able
to:
•Cr
eate
and
eva
luat
e nu
mer
ical
exp
ress
ions
.•
Crea
te a
lgeb
raic
exp
ress
ions
from
wor
d ex
pres
sions
.•
Crea
te a
lgeb
raic
exp
ress
ions
from
con
text
.•
Use
mat
hem
atica
l ter
ms t
o de
scrib
e ex
pres
sions
.
•Ap
ply
the
dist
ributi
ve p
rope
rty
and
othe
r ope
ratio
ns to
alg
ebra
ic e
xpre
s-sio
ns to
det
erm
ine
equi
vale
nce.
•Si
mpl
ify a
lgeb
raic
exp
ress
ions
.
•Re
cogn
ize e
quiv
alen
t exp
ress
ions
.
Poss
ible
task
s*Ill
umin
ation
s (N
CTM
) Bui
ldin
g Br
idge
s; V
alen
tine’
s Car
ds
Gra
de 6
Mat
hem
atics
Pla
nnin
g G
uide
– S
Y12-
13Bi
g Id
ea: E
xpre
ssio
ns a
nd E
quati
ons
*The
Gra
de 6
mat
hem
atics
tool
set i
nclu
des t
his a
nd o
ther
reso
urce
s and
can
be
foun
d on
line
at h
ttps:
//oc
s.cp
s.k1
2.il.
us/s
ites/
IKM
C/de
faul
t.asp
x an
d on
the
Depa
rtm
ent o
f Mat
hem
atics
and
Sci
ence
web
site
(cm
si.cp
s.k1
2.il.
us).
CPS Mathematics Content Framework_Version 1.0 Page 37
Plan
ning
Gui
des
Topi
c: S
ingl
e Va
riabl
e Ex
pres
sion
s, E
quati
ons,
and
Ineq
ualiti
esCC
SS-M
Con
tent
Sta
ndar
ds
6.EE
.5
Und
erst
and
solv
ing
an e
quati
on o
r ine
qual
ity a
s a p
roce
ss o
f ans
wer
ing
a qu
estio
n: w
hich
val
ues f
rom
a sp
ecifi
ed se
t, if
any,
mak
e th
e eq
uatio
n or
ineq
ualit
y tr
ue?
Use
su
bstit
ution
to d
eter
min
e w
heth
er a
giv
en n
umbe
r in
a sp
ecifi
ed se
t mak
es a
n eq
uatio
n or
ineq
ualit
y tr
ue.
6.EE
.6
Use
var
iabl
es to
repr
esen
t num
bers
and
writ
e ex
pres
sions
whe
n so
lvin
g re
al-w
orld
or m
athe
mati
cal p
robl
ems;
und
erst
and
that
a v
aria
ble
can
repr
esen
t an
unkn
own
, or,
de
pend
ing
on th
e pu
rpos
e at
han
d, a
ny n
umbe
r in
a sp
ecifi
ed se
t.
6.EE
.7
Solv
e re
al-w
orld
and
mat
hem
atica
l pro
blem
s by
writi
ng a
nd so
lvin
g eq
uatio
ns o
f the
form
x +
p =
q a
nd p
x =
q fo
r cas
es in
whi
ch p
, q, a
nd x
are
all
nonn
egati
ve ra
tiona
l nu
mbe
rs.
6.EE
.8
Writ
e an
ineq
ualit
y of
the
form
x >
c o
r x <
c to
repr
esen
t a c
onst
rain
t or c
ondi
tion
in a
real
wor
ld o
r mat
hem
atica
l pro
blem
. Rec
ogni
ze th
at in
equa
lities
of t
he fo
rm x
> c
hav
e in
finite
ly m
any
solu
tions
; rep
rese
nt so
lutio
ns o
f suc
h in
equa
lities
on
num
ber l
ine
diag
ram
s.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds f
or M
athe
mati
cal P
racti
ceHo
w it
app
lies…
MP4
– M
odel
ing
with
mat
hem
atics
.Cr
eatin
g a
gene
raliz
ed re
pres
enta
tion
of a
situ
ation
(an
equa
tion
or in
equa
lity)
MP7
– L
ook
for a
nd m
akin
g us
e of
stru
ctur
e.
Und
erst
andi
ng a
nd u
sing
a “v
aria
ble”
as a
pla
ceho
lder
for a
n un
know
n in
a
serie
s of c
alcu
latio
ns a
nd th
en u
ndoi
ng th
ose
calc
ulati
ons t
o fin
d th
e va
lue
of
the
unkn
own.
Ana
lyzin
g a
varia
ble
expr
essio
n on
one
side
of a
n eq
uatio
n an
d re
cogn
izing
that
it re
pres
ents
the
num
ber.
Key
Idea
s and
Term
s fo
r “Si
ngle
Var
iabl
e
Expr
essio
ns, E
quati
ons,
an
d In
equa
lities
”
Key
Idea
sKe
y Te
rms
•Re
ason
abo
ut si
mpl
e sin
gle
varia
ble
expr
essio
ns
•Re
ason
abo
ut (a
nd so
lve)
sim
ple
singl
e va
riabl
e eq
uatio
ns
•Re
ason
abo
ut (a
nd so
lve)
sim
ple
singl
e va
riabl
e in
equa
lities
•U
nder
stan
d th
at a
solu
tion
is a
valu
e th
at w
hen
subs
titut
ed fo
r a
varia
ble,
mak
es th
e eq
uatio
n or
ineq
ualit
y tr
ue.
Equa
tion,
ineq
ualit
y, se
t, su
bstit
ution
, var
iabl
e, so
lutio
n
Term
s sho
uld
be d
eepl
y un
ders
tood
with
in th
e co
ntex
t of t
heir
use.
Not
to b
e co
nsid
ered
stan
dalo
ne v
ocab
ular
y ex
erci
ses.
Prio
r Kno
wle
dge
•Fl
uenc
y w
ith a
rithm
etic
oper
ation
s and
thei
r con
necti
ons t
o re
al
wor
ld c
onte
xts
•Ex
perie
nce
in si
mpl
ifyin
g nu
mer
ical
exp
ress
ions
follo
win
g th
e or
der o
f ope
ratio
ns
•Tr
ansla
te fr
om a
ver
bal d
escr
iptio
n of
a li
st o
f ope
ratio
ns to
an
arith
meti
c re
pres
enta
tion
with
num
bers
•Fl
uenc
y w
ith m
athe
mati
cal t
erm
s rep
rese
nting
ope
ratio
ns
Stud
ents
will
be
able
to:
•Cr
eate
alg
ebra
ic e
xpre
ssio
ns fr
om c
onte
xt.
•Cr
eate
and
solv
e al
gebr
aic
equa
tions
from
con
text
.•
Crea
te in
equa
lities
to re
pres
ent a
con
stra
int o
r con
ditio
n fr
om
cont
ext.
•U
se su
bstit
ution
to d
eter
min
e if
a so
lutio
n m
akes
an
ineq
ualit
y tr
ue•
Repr
esen
t sol
ution
s of i
nequ
aliti
es o
n nu
mbe
r lin
e di
agra
ms
Poss
ible
task
s*Fr
om C
ogni
tive
Tuto
r Alg
ebra
1: C
ompu
ter G
ames
, CDs
, and
DVD
S; S
ellin
g Ca
rs; A
Par
k Ra
nger
’s W
ork
is N
ever
Don
e
Gra
de 6
Mat
hem
atics
Pla
nnin
g G
uide
– S
Y12-
13Bi
g Id
ea: E
xpre
ssio
ns a
nd E
quati
ons
*The
Gra
de 6
mat
hem
atics
tool
set i
nclu
des t
his a
nd o
ther
reso
urce
s and
can
be
foun
d on
line
at h
ttps:
//oc
s.cp
s.k1
2.il.
us/s
ites/
IKM
C/de
faul
t.asp
x an
d on
the
Depa
rtm
ent o
f Mat
hem
atics
and
Sci
ence
web
site
(cm
si.cp
s.k1
2.il.
us).
Chicago Public SchoolsPage 38
Topi
c: G
raph
ing
in th
e Co
ordi
nate
Pla
neCC
SS-M
Con
tent
Sta
ndar
ds
6.N
S.8
Solv
e re
al-w
orld
and
mat
hem
atica
l pro
blem
s by
grap
hing
poi
nts i
n al
l fou
r qua
dran
ts o
f the
coo
rdin
ate
plan
e. In
clud
e us
e of
coo
rdin
ates
and
abs
olut
e va
lue
to fi
nd d
istan
ces
betw
een
poin
ts w
ith th
e sa
me
first
coo
rdin
ate
or th
e sa
me
seco
nd c
oord
inat
e.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds f
or M
athe
mati
cal P
racti
ceHo
w it
app
lies…
MP4
– M
odel
with
mat
hem
atics
.U
sing
a gr
aph
to re
pres
ent a
set o
f dat
a cr
eate
s a v
isual
mod
el th
at a
llow
s fo
r dra
win
g ge
nera
lized
con
clus
ions
Key
Idea
s and
Term
s fo
r “Gr
aphi
ng in
the
Co
ordi
nate
Pla
ne “
Key
Idea
sKe
y Te
rms
•Gr
aphi
ng re
late
d pa
irs o
f val
ues i
n or
der t
o un
ders
tand
or r
epre
sent
th
e re
latio
nshi
p be
twee
n th
e qu
antiti
es
Coor
dina
te p
lane
, qua
dran
t, x-
axis,
y-a
xis,
orig
in, a
bsol
ute
valu
e
Term
s sho
uld
be d
eepl
y un
ders
tood
with
in th
e co
ntex
t of t
heir
use.
Not
to
be c
onsid
ered
stan
dalo
ne v
ocab
ular
y ex
erci
ses.
Prio
r Kno
wle
dge
•Gr
aph
poin
ts o
n th
e co
ordi
nate
pla
ne in
the
first
qua
dran
t (5.
G.1
&
5.G.
2)
Stud
ents
will
be
able
to:
•Gr
aph
orde
red
pairs
in a
ny q
uadr
ant.
•De
term
ine
the
dist
ance
bet
wee
n tw
o co
ordi
nate
s alig
ned
verti
cally
or
hor
izont
ally.
•U
se a
bsol
ute
valu
e to
find
the
dist
ance
bet
wee
n tw
o co
ordi
nate
s an
d re
cogn
ize a
bsol
ute
valu
e as
a w
ay to
acc
ount
for d
irecti
on w
hen
dete
rmin
ing
dist
ance
s.
Poss
ible
task
s*Co
ordi
nate
Pla
ne P
ictu
res;
CM
E Al
gebr
a 1
less
ons 3
.1 a
nd 3
.3
Gra
de 6
Mat
hem
atics
Pla
nnin
g G
uide
– S
Y12-
13Bi
g Id
ea: E
xpre
ssio
ns a
nd E
quati
ons
*The
Gra
de 6
mat
hem
atics
tool
set i
nclu
des t
his a
nd o
ther
reso
urce
s and
can
be
foun
d on
line
at h
ttps:
//oc
s.cp
s.k1
2.il.
us/s
ites/
IKM
C/de
faul
t.asp
x an
d on
the
Depa
rtm
ent o
f Mat
hem
atics
and
Sci
ence
web
site
(cm
si.cp
s.k1
2.il.
us).
CPS Mathematics Content Framework_Version 1.0 Page 39
Plan
ning
Gui
des
Grade 7 Mathematics Planning Guide – SY12-13Introduction
The purpose of this Planning Guide is to define the scope of the Common Core State Standards for Mathematics (CCSS-M) Content Standards to be taught in the 2012-2013 school year. It has been designed by CPS teachers to be useful to CPS teachers during the three-year transition to full implementation of the CCSS-M.
In Year 1 of the transition to CCSS-M, 2012-2013, grades 6-8 mathematics teachers will be teaching:
� To the former ILS before the ISAT in March. The expectation is for schools and teachers to continue using their current mathematics instructional materials and to address the Illinois Assessment Frameworks from the beginning of the year until the ISAT is administered. In addition to this content, teachers will explicitly integrate the Standards for Mathematical Practice into their lessons on a regular basis.
� To the new CCSS-M standards, per the scope described by this Planning Guide, after the ISAT, using their instructional materials and other resources to integrate mathematical practices with rigorous mathematical tasks that support the scope of content standards. A toolset to support this instruction includes (1) samples of rigorous grade-specific tasks (including samples of formative assessment options based on MARS (Mathematics Assessment Resource Service) tasks) and (2) general tools that include a sample lesson planning template, a tool for analyzing and modifying lessons/tasks; samples of modified lessons/tasks; and a list of professional resources. These tools are available at https://ocs.cps.k12.il.us/sites/IKMC/default.aspx and on the Department of Mathematics and Science website (cmsi.cps.k12.il.us).
Specifically, after the March ISAT, grades 6-8 mathematics teachers will focus on the Expressions and Equations progression, as defined in the Progressions for the Common Core State Standards in Mathematics: 6-8, Expressions and Equations (2011). This progression was chosen because the Grade 8 CCSS-M includes the “algebra of lines,” which was formerly taught in the first half of high school Algebra I courses (“algebra of lines” refers to equations, graphs of linear relationships, and systems of linear equations). Concentrating on this particular progression allows grades 6-8 classrooms to hone in on the vertical articulation of this very important concept, focusing on students’ cognitive development and the logical structure of the CCSS-M.
Please note: If the content standards in this Planning Guide are covered within the scope of a classroom’s instructional materials prior to ISAT, teachers should teach to these with the level of rigor expected by the CCSS-M, integrating the Standards for Mathematic Practice with the content. Post-ISAT is an opportunity to reinforce content standards already addressed, and to focus more deeply on the Expressions and Equations content standards that were not covered prior to ISAT.
During the first 3 quarters, teachers will explicitly integrate the Standards for Mathematical Practice (below) into their lessons on a regular basis. In the 4th quarter, instruction that integrates Standards for Mathematical Practice with targeted content standards is supported by “how to” guidance in this Planning Guide. By teaching the mathematical practices alongside the indicated content standards, students will be more likely to achieve the depth of conceptual understanding and procedural fluency expected by the CCSS-M.
Chicago Public SchoolsPage 40
The CCSS-M Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Following the ISAT, this Planning Guide will form the basis for instruction in Grade 7 mathematics classrooms. This year’s focus of learning, Expressions and Equations, is broken into three topics: 1) Linear Relationships, 2) Modeling Linear Relationships, and 3) Analyzing Linear Relationships. Topic components include a targeted set of content standards, target mathematical practices (and how they apply to the specific topic), key ideas for learning, key terms, and sample instructional tasks.
The guide assumes 9 weeks for instruction, including time for formative and summative assessments. The topics are sequenced in a way that we believe best develops and connects the mathematical content of the CCSS-M. However, teachers should review the topics and decide the order and time allocation appropriate for their classrooms, given their students, instructional materials, and other considerations. The order of the standards included in a topic does not imply a sequence of the content. Some standards may be revisited several times while addressing the topic, while others may be only partially addressed, depending on the mathematical focus of the topic.
Finally, this document reflects our current thinking about the transition to the CCSS-M. We welcome feedback about your experience with the document. Please share your thoughts with your network staff who will forward to the Department of Mathematics and Science.
CPS Mathematics Content Framework_Version 1.0 Page 41
Plan
ning
Gui
des
Gra
de 7
Mat
hem
atics
Pla
nnin
g G
uide
– S
Y12-
13Bi
g Id
ea: E
xpre
ssio
ns a
nd E
quati
ons
Big
Idea
Ass
essm
ent:
Expr
essi
ons a
nd E
quati
ons
See
7th G
rade
Too
lset
*: F
ishi
ng A
dven
ture
s and
Sal
es T
ax
Topi
c: L
inea
r Rel
ation
ship
sCC
SS-M
Con
tent
Sta
ndar
ds
7.EE
.1
Appl
y pr
oper
ties o
f ope
ratio
ns a
s str
ateg
ies t
o ad
d, su
btra
ct, f
acto
r, an
d ex
pand
line
ar e
xpre
ssio
ns w
ith ra
tiona
l coe
ffici
ents
7.EE
.2
Und
erst
and
that
rew
riting
an
expr
essio
n in
diff
eren
t for
ms i
n a
prob
lem
con
text
can
shed
ligh
t on
the
prob
lem
and
how
the
quan
tities
in it
are
rela
ted.
7.N
S.1d
Ap
ply
prop
ertie
s of o
pera
tions
as s
trat
egie
s to
add
and
subt
ract
real
num
bers
.7.
NS.
1c
Und
erst
and
subt
racti
on o
f rati
onal
num
bers
as a
ddin
g th
e ad
ditiv
e in
vers
e, p
– q
= p
+ (-
q).
Show
that
the
dist
ance
bet
wee
n tw
o ra
tiona
l num
bers
on
the
num
ber l
ine
is th
e ab
solu
te
valu
e of
thei
r diff
eren
ce, a
nd a
pply
this
prin
cipl
e in
real
-wor
ld c
onte
xts.
7.N
S.2a
U
nder
stan
d th
at m
ultip
licati
on is
ext
ende
d fr
om fr
actio
ns to
ratio
nal n
umbe
rs b
y re
quiri
ng th
at o
pera
tions
con
tinue
to sa
tisfy
the
prop
ertie
s of o
pera
tions
, par
ticul
arly
the
dist
ributi
ve
prop
erty
, lea
ding
to p
rodu
cts s
uch
as (-
1)(-1
) = 1
and
the
rule
s for
mul
tiply
ing
signe
d nu
mbe
rs.
Inte
rpre
t pro
duct
s of r
ation
al n
umbe
rs b
y de
scrib
ing
real
-wor
ld c
onte
xts.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds f
or M
athe
mati
cal P
racti
ceHo
w it
app
lies…
MP2
- Re
ason
abs
trac
tly a
nd q
uanti
tativ
ely.
Conn
ectin
g di
ffere
nt c
ompo
nent
s of a
n ex
pres
sion
to th
e co
ntex
t it r
epre
sent
s and
tr
ansf
orm
ing
an e
xpre
ssio
n so
that
it re
veal
s the
des
ired
info
rmati
on.
MP4
- M
odel
with
mat
hem
atics
.Cr
eatin
g an
alg
ebra
ic e
xpre
ssio
n or
equ
ation
that
repr
esen
ts a
situ
ation
or p
robl
em
pres
ente
d
MP7
- Lo
ok fo
r and
mak
e us
e of
stru
ctur
e.Vi
ewin
g 3(
x+12
) as 3
of s
omet
hing
cal
led
(x +
12)
pro
vide
s und
erst
andi
ng o
f how
to e
xpan
d th
e ex
pres
sion
to 3
x +
36 w
hich
allo
ws f
or in
terp
reta
tion
of th
e 3
as a
uni
t rat
e or
slop
e.
Key
Idea
s and
Term
s fo
r “Li
near
Rel
ation
ship
s”Ke
y Id
eas
Key
Term
s
•Ap
ply
know
ledg
e of
arit
hmeti
c op
erati
ons a
nd p
rope
rties
of n
umbe
rs to
exp
res-
sions
with
var
iabl
es to
cre
ate
diffe
rent
but
equ
ival
ent e
xpre
ssio
ns
•Cr
eate
and
inte
rpre
t equ
ival
ent e
xpre
ssio
ns
Expr
essio
n, e
quiv
alen
t, ex
pand
, sim
plify
, rati
onal
num
ber,
di
strib
utive
pro
pert
y
Term
s sho
uld
be d
eepl
y un
ders
tood
with
in th
e co
ntex
t of t
heir
use.
Not
to b
e co
nsid
ered
stan
dalo
ne v
ocab
ular
y ex
erci
ses.
Prio
r Kno
wle
dge
•Fl
uenc
y w
ith a
rithm
etic
oper
ation
s and
thei
r con
necti
ons t
o re
al w
orld
con
text
s
•Ex
perie
nce
in si
mpl
ifyin
g nu
mer
ic e
xpre
ssio
ns fo
llow
ing
the
orde
r of o
pera
tions
•So
lvin
g pr
opor
tions
usin
g ra
tio a
nd ra
te re
ason
ing
to e
xten
d a
patte
rn
•Fa
mili
arity
with
the
num
ber l
ine
Stud
ents
will
be
able
to:
•Ex
pand
line
ar e
xpre
ssio
ns w
ith ra
tiona
l coe
ffici
ents
.•
Crea
te a
nd in
terp
ret e
quiv
alen
t exp
ress
ions
, lin
king
them
to th
e co
ntex
t of t
he
prob
lem
.•
Und
erst
and
abso
lute
val
ue a
s dist
ance
bet
wee
n va
lues
and
app
ly it
to re
al w
orld
co
ntex
ts.
•Ad
d, su
btra
ct, m
ultip
ly, a
nd d
ivid
e ra
tiona
l num
bers
.•
Und
erst
and
and
appl
y th
e di
strib
utive
pro
pert
y to
exp
and
ratio
nal e
xpre
ssio
ns.
•M
odel
num
eric
al e
xpre
ssio
ns o
n th
e nu
mbe
r lin
e.
Poss
ible
task
s*M
ARS
Task
: “Pe
dro’
s Tab
les”
; MAP
Tas
k A1
2: “
Fenc
ing”
; Illu
stra
tive
Mat
hem
atics
Tas
k: 7
EE: “
Disc
ount
ed B
ooks
” an
d “E
quiv
alen
t Exp
ress
ions
”
*The
Gra
de 7
mat
hem
atics
tool
set i
nclu
des t
his a
nd o
ther
reso
urce
s and
can
be
foun
d on
line
at h
ttps:
//oc
s.cp
s.k1
2.il.
us/s
ites/
IKM
C/de
faul
t.asp
x an
d on
the
Depa
rtm
ent o
f Mat
hem
atics
and
Sci
ence
web
site
(cm
si.cp
s.k1
2.il.
us).
Chicago Public SchoolsPage 42
Gra
de 7
Mat
hem
atics
Pla
nnin
g G
uide
– S
Y12-
13Bi
g Id
ea: E
xpre
ssio
ns a
nd E
quati
ons
Topi
c: M
odel
ing
Line
ar R
elati
onsh
ips
CCSS
-M C
onte
nt S
tand
ards
7.EE
.4a
Solv
e w
ord
prob
lem
s lea
ding
to e
quati
ons o
f the
form
px
+ q
= r a
nd p
(x +
q) =
r w
here
p, q
, and
r ar
e sp
ecifi
c ra
tiona
l num
bers
. So
lve
equa
tions
of t
hese
form
s flue
ntly.
Com
pare
an
alge
brai
c so
lutio
n to
an
arith
meti
c so
lutio
n, id
entif
ying
the
sequ
ence
of t
he o
pera
tions
use
d in
eac
h ap
proa
ch.
7.EE
.4b
Solv
e w
ord
prob
lem
s lea
ding
to in
equa
lities
of t
he fo
rm p
x +
q >
r or p
x +
q <
r, w
here
p, q
, and
r ar
e sp
ecifi
c ra
tiona
l num
bers
. Gr
aph
the
solu
tion
set o
f ine
qual
ities
and
inte
rpre
t the
m
in th
e co
ntex
ts o
f the
pro
blem
s.7.
EE.3
So
lve
mul
ti-st
ep re
al-li
fe a
nd m
athe
mati
cal p
robl
ems p
osed
with
pos
itive
and
neg
ative
ratio
nal n
umbe
rs in
any
form
; con
vert
bet
wee
n fo
rms a
s app
ropr
iate
; and
ass
ess t
he
reas
onab
lene
ss o
f ans
wer
s usin
g m
enta
l com
puta
tion
and
estim
ation
stra
tegi
es.
7.RP
.2
Reco
gnize
and
repr
esen
t pro
porti
onal
rela
tions
hips
bet
wee
n qu
antiti
es.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds f
or M
athe
mati
cal P
racti
ceHo
w it
app
lies…
MP1
– M
ake
sens
e of
the
prob
lem
and
per
seve
re in
solv
ing
it.Th
roug
h de
finin
g w
hat t
he w
ord
prob
lem
s are
ask
ing
(7.E
E.4a
/b);
com
parin
g an
al
gebr
aic
solu
tion
to a
n ar
ithm
etic
solu
tion
(7.E
E.4a
) and
ass
essin
g th
e re
ason
-ab
lene
ss o
f an
answ
er (7
.EE.
3)
MP2
– R
easo
n ab
stra
ctly
and
qua
ntita
tivel
y.De
cont
extu
alizi
ng -
crea
ting
an e
quati
on o
r exp
ress
ion
that
get
s tra
nsfo
rmed
an
d th
en c
onte
xtua
lizin
g - c
onne
cting
the
new
exp
ress
ion
or e
quati
on b
ack
into
th
e co
ntex
t of t
he p
robl
em
MP4
– M
odel
with
mat
hem
atics
.Cr
eatin
g an
equ
ation
from
the
cont
ext o
f a w
ord
prob
lem
.
Key
Idea
s and
Term
s fo
r “M
odel
ing
Line
ar
Rela
tions
hips
”
Key
Idea
sKe
y Te
rms
•M
odel
/Rep
rese
nt c
onte
xtua
l lin
ear p
robl
ems w
ith:
singl
e va
riabl
e ex
pres
sions
, equ
ation
s, in
equa
lities
, tab
les,
and
gr
aphs
Uni
t rat
e, so
lutio
n, so
lutio
n se
t, m
ulti-
step
pro
blem
s, e
stim
ation
, ap
prox
imati
on, r
atio,
pro
porti
on, s
cale
fact
or, c
ompl
ex fr
actio
ns
Term
s sho
uld
be d
eepl
y un
ders
tood
with
in th
e co
ntex
t of t
heir
use.
Not
to b
e co
nsid
ered
stan
dalo
ne v
ocab
ular
y ex
erci
ses.
Prio
r Kno
wle
dge
•Kn
owle
dge
of a
nd c
apab
ility
to p
lot p
oint
s in
all f
our q
uadr
ants
of
the
coor
dina
te p
lane
•U
nder
stan
d ra
tes i
n th
e fo
rms o
f equ
ival
ent r
atios
and
tabl
es
•W
rite,
read
, and
eva
luat
e ex
pres
sions
in w
hich
lette
rs st
and
for
num
bers
•Id
entif
y w
hen
two
expr
essio
ns a
re e
quiv
alen
t
•U
nder
stan
d th
at p
ositi
ve a
nd n
egati
ve n
umbe
rs a
re u
sed
toge
ther
to d
e-sc
ribe
quan
tities
hav
ing
oppo
site
dire
ction
s or v
alue
s
Stud
ents
will
be
able
to:
•Re
pres
ent l
inea
r rel
ation
ship
s in
vario
us fo
rms,
incl
udin
g ta
bles
, gr
aphs
, equ
ation
s, e
xpre
ssio
ns, a
nd in
equa
lities
.•
Use
pro
perti
es o
f ope
ratio
ns to
gen
erat
e eq
uiva
lent
exp
ress
ions
.
•So
lve
real
-life
and
mat
hem
atica
l pro
blem
s usin
g nu
mer
ical
and
alg
ebra
ic
equa
tions
•Gr
aph
solu
tions
and
solu
tion
sets
of e
quati
ons a
nd in
equa
lities
, and
inte
r-pr
et th
em in
the
cont
ext o
f the
pro
blem
s
Poss
ible
task
s*M
AP T
ask
A05
“Bas
ebal
l Jer
seys
”; C
MP2
8th
Sha
pes o
f Alg
ebra
: Pro
blem
5.2
; Illu
stra
tive
Mat
hem
atics
: “Sh
rinki
ng”;
and
“Art
Cla
ss”
*The
Gra
de 7
mat
hem
atics
tool
set i
nclu
des t
his a
nd o
ther
reso
urce
s and
can
be
foun
d on
line
at h
ttps:
//oc
s.cp
s.k1
2.il.
us/s
ites/
IKM
C/de
faul
t.asp
x an
d on
the
Depa
rtm
ent o
f Mat
hem
atics
and
Sci
ence
web
site
(cm
si.cp
s.k1
2.il.
us).
CPS Mathematics Content Framework_Version 1.0 Page 43
Plan
ning
Gui
des
Gra
de 7
Mat
hem
atics
Pla
nnin
g G
uide
– S
Y12-
13Bi
g Id
ea: E
xpre
ssio
ns a
nd E
quati
ons
Topi
c: A
naly
zing
Lin
ear R
elati
onsh
ips
CCSS
-M C
onte
nt S
tand
ards
7.EE
.4a
Solv
e w
ord
prob
lem
s lea
ding
to e
quati
ons o
f the
form
px
+ q
= r a
nd p
(x +
q) =
r w
here
p, q
, and
r ar
e sp
ecifi
c ra
tiona
l num
bers
. So
lve
equa
tions
of t
hese
form
s flue
ntly.
Co
mpa
re a
nd a
lgeb
raic
solu
tion
to a
n ar
ithm
etic
solu
tion,
iden
tifyi
ng th
e se
quen
ce o
f the
ope
ratio
ns u
sed
in e
ach
appr
oach
. 7.
EE.4
b So
lve
wor
d pr
oble
ms l
eadi
ng to
ineq
ualiti
es o
f the
form
px
+ q
> r o
r px
+ q
< r,
whe
re p
, q, a
nd r
are
spec
ific
ratio
nal n
umbe
rs.
Grap
h th
e so
lutio
n se
t of i
nequ
aliti
es a
nd
inte
rpre
t the
m in
the
cont
exts
of t
he p
robl
ems.
7.EE
.3
Solv
e m
ulti-
step
real
-life
and
mat
hem
atica
l pro
blem
s pos
ed w
ith p
ositi
ve a
nd n
egati
ve ra
tiona
l num
bers
in a
ny fo
rm; c
onve
rt b
etw
een
form
s as a
ppro
pria
te; a
nd a
sses
s the
re
ason
able
ness
of a
nsw
ers u
sing
men
tal c
ompu
tatio
n an
d es
timati
on st
rate
gies
.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds f
or M
athe
mati
cal P
racti
ceHo
w it
app
lies…
MP1
- Mak
e se
nse
of th
e pr
oble
m a
nd p
erse
vere
in so
lvin
g it.
Thro
ugh
defin
ing
wha
t the
wor
d pr
oble
ms a
re a
skin
g (7
.EE.
4a/b
);
com
parin
g an
alg
ebra
ic so
lutio
n to
an
arith
meti
c so
lutio
n (7
.EE.
4a) a
nd
asse
ssin
g th
e re
ason
able
ness
of a
n an
swer
(7.E
E.3)
MP2
- Rea
son
abst
ract
ly a
nd q
uanti
tativ
ely.
Deco
ntex
tual
izing
- cr
eatin
g an
equ
ation
or e
xpre
ssio
n th
at g
ets
tran
sfor
med
and
then
con
text
ualiz
ing
- con
necti
ng th
e ne
w e
xpre
ssio
n or
eq
uatio
n ba
ck in
to th
e co
ntex
t of t
he p
robl
em
MP4
– M
odel
ing
with
mat
hem
atics
.Cr
eatin
g an
equ
ation
from
the
cont
ext o
f a w
ord
prob
lem
.
Key
Idea
s and
Term
s fo
r “An
alyz
ing
Line
ar
Rela
tions
hips
“
Key
Idea
sKe
y Te
rms
•An
alyz
e an
d in
terp
ret s
ingl
e va
riabl
e ex
pres
sions
, equ
ation
s, in
-eq
ualiti
es, t
able
s, a
nd g
raph
s to
solv
e pr
oble
ms i
n co
ntex
t. Ex
pres
sion,
equ
ation
, ine
qual
ity, r
ation
al n
umbe
rs, s
oluti
on
Term
s sho
uld
be d
eepl
y un
ders
tood
with
in th
e co
ntex
t of t
heir
use.
Not
to
be c
onsid
ered
stan
dalo
ne v
ocab
ular
y ex
erci
ses.
Prio
r Kno
wle
dge
•Kn
owle
dge
of a
nd c
apab
ility
to p
lot p
oint
s in
all f
our q
uadr
ants
of t
he
coor
dina
te p
lane
•U
nder
stan
d ra
tes i
n th
e fo
rms o
f equ
ival
ent r
atios
and
tabl
es
•W
rite,
read
, and
eva
luat
e ex
pres
sions
in w
hich
lette
rs st
and
for
num
bers
•Id
entif
y w
hen
two
expr
essio
ns a
re e
quiv
alen
t
•U
nder
stan
d th
at p
ositi
ve a
nd n
egati
ve n
umbe
rs a
re u
sed
toge
ther
to
desc
ribe
quan
tities
hav
ing
oppo
site
dire
ction
s or v
alue
s
Stud
ents
will
be
able
to:
•So
lve
linea
r equ
ation
s in
one
varia
ble.
•Id
entif
y lin
ear e
quati
ons w
ith o
ne, i
nfini
tely
man
y, or
no
solu
tion.
•Tr
ansf
orm
an
expr
essio
n or
equ
ation
into
alte
rnat
e fo
rms.
•Fi
nd th
e so
lutio
n to
a sy
stem
of e
quati
ons f
rom
a g
raph
.
•So
lve
syst
ems a
lgeb
raic
ally
usin
g su
bstit
ution
and
elim
inati
on/
com
bina
tion.
•U
nder
stan
d th
e so
lutio
n of
a sy
stem
of e
quati
ons i
n th
e co
ntex
t of t
he
prob
lem
and
writ
e as
an
orde
red
pair.
Poss
ible
task
s*In
side
Mat
hem
atics
Pro
blem
s of t
he M
onth
: “Gr
owin
g St
airc
ases
” an
d “S
quirr
elin
g it
Away
”; C
PM –
Eng
agin
g In
equa
lities
*The
Gra
de 7
mat
hem
atics
tool
set i
nclu
des t
his a
nd o
ther
reso
urce
s and
can
be
foun
d on
line
at h
ttps:
//oc
s.cp
s.k1
2.il.
us/s
ites/
IKM
C/de
faul
t.asp
x an
d on
the
Depa
rtm
ent o
f Mat
hem
atics
and
Sci
ence
web
site
(cm
si.cp
s.k1
2.il.
us).
Chicago Public SchoolsPage 44
Grade 8 Mathematics Planning Guide – SY12-13Introduction
The purpose of this Planning Guide is to define the scope of the Common Core State Standards for Mathematics (CCSS-M) Content Standards to be taught in the 2012-2013 school year. It has been designed by CPS teachers to be useful to CPS teachers during the three-year transition to full implementation of the CCSS-M.
In Year 1 of the transition to CCSS-M, 2012-2013, grades 6-8 mathematics teachers will be teaching:
� To the former ILS before the ISAT in March. The expectation is for schools and teachers to continue using their current mathematics instructional materials and to address the Illinois Assessment Frameworks from the beginning of the year until the ISAT is administered. In addition to this content, teachers will explicitly integrate the Standards for Mathematical Practice into their lessons on a regular basis.
� To the new CCSS-M standards, per the scope described by this Planning Guide, after the ISAT, using their instructional materials and other resources to integrate mathematical practices with rigorous mathematical tasks that support the scope of content standards. A toolset to support this instruction includes (1) samples of rigorous grade-specific tasks (including samples of formative assessment options based on MARS (Mathematics Assessment Resource Service) tasks) and (2) general tools that include a sample lesson planning template, a tool for analyzing and modifying lessons/tasks; samples of modified lessons/tasks; and a list of professional resources. These tools are available at https://ocs.cps.k12.il.us/sites/IKMC/default.aspx and on the Department of Mathematics and Science website (http://cmsi.cps.k12.il.us).
Specifically, after the March ISAT, grades 6-8 mathematics teachers will focus on the Expressions and Equations progression, as defined in the Progressions for the Common Core State Standards in Mathematics: 6-8, Expressions and Equations (2011). This progression was chosen because the Grade 8 CCSS-M includes the “algebra of lines,” which was formerly taught in the first half of high school Algebra I courses (“algebra of lines” refers to equations, graphs of linear relationships, and systems of linear equations). Concentrating on this particular progression allows grades 6-8 classrooms to hone in on the vertical articulation of this very important concept, focusing on students’ cognitive development and the logical structure of the CCSS-M.
Please note: If the content standards in this Planning Guide are covered within the scope of a classroom’s instructional materials prior to ISAT, teachers should teach to these with the level of rigor expected by the CCSS-M, integrating the Standards for Mathematic Practice with the content. Post-ISAT is an opportunity to reinforce content standards already addressed, and to focus more deeply on the Expressions and Equations content standards that were not covered prior to ISAT.
During the first 3 quarters, teachers will explicitly integrate the Standards for Mathematical Practice (below) into their lessons on a regular basis. In the 4th quarter, instruction that integrates Standards for Mathematical Practice with targeted content standards is supported by “how to” guidance in this Planning Guide. By teaching the mathematical practices alongside the indicated content standards, students will be more likely to achieve the depth of conceptual understanding and procedural fluency expected by the CCSS-M.
CPS Mathematics Content Framework_Version 1.0 Page 45
Plan
ning
Gui
des
The CCSS-M Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Following the ISAT, this Planning Guide will form the basis for instruction in Grade 8 mathematics classrooms. This year’s focus of learning, Expressions and Equations, is broken into three topics: 1) What is a Function? 2) Linear Functions and Equations and 3) Solving Linear Equations and Systems. Topic components include a targeted set of content standards, target mathematical practices (and how they apply to the specific topic), key ideas for learning, key terms, and sample instructional tasks.
The guide assumes 9 weeks for instruction, including time for formative and summative assessments. The topics are sequenced in a way that we believe best develops and connects the mathematical content of the CCSS-M. However, teachers should review the topics and decide the order and time allocation appropriate for their classrooms, given their students, instructional materials, and other considerations. The order of the standards included in a topic does not imply a sequence of the content. Some standards may be revisited several times while addressing the topic, while others may be only partially addressed, depending on the mathematical focus of the topic.
Finally, this document reflects our current thinking about the transition to the CCSS-M. We welcome feedback about your experience with the document. Please share your thoughts with your network staff who will forward to the Department of Mathematics and Science.
Chicago Public SchoolsPage 46
Big
Idea
Ass
essm
ent:
Expr
essi
ons a
nd E
quati
ons
See
the
8th G
rade
Too
lset
*: M
ARS
task
: Sor
ting
Func
tions
, Sum
mati
ve fo
r thi
s Big
Idea
Topi
c: W
hat i
s a F
uncti
on?
CCSS
-M C
onte
nt S
tand
ards
8.F.1
U
nder
stan
d th
at a
func
tion
is a
rule
that
ass
igns
to e
ach
inpu
t exa
ctly
one
out
put.
The
grap
h of
a fu
nctio
n is
the
set o
f ord
ered
pai
rs c
onsis
ting
of a
n in
put a
nd th
e co
rres
pond
ing
outp
ut.
8.EE
.5 G
raph
pro
porti
onal
rela
tions
hips
, int
erpr
eting
the
unit
rate
as t
he sl
ope
of th
e gr
aph.
Com
pare
two
diffe
rent
pro
porti
onal
rela
tions
hips
repr
esen
ted
in d
iffer
ent w
ays.
8.F.2
Co
mpa
re p
rope
rties
of t
wo
func
tions
eac
h re
pres
ente
d in
a d
iffer
ent w
ay (a
lgeb
raic
ally,
gra
phic
ally,
num
eric
ally
in ta
bles
, or b
y ve
rbal
des
crip
tions
).8.
F.5
Desc
ribe
qual
itativ
ely
the
func
tiona
l rel
ation
ship
bet
wee
n tw
o qu
antiti
es b
y an
alyz
ing
a gr
aph
(e.g
., w
here
the
func
tion
is in
crea
sing
or d
ecre
asin
g, li
near
or n
onlin
ear)
. Ske
tch
a gr
aph
that
exh
ibits
the
qual
itativ
e fe
atur
es o
f a fu
nctio
n th
at h
as b
een
desc
ribed
ver
bally
.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds f
or M
athe
mati
cal P
racti
ceHo
w it
app
lies…
MP7
- Loo
k fo
r and
mak
e us
e of
stru
ctur
e.Ho
w d
iffer
ent c
ompo
nent
s of a
n eq
uatio
n aff
ect o
r are
repr
esen
ted
in th
e gr
aph
of th
e eq
uatio
n; h
ow c
ompo
nent
s of t
he g
raph
can
lead
to d
evel
op-
ing
an e
quati
on
MP8
- Lo
ok fo
r and
exp
ress
regu
larit
y in
repe
ated
reas
onin
g.
Thro
ugh
draw
ing
the
grap
hs o
f var
ious
equ
ation
s, st
uden
ts w
ill re
cogn
ize
the
effec
t of d
iffer
ent p
ortio
ns o
f the
equ
ation
on
the
grap
hs; t
hrou
gh
repe
ating
cal
cula
tions
, stu
dent
s exp
ress
the
gene
raliz
ed se
t of o
pera
tions
an
d cr
eate
an
equa
tion.
Key
Idea
s and
Term
s fo
r “W
hat i
s a F
uncti
on?”
Key
Idea
sKe
y Te
rms
•De
finiti
on a
nd c
hara
cter
istics
of d
iffer
ent t
ypes
of f
uncti
ons
•Re
latio
nshi
p be
twee
n gr
aph
and
equa
tion
•De
velo
pmen
t and
con
siste
nt u
se o
f the
lang
uage
of f
uncti
ons
Func
tion,
inpu
t, ou
tput
, ind
epen
dent
var
iabl
e, d
epen
dent
var
iabl
e, ra
te o
f ch
ange
, Car
tesia
n pl
ane,
slop
e, in
terc
ept,
prop
ortio
nal
Term
s sho
uld
be d
eepl
y un
ders
tood
with
in th
e co
ntex
t of t
heir
use.
Not
to
be c
onsid
ered
stan
dalo
ne v
ocab
ular
y ex
erci
ses.
Prio
r Kno
wle
dge
•U
se o
f lett
ers a
s var
iabl
es ra
ther
than
box
es, e
tc. a
s in
earli
er g
rade
s
•Fa
mili
arity
with
gra
phin
g in
all
four
qua
dran
ts o
f the
Car
tesia
n pl
ane
•U
nder
stan
ding
of t
he m
eani
ng o
f ope
ratio
ns, p
artic
ular
ly w
ith in
tege
rs
Stud
ents
will
be
able
to:
•M
ove
betw
een
tabl
es o
f val
ue a
nd fu
nctio
n ru
les t
o re
pres
ent
func
tions
.
•Gr
aph
func
tions
usin
g in
put-o
utpu
t val
ues a
s ord
ered
pai
rs a
nd
iden
tify
type
of f
uncti
on th
roug
h th
e sh
ape
of th
e gr
aph.
•M
atch
cor
resp
ondi
ng ta
bles
, gra
phs,
equ
ation
s.
•Dr
aw a
gra
ph g
iven
ver
bal d
escr
iptio
ns.
•Id
entif
y sh
apes
of g
raph
s of p
aren
t fun
ction
s (lin
ear,
quad
ratic
, cub
ic,
expo
nenti
al, e
tc.)
and
the
effec
t of d
iffer
ent c
ompo
nent
s of t
he
equa
tion
(lead
ing
coeffi
cien
t, co
nsta
nt te
rm, e
tc.).
•Gr
aph
prop
ortio
nal r
elati
onsh
ips.
•Co
mpa
re d
iffer
ent p
ropo
rtion
al re
latio
nshi
ps re
pres
ente
d in
diff
eren
t fo
rms.
Poss
ible
task
(s)*
CME
Alge
bra
1 te
xt,
Chap
ter 3
, for
shap
es o
f gra
phs o
f par
ent f
uncti
ons
Gra
de 8
Mat
hem
atics
Pla
nnin
g G
uide
– S
Y12-
13Bi
g Id
ea: E
xpre
ssio
ns a
nd E
quati
ons
*The
Gra
de 8
mat
hem
atics
tool
set i
nclu
des t
his a
nd o
ther
reso
urce
s and
can
be
foun
d on
line
at h
ttps:
//oc
s.cp
s.k1
2.il.
us/s
ites/
IKM
C/de
faul
t.asp
x an
d on
the
Depa
rtm
ent o
f Mat
hem
atics
and
Sci
ence
web
site
(cm
si.cp
s.k1
2.il.
us).
CPS Mathematics Content Framework_Version 1.0 Page 47
Plan
ning
Gui
des
Topi
c: L
inea
r Fun
ction
s and
Equ
ation
sCC
SS-M
Con
tent
Sta
ndar
ds8.
EE.6
U
se si
mila
r tria
ngle
s to
expl
ain
why
the
slope
m is
the
sam
e be
twee
n an
y tw
o di
stinc
t poi
nts o
n a
non-
verti
cal l
ine
in th
e co
ordi
nate
pla
ne; d
eriv
e th
e eq
uatio
n y
= m
x fo
r a li
ne th
roug
h th
e or
igin
and
the
equa
tion
y =
mx
+ b
for a
line
inte
rcep
ting
the
verti
cal a
xis a
t b8.
F.3
Inte
rpre
t the
equ
ation
y =
mx
+ b
as d
efini
ng a
line
ar fu
nctio
n, w
hose
gra
ph is
a st
raig
ht li
ne; g
ive
exam
ples
of f
uncti
ons t
hat a
re n
ot li
near
. 8.
F.4
Cons
truc
t a fu
nctio
n to
mod
el a
line
ar re
latio
nshi
p be
twee
n tw
o qu
antiti
es. D
eter
min
e th
e ra
te o
f cha
nge
and
initi
al v
alue
of t
he fu
nctio
n fr
om a
des
crip
tion
of a
rela
tions
hip
or fr
om
two
(x, y
) val
ues,
incl
udin
g re
adin
g th
ese
from
a ta
ble
or fr
om a
gra
ph. I
nter
pret
the
rate
of c
hang
e an
d in
itial
val
ue o
f a li
near
func
tion
in te
rms o
f the
situ
ation
it m
odel
s, a
nd in
term
s of
its g
raph
or a
tabl
e of
val
ues.
8.SP
.3
Use
the
equa
tion
of a
line
ar m
odel
to so
lve
prob
lem
s in
the
cont
ext o
f biv
aria
te m
easu
rem
ent d
ata,
inte
rpre
ting
the
slope
and
inte
rcep
t.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds f
or M
athe
mati
cal P
racti
ceHo
w it
app
lies…
MP2
- Re
ason
ing
abst
ract
ly a
nd q
uanti
tativ
ely.
Deco
ntex
tual
izing
is w
orki
ng st
rictly
with
the
equa
tion
and
the
grap
h;
cont
extu
alizi
ng is
inte
rpre
ting
the
slope
as t
he ra
te o
f cha
nge
and
the
y-in
terc
ept a
s the
initi
al v
alue
of t
he c
onte
xt o
r situ
ation
. Th
is al
so in
volv
es
unde
rsta
ndin
g w
hen
a gr
aph
shou
ld b
e co
ntinu
ous o
r whe
n it
will
not
co
ntinu
e fr
om th
e fir
st q
uadr
ant t
o th
e se
cond
, thi
rd, o
r fou
rth.
MP7
- Loo
k fo
r and
mak
e us
e of
stru
ctur
e.Ho
w d
iffer
ent c
ompo
nent
s of a
n eq
uatio
n aff
ect o
r are
repr
esen
ted
in th
e gr
aph
of th
e eq
uatio
n or
aris
e fr
om c
onte
xt
MP8
- Lo
ok fo
r and
exp
ress
regu
larit
y in
repe
ated
reas
onin
g.Re
peat
ed c
alcu
latio
n of
slop
e le
ads t
o an
equ
ation
for t
he g
raph
of a
line
or
a li
near
con
text
.
Key
Idea
s and
Term
s fo
r “Li
near
Fun
ction
s and
Eq
uatio
ns”
Key
Idea
sKe
y Te
rms
•Eq
uatio
ns a
nd g
raph
s of l
ines
•Bu
ildin
g lin
ear e
quati
ons f
rom
a c
onte
xt
Slop
e, in
terc
epts
, slo
pe-in
terc
ept f
orm
, pro
porti
onal
Term
s sho
uld
be d
eepl
y un
ders
tood
with
in th
e co
ntex
t of t
heir
use.
Not
to
be c
onsid
ered
stan
dalo
ne v
ocab
ular
y ex
erci
ses.
Prio
r Kno
wle
dge
•Fl
uenc
y w
ith a
rithm
etic
oper
ation
s and
the
conn
ectio
ns to
real
wor
ld c
onte
xts
•Ex
posu
re to
pro
porti
onal
rela
tions
hips
and
reas
onin
g•
Und
erst
andi
ng th
at a
gra
ph is
a p
ictu
re o
f the
set o
f coo
rdin
ate
pairs
that
satis
fy a
n eq
uatio
n
Stud
ents
will
be
able
to:
•Tr
ansla
te fr
om E
nglis
h to
alg
ebra
ic e
xpre
ssio
ns/e
quati
ons.
•U
se si
mila
r tria
ngle
s to
dete
rmin
e w
hy sl
ope
is sa
me
betw
een
colli
near
po
ints
.•
Deriv
e a
linea
r equ
ation
from
a c
onte
xt a
nd tr
ansf
orm
it in
to y
=mx
or
y=m
x+b
and
inte
rpre
t the
mea
ning
of e
ach
com
pone
nt o
f the
equ
ation
.•
Reco
gnize
non
-line
ar e
quati
ons.
•De
term
ine
rate
of c
hang
e in
line
ar re
latio
nshi
ps.
•M
ove
fluen
tly b
etw
een
a ta
ble,
a g
raph
and
an
equa
tion.
•De
scrib
e th
e re
latio
nshi
p be
twee
n tw
o va
riabl
es.
•In
terp
ret m
eani
ng o
f slo
pe a
nd y
-inte
rcep
t in
cont
exts
.
Poss
ible
task
(s)*
Tria
ngle
Tas
k (N
YC E
E U
nit)
; CM
E le
sson
4.1
; Aus
sie F
ir Tr
ees (
NYC
EE
unit)
; Squ
are
Patte
rns (
MAR
S); E
Z Co
aste
rs (N
YC E
E U
nit)
Two
Oil
Tank
s(N
YC E
E U
nit)
Sum
mati
ve fo
r thi
s top
ic a
nd sl
ight
brid
ge to
syst
ems
Gra
de 8
Mat
hem
atics
Pla
nnin
g G
uide
– S
Y12-
13Bi
g Id
ea: E
xpre
ssio
ns a
nd E
quati
ons
*The
Gra
de 8
mat
hem
atics
tool
set i
nclu
des t
his a
nd o
ther
reso
urce
s and
can
be
foun
d on
line
at h
ttps:
//oc
s.cp
s.k1
2.il.
us/s
ites/
IKM
C/de
faul
t.asp
x an
d on
the
Depa
rtm
ent o
f Mat
hem
atics
and
Sci
ence
web
site
(cm
si.cp
s.k1
2.il.
us).
Chicago Public SchoolsPage 48
Topi
c: S
olvi
ng L
inea
r Equ
ation
s and
Sys
tem
sCC
SS-M
Con
tent
Sta
ndar
ds
8.EE
.7
Solv
e lin
ear e
quati
ons i
n on
e va
riabl
e.
a. G
ive
exam
ples
of l
inea
r equ
ation
s in
one
varia
ble
with
one
solu
tion,
infin
itely
man
y so
lutio
ns, o
r no
solu
tions
. Sho
w w
hich
of t
hese
pos
sibili
ties i
s the
cas
e by
succ
essiv
ely
tran
sfor
min
g th
e gi
ven
equa
tion
into
sim
pler
form
s, u
ntil a
n eq
uiva
lent
equ
ation
of t
he fo
rm x
= a
, a =
a, o
r a =
b re
sults
(whe
re a
and
b a
re d
iffer
ent n
umbe
rs).
b. S
olve
line
ar e
quati
ons w
ith ra
tiona
l num
ber c
oeffi
cien
ts, i
nclu
ding
equ
ation
s who
se so
lutio
ns re
quire
exp
andi
ng e
xpre
ssio
ns u
sing
the
dist
ributi
ve p
rope
rty
and
colle
cting
lik
e te
rms.
8.EE
.8
Anal
yze
and
solv
e pa
irs o
f sim
ulta
neou
s lin
ear e
quati
ons.
a. U
nder
stan
d th
at so
lutio
ns to
a sy
stem
of t
wo
linea
r equ
ation
s in
two
varia
bles
cor
resp
ond
to p
oint
s of i
nter
secti
on o
f the
ir gr
aphs
, bec
ause
poi
nts o
f int
erse
ction
satis
fy b
oth
equa
tions
sim
ulta
neou
sly.
b. S
olve
syst
ems o
f tw
o lin
ear e
quati
ons i
n tw
o va
riabl
es a
lgeb
raic
ally,
and
esti
mat
e so
lutio
ns b
y gr
aphi
ng th
e eq
uatio
ns. S
olve
sim
ple
case
s by
insp
ectio
n.
c. S
olve
real
-wor
ld a
nd m
athe
mati
cal p
robl
ems l
eadi
ng to
two
linea
r equ
ation
s in
two
varia
bles
.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds f
or M
athe
mati
cal P
racti
ceHo
w it
app
lies…
MP4
- M
odel
with
mat
hem
atics
.Cr
eatin
g an
equ
ation
that
repr
esen
ts th
e sit
uatio
n an
d ca
ptur
es th
e
gene
raliz
ed si
tuati
on o
f the
con
text
.
MP7
- Loo
k fo
r and
mak
e us
e of
stru
ctur
e.Ho
w d
iffer
ent c
ompo
nent
s of a
n eq
uatio
n aff
ect o
r are
repr
esen
ted
in th
e gr
aph
of th
e eq
uatio
n or
aris
e fr
om c
onte
xt a
nd u
se v
ario
us re
pres
enta
-tio
ns o
f lin
ear r
elati
onsh
ips a
nd id
entif
y be
nefit
s to
usin
g ea
ch fo
rm.
Key
Idea
s and
Term
s fo
r “So
lvin
g Li
near
Equ
a-tio
ns a
nd S
yste
ms“
Key
Idea
sKe
y Te
rms
•So
lvin
g lin
ear e
quati
ons
•So
lvin
g sy
stem
s of l
inea
r equ
ation
s
Brea
k-ev
en p
oint
, int
erse
ction
poi
nt, s
ubsti
tutio
n, e
limin
ation
, lik
e te
rms,
eq
uiva
lenc
e, d
istrib
utive
pro
pert
y, so
lutio
n
Term
s sho
uld
be d
eepl
y un
ders
tood
with
in th
e co
ntex
t of t
heir
use.
Not
to
be c
onsid
ered
stan
dalo
ne v
ocab
ular
y ex
erci
ses.
Prio
r Kno
wle
dge
•Su
bstit
uting
val
ues f
or v
aria
bles
to e
valu
ate
expr
essio
ns (v
alue
of 3
x +4
if x
=8)
•Ap
ply
the
dist
ributi
ve p
rope
rty
•U
nder
stan
d eq
uiva
lent
exp
ress
ions
•Co
mbi
ning
like
term
s
Stud
ents
will
be
able
to:
•So
lve
linea
r equ
ation
s in
one
varia
ble.
•Id
entif
y lin
ear e
quati
ons w
ith o
ne, i
nfini
tely
man
y, or
no
solu
tion.
•Tr
ansf
orm
an
expr
essio
n or
equ
ation
into
alte
rnat
e fo
rms.
•Fi
nd th
e so
lutio
n to
a sy
stem
of e
quati
ons f
rom
a g
raph
.
•So
lve
syst
ems a
lgeb
raic
ally
usin
g su
bstit
ution
and
elim
inati
on/
com
bina
tion.
•U
nder
stan
d th
e so
lutio
n of
a sy
stem
of e
quati
ons i
n th
e co
ntex
t of t
he
prob
lem
and
writ
e as
an
orde
red
pair.
Poss
ible
task
(s)*
CME
less
ons 4
.10,
4.1
2; C
arne
gie
car r
enta
ls fr
om tw
o co
mpa
nies
; Cel
l Pho
ne P
lans
Gra
de 8
Mat
hem
atics
Pla
nnin
g G
uide
– S
Y12-
13Bi
g Id
ea: E
xpre
ssio
ns a
nd E
quati
ons
*The
Gra
de 8
mat
hem
atics
tool
set i
nclu
des t
his a
nd o
ther
reso
urce
s and
can
be
foun
d on
line
at h
ttps:
//oc
s.cp
s.k1
2.il.
us/s
ites/
IKM
C/de
faul
t.asp
x an
d on
the
Depa
rtm
ent o
f Mat
hem
atics
and
Sci
ence
web
site
(cm
si.cp
s.k1
2.il.
us).
CPS Mathematics Content Framework_Version 1.0 Page 49
Plan
ning
Gui
des
Algebra I Planning Guide – SY12-13 Introduction
The purpose of this Algebra I Planning Guide is to define the scope of the Common Core State Standards for Mathematics (CCSS-M) content standards to be taught in the 2012-2013 school year. It has been designed by CPS teachers to be useful to CPS teachers during the three-year transition to full implementation of the CCSS-M.
Algebra I typically extends the mathematics that students learn in the middle grade courses. However, in CPS, prior to CCSS-M, “algebra of lines” has not been fully introduced in the middle grades. (“Algebra of lines” refers to equations, graphs of linear relationships, and systems of linear equations.) Therefore, the Algebra I Planning Guide (this document) includes this concept. To accommodate the time required to teach “algebra of lines,” the guide does not include the Statistics and Probability content outlined in the traditional Algebra I course in the Appendix A: Designing High School Mathematics Courses on the Common Core State Standards (2011). This scope of CCSS-M content standards for this first year of transition will still be rather ambitious for Algebra I.
This Planning Guide is structured around six Big Ideas (below). For each Big Idea, a summative assessment is referenced, and topics provide the detailed components to support instruction. Topic components include a targeted set of content standards, target mathematical practices (and how they apply to the specific topic), and sample instructional tasks. The sample tasks have been selected for their rigor and their potential usefulness as formative assessments for the topic. A toolset to support instruction includes: a lesson planning template; samples of rigorous tasks; samples of formative assessment options based on MARS (Mathematics Assessment Resource Service) tasks; tools for examining and modifying lessons/tasks to increase rigor; samples of modified lessons/tasks; and a recommended list of professional resources. These tools are available at https://ocs.cps.k12.il.us/sites/IKMC/default.aspx and on the Department of Mathematics and Science website (http://cmsi.cps.k12.il.us).
The guide assumes 159 days for instruction, including time for formative and summative assessments. The Big Ideas are sequenced in a way that we believe best develops and connects the mathematical content of the CCSS-M. However, teachers should review the Big Ideas and decide the order and time allocation appropriate for their classrooms, given their students, instructional materials, and other considerations. The order of the standards presented in a topic does not imply a sequence of the content. Some standards may be revisited several times while addressing the topic, while others may be only partially addressed, depending on the topic’s mathematical focus.
Throughout Algebra I, students should continue to develop proficiency in the Common Core’s Standards for Mathematical Practice (below). Teachers should integrate the instruction of content standards with mathematical practices. This Planning Guide includes “how to” guidance to support this approach to integrated instruction. When the mathematical practices are taught alongside the indicated content standards, students will be more likely to achieve the depth of conceptual understanding and procedural fluency that are expected by the CCSS-M.
Chicago Public SchoolsPage 50
The CCSS-M Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning
Finally, this document reflects our current thinking about the transition to the CCSS-M. We welcome feedback about your experience with the document. Please share your thoughts with your network staff who will forward to the Department of Mathematics and Science.
BIG IDEAS in ALGEBRA I, YEAR 1 PAGE
Foundations for Algebra .......................................................................................................................................51
Introduction to Functions and Their Rules .......................................................................................................... 55
Linear Equations and Inequalities ..................................................................................................................... 57
Modeling with Linear Functions ...........................................................................................................................59
Solving Systems of Equations and Inequalities ................................................................................................... 62
Non-linear Functions and Equations ....................................................................................................................65
CPS Mathematics Content Framework_Version 1.0 Page 51
Plan
ning
Gui
des
Big
Idea
Ass
essm
ent:
Foun
datio
ns fo
r Alg
ebra
See
Alge
bra
I Too
lset
*: C
entr
al P
ark
Topi
c: O
pera
tions
on
Ratio
nal N
umbe
rsCC
SS-M
Con
tent
Sta
ndar
ds6-
NS.
7c
Und
erst
and
the
abso
lute
val
ue o
f a ra
tiona
l num
ber a
s its
dist
ance
from
0 o
n th
e nu
mbe
r lin
e; in
terp
ret a
bsol
ute
valu
e as
mag
nitu
de fo
r a p
ositi
ve o
r a n
egati
ve q
uanti
ty in
a
real
-wor
ld si
tuati
on.
7-N
S.1b
Und
erst
and
p +
q as
the
num
ber l
ocat
ed a
dist
ance
of |
q| fr
om p
, in
the
positi
ve o
r neg
ative
dire
ction
dep
endi
ng o
n w
heth
er q
is p
ositi
ve o
r neg
ative
. Sh
ow th
at a
num
ber a
nd
its o
ppos
ite h
ave
a su
m o
f 0 (a
re a
dditi
ve in
vers
es).
Inte
rpre
t sum
s of r
ation
al n
umbe
rs b
y de
scrib
ing
real
-wor
ld c
onte
xts.
7-N
S.1c
U
nder
stan
d th
at su
btra
ction
of r
ation
al n
umbe
rs a
s add
ing
the
addi
tive
inve
rse,
p –
q =
p +
(-q)
. Sh
ow th
at th
e di
stan
ce o
f tw
o ra
tiona
l num
bers
on
the
num
ber l
ine
is th
e ab
solu
te v
alue
of t
heir
diffe
renc
e, a
nd a
pply
this
prin
cipl
e in
real
-wor
ld c
onte
xts.
7-N
S.1d
App
ly p
rope
rties
of o
pera
tions
as s
trat
egie
s to
add
and
subt
ract
ratio
nal n
umbe
rs.
7-N
S.2a
U
nder
stan
d th
at m
ultip
licati
on is
ext
ende
d fr
om fr
actio
ns to
ratio
nal n
umbe
rs b
y re
quiri
ng th
at o
pera
tions
con
tinue
to sa
tisfy
the
prop
ertie
s of o
pera
tions
, par
ticul
arly
the
dist
ributi
ve p
rope
rty,
lead
ing
to p
rodu
cts s
uch
as (-
1)(-1
) = 1
and
the
rule
s for
mul
tiply
ing
signe
d nu
mbe
rs.
Inte
rpre
t pro
duct
s of r
ation
al n
umbe
rs b
y de
scrib
ing
real
-wor
ld
cont
exts
.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…
MP2
- Re
ason
abs
trac
tly a
nd q
uanti
tativ
ely.
Man
ipul
ate
the
mat
hem
atica
l rep
rese
ntati
on b
y sh
owin
g th
e pr
oces
s co
nsid
erin
g th
e m
eani
ng o
f the
qua
ntitie
s inv
olve
d.
MP3
- Co
nstr
uct v
iabl
e ar
gum
ents
and
criti
que
the
reas
onin
g of
oth
ers.
Justi
fy (o
rally
and
in w
ritten
form
) the
app
roac
h us
ed, i
nclu
ding
how
it fi
ts
in th
e co
ntex
t fro
m w
hich
the
data
aro
se.
MP7
- Lo
ok fo
r and
mak
e us
e of
stru
ctur
e.U
se p
atter
ns o
r str
uctu
re to
mak
e se
nse
of m
athe
mati
cs a
nd c
onne
ct
prio
r kno
wle
dge
to si
mila
r situ
ation
s and
ext
end
to n
ovel
situ
ation
s.
Prio
r Kno
wle
dge
•Th
e nu
mbe
r lin
e
•Ra
tiona
l num
bers
•Ad
ditiv
e in
vers
e
•Pr
oper
ties o
f ope
ratio
ns
•Pr
oper
ties o
f ope
ratio
ns, e
sp. t
he d
istrib
utive
pro
pert
y
Stud
ents
will
be
able
to:
•Fi
nd th
e ab
solu
te v
alue
of a
ratio
nal n
umbe
r.
•Ex
plai
n th
e m
eani
ng o
f abs
olut
e va
lue
as it
s dist
ance
from
0 o
n th
e nu
mbe
r lin
e.
•Ad
d ra
tiona
l num
bers
.
•U
se th
e nu
mbe
r lin
e an
d th
e de
finiti
on o
f abs
olut
e va
lue
to e
xpla
in
how
to a
dd ra
tiona
l num
bers
.
•Ex
plai
n w
hy th
e su
m o
f tw
o op
posit
e nu
mbe
rs is
0.
•Ap
ply
the
skill
of a
ddin
g ra
tiona
l num
bers
to th
e re
al w
orld
.
•Su
btra
ct ra
tiona
l num
bers
.
•U
se th
e co
ncep
ts o
f abs
olut
e va
lue
and
addi
tive
inve
rse
to e
xpla
in
how
to su
btra
ct ra
tiona
l num
bers
.
•Ap
ply
the
skill
of s
ubtr
actin
g ra
tiona
l num
bers
to th
e re
al w
orld
.
•U
se p
rope
rties
of o
pera
tions
whe
n ad
ding
and
subt
racti
ng ra
tiona
l nu
mbe
rs.
•M
ultip
ly ra
tiona
l num
bers
.
•In
terp
ret p
rodu
cts o
f rati
onal
num
bers
in re
al w
orld
con
text
s.
Poss
ible
task
(s)*
*The
Alg
ebra
I to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
Alge
bra
I Pla
nnin
g G
uide
– S
Y12-
13
Big
Idea
: Fou
ndati
ons f
or A
lgeb
ra
Chicago Public SchoolsPage 52
Topi
c: W
orki
ng w
ith E
xpre
ssio
nsCC
SS-M
Con
tent
Sta
ndar
ds6-
EE.2
c Ev
alua
te e
xpre
ssio
ns a
t spe
cific
val
ues o
f the
ir va
riabl
es. I
nclu
de e
xpre
ssio
ns th
at a
rise
from
form
ulas
use
d in
real
-wor
ld p
robl
ems.
Per
form
arit
hmeti
c op
erati
ons,
incl
udin
g th
ose
invo
lvin
g w
hole
-num
ber e
xpon
ents
, in
the
conv
entio
nal o
rder
whe
n th
ere
are
no p
aren
thes
es to
spec
ify a
par
ticul
ar o
rder
(Ord
er o
f Ope
ratio
ns).
6-EE
.3
Appl
y th
e pr
oper
ties o
f ope
ratio
ns to
gen
erat
e eq
uiva
lent
exp
ress
ions
.6-
EE.4
Id
entif
y w
hen
two
expr
essio
ns a
re e
quiv
alen
t (i.e
., w
hen
two
expr
essio
ns n
ame
the
sam
e nu
mbe
r reg
ardl
ess o
f whi
ch v
alue
is su
bstit
uted
into
them
).7-
EE.1
Ap
ply
prop
ertie
s of o
pera
tions
as s
trat
egie
s to
add,
subt
ract
, fac
tor a
nd e
xpan
d lin
ear e
xpre
ssio
ns w
ith ra
tiona
l coe
ffici
ents
.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…
MP1
- M
ake
sens
e of
pro
blem
s and
per
seve
re in
solv
ing
them
.Ev
alua
te p
rogr
ess t
owar
d th
e so
lutio
n an
d m
ake
revi
sions
if n
eces
sary
.
MP2
- Re
ason
abs
trac
tly a
nd q
uanti
tativ
ely.
Reco
gnize
the
rela
tions
hips
bet
wee
n nu
mbe
rs/q
uanti
ties w
ithin
the
pr
oces
s to
eval
uate
a p
robl
em.
MP6
- Att
end
to p
reci
sion.
Calc
ulat
e an
swer
s effi
cien
tly a
nd a
ccur
atel
y an
d la
bel t
hem
app
ropr
iate
ly.
Prio
r Kno
wle
dge
•O
rder
of o
pera
tions
•M
eani
ng o
f exp
onen
ts
•Fo
ur o
pera
tions
on
ratio
nal n
umbe
rs
•Pr
oper
ties o
f ope
ratio
ns
•M
eani
ng o
f “eq
uiva
lent
”
•Ev
alua
ting
expr
essio
ns
Stud
ents
will
be
able
to:
•Ev
alua
te a
lgeb
raic
exp
ress
ions
, inc
ludi
ng th
ose
with
exp
onen
ts a
nd
thos
e re
quiri
ng k
now
ledg
e of
ord
er o
f ope
ratio
ns.
•Ap
ply
the
prop
ertie
s of o
pera
tions
to g
ener
ate
equi
vale
nt e
xpre
ssio
ns.
•Re
cogn
ize e
quiv
alen
t exp
ress
ions
.
•U
se p
rope
rties
of o
pera
tions
to g
ener
ate
equi
vale
nt e
xpre
ssio
ns.
Poss
ible
task
(s)*
A M
illio
n Do
llars
F
rom
: Mat
hem
atics
Ass
essm
ent P
roje
ct
Alge
bra
I Pla
nnin
g G
uide
– S
Y12-
13
Big
Idea
: Fou
ndati
ons f
or A
lgeb
ra
*The
Alg
ebra
I to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
CPS Mathematics Content Framework_Version 1.0 Page 53
Plan
ning
Gui
des
Alge
bra
I Pla
nnin
g G
uide
– S
Y12-
13
Big
Idea
: Fou
ndati
ons f
or A
lgeb
ra
To
pic:
Intr
oduc
tion
to E
quati
ons a
nd F
uncti
ons
CCSS
-M C
onte
nt S
tand
ards
7-EE
.4
Use
var
iabl
es to
repr
esen
t qua
ntitie
s in
a re
al-w
orld
or m
athe
mati
cal p
robl
em, a
nd c
onst
ruct
sim
ple
equa
tions
and
ineq
ualiti
es to
solv
e pr
oble
ms b
y re
ason
ing
abou
t the
qu
antiti
es.
A-CE
D.2
Crea
te e
quati
ons i
n tw
o or
mor
e va
riabl
es to
repr
esen
t rel
ation
ship
s bet
wee
n qu
antiti
es; g
raph
equ
ation
s on
coor
dina
te a
xes w
ith la
bels
and
scal
es.
6-EE
.9
Use
var
iabl
es to
repr
esen
t tw
o qu
antiti
es in
a re
al-w
orld
pro
blem
that
cha
nge
in re
latio
nshi
p to
one
ano
ther
; writ
e an
equ
ation
to e
xpre
ss o
ne q
uanti
ty, t
houg
ht o
f as t
he
depe
nden
t var
iabl
e, in
term
s of t
he o
ther
qua
ntity
, tho
ught
of a
s the
inde
pend
ent v
aria
ble.
Ana
lyze
the
rela
tions
hip
betw
een
the
depe
nden
t and
inde
pend
ent v
aria
bles
usin
g gr
aphs
and
tabl
es, a
nd re
late
thes
e to
the
equa
tion.
7-RP
.2a
Deci
de w
heth
er tw
o qu
antiti
es a
re in
a p
ropo
rtion
al re
latio
nshi
p, e
.g.,
by te
sting
for e
quiv
alen
t rati
os in
a ta
ble
or g
raph
ing
on a
coo
rdin
ate
plan
e an
d ob
serv
ing
whe
ther
the
grap
h is
a st
raig
ht li
ne th
roug
h th
e or
igin
.7-
RP.2
b Id
entif
y th
e co
nsta
nt o
f pro
porti
onal
ity (u
nit r
ate)
in ta
bles
, gra
phs,
equ
ation
s, d
iagr
ams,
and
ver
bal d
escr
iptio
ns o
f pro
porti
onal
rela
tions
hips
.7-
RP.2
d Ex
plai
n w
hat a
poi
nt (x
, y) o
n th
e gr
aph
of a
pro
porti
onal
rela
tions
hip
mea
ns in
term
s of t
he si
tuati
on, w
ith sp
ecia
l atte
ntion
to th
e po
ints
(0, 0
) and
(1, r
) whe
re r
is th
e un
it ra
te.
8-EE
.5
Grap
h pr
opor
tiona
l rel
ation
ship
s, in
terp
retin
g th
e un
it ra
te a
s the
slop
e of
the
grap
h. C
ompa
re tw
o di
ffere
nt p
ropo
rtion
al re
latio
nshi
ps re
pres
ente
d in
diff
eren
t way
s.F-
IF.1
Und
erst
and
that
a fu
nctio
n fr
om o
ne se
t (ca
lled
the
dom
ain)
to a
noth
er se
t (ca
lled
the
rang
e) a
ssig
ns to
eac
h el
emen
t of t
he d
omai
n ex
actly
one
ele
men
t of t
he ra
nge.
If f
is a
func
tion
and
x is
an e
lem
ent o
f its
dom
ain,
then
f(x)
den
otes
the
outp
ut o
f f c
orre
spon
ding
to th
e in
put x
. The
gra
ph o
f f is
the
grap
h of
the
equa
tion
y =
f(x).
F-IF.
2 U
se fu
nctio
n no
tatio
n, e
valu
ate
func
tions
for i
nput
s in
thei
r dom
ains
, and
inte
rpre
t sta
tem
ents
that
use
func
tion
nota
tion
in te
rms o
f a c
onte
xt.
F- IF
.5
Rela
te th
e do
mai
n of
a fu
nctio
n to
its g
raph
and
, whe
re a
pplic
able
, to
the
quan
titati
ve re
latio
nshi
p it
desc
ribes
. For
exa
mpl
e, if
the
func
tion
h(n)
giv
es th
e nu
mbe
r of
pers
on-h
ours
it ta
kes t
o as
sem
ble
n en
gine
s in
a fa
ctor
y, th
en th
e po
sitive
inte
gers
wou
ld b
e an
app
ropr
iate
dom
ain
for t
he fu
nctio
n.★
★ Sp
ecifi
c m
odel
ing
stan
dard
(ver
sus a
n ex
ampl
e of
the
mod
elin
g St
anda
rd fo
r Mat
hem
atica
l Pra
ctice
)
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…
MP2
- Re
ason
abs
trac
tly a
nd q
uanti
tativ
ely.
Eval
uate
pro
gres
s tow
ard
the
solu
tion
and
mak
e re
visio
ns if
nec
essa
ry.
MP3
- Co
nstr
uct v
iabl
e ar
gum
ents
and
criti
que
the
reas
onin
g of
oth
ers.
Use
obs
erva
tions
and
prio
r kno
wle
dge
(sta
ted
assu
mpti
ons,
defi
nitio
ns,
and
prev
ious
est
ablis
hed
resu
lts) t
o m
ake
conj
ectu
res a
nd c
onst
ruct
ar
gum
ents
.
MP6
- Att
end
to p
reci
sion.
Calc
ulat
e an
swer
s effi
cien
tly a
nd a
ccur
atel
y an
d la
bel t
hem
app
ropr
iate
ly.
Prio
r Kno
wle
dge
•M
eani
ng o
f “va
riabl
e”
•M
eani
ng o
f equ
al a
nd in
equa
lity
signs
•Tr
ansla
ting
wor
ds in
to sy
mbo
ls
•Re
ason
ing
from
wor
ds to
sym
bols
•M
eani
ng o
f dep
ende
nt a
nd in
depe
nden
t var
iabl
es
•M
eani
ng o
f “pr
opor
tiona
l”
•Eq
uiva
lent
ratio
s
•Gr
aphi
ng li
near
rela
tions
hips
•Co
ncep
t of “
unit
rate
” or
slop
e
•M
eani
ng o
f pro
porti
onal
rela
tions
hip
•Th
e co
ordi
nate
pla
ne
•Th
e co
ncep
t of s
lope
•Ev
alua
ting
expr
essio
ns
•M
eani
ng o
f y =
f(x)
•M
eani
ng o
f dom
ain
Chicago Public SchoolsPage 54
Stud
ents
will
be
able
to:
•Cr
eate
equ
ation
s or i
nequ
aliti
es to
des
crib
e sit
uatio
ns in
real
-wor
ld o
r m
athe
mati
cal p
robl
ems.
•W
rite
equa
tions
in tw
o va
riabl
es to
des
crib
e re
al w
orld
situ
ation
s.
•Gr
aph
a re
latio
nshi
p be
twee
n tw
o va
riabl
es in
the
coor
dina
te p
lane
.
•An
alyz
e th
e re
latio
nshi
p be
twee
n th
e tw
o va
riabl
es u
sing
both
tabl
es a
nd
grap
hs.
•U
se g
raph
ing
calc
ulat
or to
ana
lyze
rela
tions
hip.
•De
cide
whe
ther
a re
latio
nshi
p be
twee
n tw
o qu
antiti
es is
pro
porti
onal
by
a)
Usin
g ra
tios
b)
Anal
yzin
g th
e gr
aph
of th
e re
latio
nshi
p
•Id
entif
y th
e co
nsta
nt o
f pro
porti
onal
ity in
a p
ropo
rtion
al re
latio
nshi
p be
twee
n tw
o va
riabl
es b
y ex
amin
ing:
a)
Tabl
es
b)
Grap
hs
c)
Equa
tions
d)
Diag
ram
s
e)
Verb
al d
escr
iptio
n
•In
terp
ret t
he m
eani
ng o
f poi
nts o
n th
e gr
aph
of a
pro
porti
onal
rela
tion-
ship
, esp
ecia
lly th
e m
eani
ng o
f the
poi
nts (
0, 0
) and
(1, r
).
•Gr
aph
a pr
opor
tiona
l rel
ation
ship
and
und
erst
and
that
the
unit
rate
in
the
prop
ortio
n is
the
slope
of t
he g
raph
of t
he re
latio
nshi
p.
•Co
mpa
re tw
o di
ffere
nt p
ropo
rtion
al re
latio
nshi
ps w
heth
er re
pres
ente
d gr
aphi
cally
, in
tabl
es, t
houg
h eq
uatio
ns o
r in
verb
al d
escr
iptio
ns.
•De
fine
func
tion
in te
rms o
f dom
ain
and
rang
e, in
put a
nd o
utpu
t,
nota
tion.
•W
rite
func
tions
usin
g fu
nctio
ns n
otati
on.
•Ev
alua
te fu
nctio
ns.
•U
nder
stan
d ho
w to
dec
ide
upon
the
dom
ain
of a
func
tion
as it
rela
tes t
o a
real
-wor
ld c
onte
xt.
Poss
ible
task
(s)*
The
Whe
el S
hop
Fro
m: I
nsid
e M
athe
mati
cs
Alge
bra
I Pla
nnin
g G
uide
– S
Y12-
13
Big
Idea
: Fou
ndati
ons f
or A
lgeb
ra
*The
Alg
ebra
I to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
CPS Mathematics Content Framework_Version 1.0 Page 55
Plan
ning
Gui
des
Big
Idea
Ass
essm
ent:
Intr
oduc
tion
to F
uncti
ons a
nd T
heir
Rule
sSe
e Al
gebr
a I T
ools
et*:
Cel
l Pho
nes (
from
Illu
stra
tive
Mat
hem
atics
)
Topi
c: R
epre
senti
ng M
athe
mati
cal R
elati
onsh
ips i
n M
ultip
le W
ays
CCSS
-M C
onte
nt S
tand
ards
A-CE
D.2
Crea
te e
quati
ons i
n tw
o or
mor
e va
riabl
es to
repr
esen
t rel
ation
ship
s bet
wee
n qu
antiti
es; g
raph
equ
ation
s on
coor
dina
te a
xes w
ith la
bels
and
scal
es.
N-Q
.1
Use
uni
ts a
s a w
ay to
und
erst
and
prob
lem
s and
to g
uide
the
solu
tion
of m
ulti-
step
pro
blem
s; c
hoos
e an
d in
terp
ret u
nits
con
siste
ntly
in fo
rmul
as; c
hoos
e an
d in
terp
ret t
he sc
ale
and
the
orig
in in
gra
phs a
nd d
ata
disp
lays
.F-
IF.1
Und
erst
and
that
a fu
nctio
n fr
om o
ne se
t (ca
lled
the
dom
ain)
to a
noth
er se
t (ca
lled
the
rang
e) a
ssig
ns to
eac
h el
emen
t of t
he d
omai
n ex
actly
one
ele
men
t of t
he ra
nge.
If f
is a
func
tion
and
x is
an e
lem
ent o
f its
dom
ain,
then
f(x)
den
otes
the
outp
ut o
f f c
orre
spon
ding
to th
e in
put x
. The
gra
ph o
f f is
the
grap
h of
the
equa
tion
y =
f(x).
6-EE
.9
Use
var
iabl
es to
repr
esen
t tw
o qu
antiti
es in
a re
al-w
orld
pro
blem
that
cha
nge
in re
latio
nshi
p to
one
ano
ther
; writ
e an
equ
ation
to e
xpre
ss o
ne q
uanti
ty, t
houg
ht o
f as t
he
depe
nden
t var
iabl
e, in
term
s of t
he o
ther
qua
ntity
, tho
ught
of a
s the
inde
pend
ent v
aria
ble.
Ana
lyze
the
rela
tions
hip
betw
een
the
depe
nden
t and
inde
pend
ent v
aria
bles
usin
g gr
aphs
and
tabl
es, a
nd re
late
thes
e to
the
equa
tion.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…
MP1
- M
ake
sens
e of
pro
blem
s and
per
seve
re in
solv
ing
them
.An
alyz
e gi
ven
info
rmati
on to
dev
elop
pos
sible
stra
tegi
es fo
r sol
ving
the
prob
lem
.
MP2
- Re
ason
abs
trac
tly a
nd q
uanti
tativ
ely.
Man
ipul
ate
the
mat
hem
atica
l rep
rese
ntati
on b
y sh
owin
g th
e pr
oces
s co
nsid
erin
g th
e m
eani
ng o
f the
qua
ntitie
s inv
olve
d.
MP7
- Lo
ok fo
r and
mak
e us
e of
stru
ctur
e.U
se p
atter
ns o
r str
uctu
re to
mak
e se
nse
of m
athe
mati
cs a
nd c
onne
ct
prio
r kno
wle
dge
to si
mila
r situ
ation
s and
ext
end
to n
ovel
situ
ation
s.
MP8
- Lo
ok fo
r and
exp
ress
regu
larit
y in
repe
ated
reas
onin
g.Ge
nera
lize
the
proc
ess t
o cr
eate
a sh
ortc
ut w
hich
may
lead
to d
evel
op-
ing
rule
s or c
reati
ng a
form
ula.
Prio
r Kno
wle
dge
•U
se v
aria
bles
to re
pres
ent n
umbe
rs.
•W
rite
expr
essio
ns w
hen
solv
ing
a re
al w
orld
or m
athe
mati
cal p
robl
em.
•So
lve
equa
tions
.
•Ap
ply
prop
ertie
s of o
pera
tions
.
Stud
ents
will
be
able
to:
•So
lve
real
-wor
ld p
robl
ems a
nd m
odel
real
-wor
ld si
tuati
ons u
sing
pat-
tern
s and
mat
hem
atica
l rel
ation
ship
s.
•M
ake
conn
ectio
ns a
mon
g re
pres
enta
tions
of m
athe
mati
cal r
elati
on-
ship
s, u
sing
wor
ds, p
ictu
res,
tabl
es, g
raph
s, a
nd a
lgeb
raic
rule
s.
•Co
mpa
re a
nd c
ontr
ast t
o de
term
ine
the
adva
ntag
es a
nd li
mita
tions
of
usin
g a
parti
cula
r rep
rese
ntati
on to
ans
wer
a q
uesti
on.
•An
alyz
e an
d cr
eate
equ
ival
ent a
lgeb
raic
exp
ress
ions
and
rule
s.
•Pl
ot p
oint
s and
scal
e ax
es.
•Ke
ep tr
ack
of st
eps o
r pro
cess
es to
refle
ct a
nd o
bser
ve p
atter
ns th
at
will
aid
in th
e fo
rmul
ation
of e
quati
ons t
hat a
rise
from
func
tions
.
•U
nder
stan
d th
e co
ncep
t of a
func
tiona
l rel
ation
ship
as w
ell a
s the
ba
sic a
spec
ts o
f lin
ear r
elati
onsh
ips.
•De
mon
stra
te a
n un
ders
tand
ing
of “
reas
onab
le in
puts
” an
d di
scre
te
and
conti
nuou
s dat
a.
Poss
ible
task
(s)*
Prin
ting
Tick
ets
Fro
m: M
athe
mati
cs A
sses
smen
t Pro
ject
*The
Alg
ebra
I to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
Alge
bra
I Pla
nnin
g G
uide
– S
Y12-
13
Big
Idea
: Int
rodu
ction
to F
uncti
ons a
nd T
heir
Rule
s
Chicago Public SchoolsPage 56
Alge
bra
I Pla
nnin
g G
uide
– S
Y12-
13
Big
Idea
: Int
rodu
ction
to F
uncti
ons a
nd T
heir
Rule
s
Topi
c: F
urth
er E
xam
inati
on o
f Fun
ction
s and
Equ
ation
sCC
SS-M
Con
tent
Sta
ndar
dsF-
IF.2
Use
func
tion
nota
tion,
eva
luat
e fu
nctio
ns fo
r inp
uts i
n th
eir d
omai
ns, a
nd in
terp
ret s
tate
men
ts th
at u
se fu
nctio
n no
tatio
n in
term
s of a
con
text
.F-
IF.5
Rela
te th
e do
mai
n of
a fu
nctio
n to
its g
raph
and
, whe
re a
pplic
able
, to
the
quan
titati
ve re
latio
nshi
p it
desc
ribes
. F
or e
xam
ple,
if th
e fu
nctio
n h
(n) g
ives
the
num
ber o
f per
son-
hour
s it t
akes
to a
ssem
ble
n en
gine
s in
a fa
ctor
y, th
en th
e po
sitive
inte
gers
wou
ld b
e an
app
ropr
iate
dom
ain
for t
he fu
nctio
n.7-
RP.2
a De
cide
whe
ther
two
quan
tities
are
in a
pro
porti
onal
rela
tions
hip,
e.g
., by
testi
ng fo
r equ
ival
ent r
atios
in a
tabl
e or
gra
phin
g on
a c
oord
inat
e pl
ane
and
obse
rvin
g w
heth
er th
e gr
aph
is a
stra
ight
lin
e th
roug
h th
e or
igin
.7-
RP.2
b Id
entif
y th
e co
nsta
nt o
f pro
porti
onal
ity (u
nit r
ate)
in ta
bles
, gra
phs,
equ
ation
s, d
iagr
ams,
and
ver
bal d
escr
iptio
ns o
f pro
porti
onal
rela
tions
hips
.7-
RP.2
d Ex
plai
n w
hat a
poi
nt (x
, y) o
n th
e gr
aph
of a
pro
porti
onal
rela
tions
hip
mea
ns in
term
s of t
he si
tuati
on, w
ith sp
ecia
l atte
ntion
to th
e po
ints
(0, 0
) and
(1, r
) whe
re r
is th
e un
it ra
te.
8-EE
.5
Grap
h pr
opor
tiona
l rel
ation
ship
s, in
terp
retin
g th
e un
it ra
te a
s the
slop
e of
the
grap
h. C
ompa
re tw
o di
ffere
nt p
ropo
rtion
al re
latio
nshi
ps re
pres
ente
d in
diff
eren
t way
s.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…
MP3
- Co
nstr
uct v
iabl
e ar
gum
ents
and
criti
que
the
reas
onin
g of
oth
ers.
Use
obs
erva
tions
and
prio
r kno
wle
dge
(sta
ted
assu
mpti
ons,
defi
nitio
ns,
and
prev
ious
est
ablis
hed
resu
lts) t
o m
ake
conj
ectu
res a
nd c
onst
ruct
ar
gum
ents
.
MP4
- M
odel
with
mat
hem
atics
.U
se a
var
iety
of m
etho
ds to
mod
el, r
epre
sent
, and
solv
e re
al-w
orld
pr
oble
ms.
MP5
- U
se a
ppro
pria
te to
ols s
trat
egic
ally.
Sele
ct a
nd u
se a
ppro
pria
te to
ols t
o be
st m
odel
/sol
ve p
robl
ems.
MP6
- Att
end
to p
reci
sion.
Calc
ulat
e an
swer
s effi
cien
tly a
nd a
ccur
atel
y an
d la
bel t
hem
app
ropr
iate
ly.
Prio
r Kno
wle
dge
Repr
esen
t pro
porti
onal
rela
tions
hips
by
equa
tions
.
Stud
ents
will
be
able
to:
•Re
pres
ent f
uncti
ons u
sing
wor
ds, t
able
s, g
raph
s, a
nd sy
mbo
ls.
•Id
entif
y in
depe
nden
t var
iabl
es in
func
tiona
l rel
ation
ship
s.
•U
se fu
nctio
n no
tatio
n.
•Re
cogn
ize d
iffer
ence
bet
wee
n pr
opor
tiona
l and
non
-pro
porti
onal
sit
uatio
ns re
pres
ente
d by
line
ar fu
nctio
ns.
•De
term
ine
the
cons
tant
in d
irect
var
iatio
n sit
uatio
ns.
Poss
ible
task
(s)*
A Go
lden
Cro
wn
Fro
m: M
athe
mati
cs A
sses
smen
t Pro
ject
*The
Alg
ebra
I to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
CPS Mathematics Content Framework_Version 1.0 Page 57
Plan
ning
Gui
des
Alge
bra
I Pla
nnin
g G
uide
– S
Y12-
13
Big
Idea
: Lin
ear E
quati
ons a
nd In
equa
lities
Big
Idea
Ass
essm
ent:
Line
ar E
quati
ons a
nd In
equa
lities
See
Alge
bra
I Too
lset
*: T
he R
oad
Trip
!
Topi
c: L
inea
r Equ
ation
s and
Ineq
ualiti
esCC
SS-M
Con
tent
Sta
ndar
ds7-
EE.4
a So
lve
wor
d pr
oble
ms l
eadi
ng to
equ
ation
s of t
he fo
rm p
x +
q =
r and
p(x
+ q
) = r,
whe
re p
, q, a
nd r
are
spec
ific
ratio
nal n
umbe
rs. S
olve
equ
ation
s of t
hese
form
s flue
ntly.
Co
mpa
re a
n al
gebr
aic
solu
tion
to a
n ar
ithm
etic
solu
tion,
iden
tifyi
ng th
e se
quen
ce o
f the
ope
ratio
ns u
sed
in e
ach
appr
oach
.8-
EE.7
a Gi
ve e
xam
ples
of l
inea
r equ
ation
s in
one
varia
ble
with
one
solu
tion,
infin
itely
man
y so
lutio
ns, o
r no
solu
tions
. Sho
w w
hich
of t
hese
pos
sibili
ties i
s the
cas
e by
succ
essiv
ely
tran
sfor
min
g th
e gi
ven
equa
tion
into
sim
pler
form
s, u
ntil a
n eq
uiva
lent
equ
ation
of t
he fo
rm x
= a
, a =
a, o
r a =
b re
sults
(whe
re a
and
b a
re d
iffer
ent n
umbe
rs).
8-EE
.7b
Solv
e lin
ear e
quati
ons w
ith ra
tiona
l num
ber c
oeffi
cien
ts, i
nclu
ding
equ
ation
s who
se so
lutio
ns re
quire
exp
andi
ng e
xpre
ssio
ns u
sing
the
dist
ributi
ve p
rope
rty
and
colle
cting
like
te
rms.
A-CE
D.4
Rear
rang
e fo
rmul
as to
hig
hlig
ht a
qua
ntity
of i
nter
est,
usin
g th
e sa
me
reas
onin
g as
in so
lvin
g eq
uatio
ns. F
or e
xam
ple,
rear
rang
e O
hm’s
law
V =
IR to
hig
hlig
ht re
sista
nce
R.A-
REI.1
Ex
plai
n ea
ch st
ep in
solv
ing
a sim
ple
equa
tion
as fo
llow
ing
from
the
equa
lity
of n
umbe
rs a
sser
ted
at th
e pr
evio
us st
ep, s
tarti
ng fr
om th
e as
sum
ption
that
the
orig
inal
equ
ation
ha
s a so
lutio
n. C
onst
ruct
a v
iabl
e ar
gum
ent t
o ju
stify
a so
lutio
n m
etho
d.
A-RE
I.3
Solv
e lin
ear e
quati
ons a
nd in
equa
lities
in o
ne v
aria
ble,
incl
udin
g eq
uatio
ns w
ith c
oeffi
cien
ts re
pres
ente
d by
lette
rs.
A-RE
I.11
Expl
ain
why
the
x-co
ordi
nate
s of t
he p
oint
s whe
re th
e gr
aphs
of t
he e
quati
ons y
= f
(x )
and
y =
g (x
) int
erse
ct a
re th
e so
lutio
ns o
f the
equ
ation
f(x)
= g
(x);
find
the
solu
tions
ap
prox
imat
ely,
e.g.
, usin
g te
chno
logy
to g
raph
the
func
tions
, mak
e ta
bles
of v
alue
s, o
r find
succ
essiv
e ap
prox
imati
ons.
Incl
ude
case
s whe
re f(
x) a
nd/o
r g(x
) are
line
ar,
poly
nom
ial,
ratio
nal,
abso
lute
val
ue, e
xpon
entia
l, an
d lo
garit
hmic
func
tions
.F-
LE.2
Co
nstr
uct l
inea
r and
exp
onen
tial f
uncti
ons,
incl
udin
g ar
ithm
etic
and
geom
etric
sequ
ence
s, g
iven
a g
raph
, a d
escr
iptio
n of
a re
latio
nshi
p, o
r tw
o in
put-o
utpu
t pai
rs (i
nclu
de
read
ing
thes
e fr
om a
tabl
e).
7-EE
.4b
Solv
e w
ord
prob
lem
s lea
ding
to in
equa
lities
of t
he fo
rm p
x +
q >
r or p
x +
q <
r, w
here
p, q
, and
r ar
e sp
ecifi
c ra
tiona
l num
bers
. Gra
ph th
e so
lutio
n se
t of t
he in
equa
lity
and
inte
rpre
t it i
n th
e co
ntex
t of t
he p
robl
em.
A-CE
D.3
Repr
esen
t con
stra
ints
by
equa
tions
or i
nequ
aliti
es a
nd/o
r by
syst
ems o
f equ
ation
s and
/or i
nequ
aliti
es, a
nd in
terp
ret s
oluti
ons a
s via
ble
or n
on-v
iabl
e op
tions
in a
mod
elin
g co
ntex
t.A-
REI.3
So
lve
linea
r equ
ation
s and
ineq
ualiti
es in
one
var
iabl
e, in
clud
ing
equa
tions
and
coe
ffici
ents
repr
esen
ted
by le
tters
.A-
REI.1
2 Gr
aph
the
solu
tions
to a
line
ar in
equa
lity
in tw
o va
riabl
es a
s a h
alf p
lane
(exc
ludi
ng th
e bo
unda
ry in
the
case
of a
stric
t ine
qual
ity),
and
grap
h th
e so
lutio
n se
t to
a sy
stem
of
linea
r ine
qual
ities
in tw
o va
riabl
es a
s the
inte
rsec
tion
of th
e co
rres
pond
ing
half-
plan
es.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…
MP2
- Re
ason
abs
trac
tly a
nd q
uanti
tativ
ely.
Tran
slate
giv
en in
form
ation
to c
reat
e a
mat
hem
atica
l rep
rese
ntati
on fo
r a
conc
ept.
MP4
- M
odel
with
mat
hem
atics
.Ch
oose
a m
odel
that
is b
oth
appr
opria
te a
nd e
ffici
ent t
o ar
rive
at o
ne o
r m
ore
desir
ed so
lutio
ns.
MP7
- Lo
ok fo
r and
mak
e us
e of
stru
ctur
e.U
se p
atter
ns o
r str
uctu
re to
mak
e se
nse
of m
athe
mati
cs a
nd c
onne
ct
prio
r kno
wle
dge
to si
mila
r situ
ation
s and
ext
end
to n
ovel
situ
ation
s.
Prio
r Kno
wle
dge
•U
se v
aria
bles
to re
pres
ent n
umbe
rs
•W
rite
expr
essio
ns w
hen
solv
ing
a re
al w
orld
or m
athe
mati
cal p
robl
em
•So
lve
equa
tions
•Ap
ply
prop
ertie
s of o
pera
tions
Chicago Public SchoolsPage 58
Alge
bra
I Pla
nnin
g G
uide
– S
Y12-
13
Big
Idea
: Lin
ear E
quati
ons a
nd In
equa
lities
Stud
ents
will
be
able
to:
•Co
mm
unic
ate
mat
hem
atica
l ide
as a
nd c
oncl
usio
ns th
roug
h la
ngua
ge
and
repr
esen
tatio
n.
•U
se re
ason
ing
to m
ake
conj
ectu
res a
nd v
erify
con
clus
ions
.
•An
alyz
e sit
uatio
ns in
volv
ing
linea
r fun
ction
s and
form
ulat
e lin
ear
equa
tions
to so
lve
prob
lem
s.
•U
se d
iffer
ent m
etho
ds fo
r sol
ving
line
ar e
quati
ons:
usin
g co
ncre
te
mod
els,
gra
phs,
and
the
prop
ertie
s of e
qual
ity.
•Ch
oose
an
appr
opria
te m
etho
d, a
nd so
lve
the
equa
tions
.
•Ap
ply
tech
niqu
es fo
r sol
ving
equ
ation
s in
one
varia
ble
to so
lve
liter
al
equa
tions
.
•Co
mpa
re a
nd c
ontr
ast t
o de
term
ine
the
adva
ntag
es a
nd li
mita
tions
of
usin
g a
parti
cula
r rep
rese
ntati
on to
ans
wer
a q
uesti
on.
•An
alyz
e an
d cr
eate
equ
ival
ent a
lgeb
raic
exp
ress
ions
and
rule
s.
•W
rite
ineq
ualiti
es in
one
and
two
varia
bles
to re
pres
ent p
robl
em
situa
tions
.
•So
lve
linea
r ine
qual
ities
in o
ne v
aria
ble
usin
g ta
bles
, gra
phs,
and
al
gebr
aic
oper
ation
s.
•So
lve
a lin
ear s
yste
m o
f equ
ation
s with
two
varia
bles
.
•Gr
aph
solu
tions
to li
near
ineq
ualiti
es in
one
var
iabl
e on
a n
umbe
r lin
e.
•Gr
aph
solu
tions
to li
near
ineq
ualiti
es in
two
varia
bles
on
a co
ordi
nate
pl
ane.
•Gr
aph
solu
tions
to sy
stem
s of l
inea
r ine
qual
ities
in tw
o va
riabl
es o
n a
coor
dina
te p
lane
.
Poss
ible
task
(s)*
Tab
le T
iling
F
rom
: Mat
hem
atics
Ass
essm
ent P
roje
ct
*The
Alg
ebra
I to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
CPS Mathematics Content Framework_Version 1.0 Page 59
Plan
ning
Gui
des
Alge
bra
I Pla
nnin
g G
uide
– S
Y12-
13
Big
Idea
: Mod
elin
g w
ith L
inea
r Fun
ction
s
*The
Alg
ebra
I to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
Big
Idea
Ass
essm
ent:
Mod
elin
g w
ith L
inea
r Fun
ction
sSe
e Al
gebr
a I T
ools
et*:
Buy
ing
Chip
s and
Can
dy
Topi
c: L
inea
rity
as C
onst
ant R
ate
of C
hang
eCC
SS-M
Con
tent
Sta
ndar
ds
8-F.5
De
scrib
e qu
alita
tivel
y th
e fu
nctio
nal r
elati
onsh
ip b
etw
een
two
quan
tities
by
anal
yzin
g a
grap
h (e
.g.,
whe
re th
e fu
nctio
n is
incr
easin
g or
dec
reas
ing,
line
ar o
r non
linea
r). S
ketc
h a
grap
h th
at e
xhib
its th
e qu
alita
tive
feat
ures
of a
func
tion
that
has
bee
n de
scrib
ed v
erba
lly (e
.g.,
grap
hing
a d
istan
ce v.
tim
e st
ory)
.F-
IF.6
Calc
ulat
e an
d in
terp
ret t
he a
vera
ge ra
te o
f cha
nge
of a
func
tion
(pre
sent
ed sy
mbo
lical
ly o
r as a
tabl
e) o
ver a
spec
ified
inte
rval
. Esti
mat
e th
e ra
te o
f cha
nge
from
a g
raph
.F-
LE.1
.b R
ecog
nize
situ
ation
s in
whi
ch o
ne q
uanti
ty c
hang
es a
t a c
onst
ant r
ate
per u
nit i
nter
val r
elati
ve to
ano
ther
.7-
RP.2
a De
cide
whe
ther
two
quan
tities
are
in a
pro
porti
onal
rela
tions
hip,
e.g
., by
testi
ng fo
r equ
ival
ent r
atios
in a
tabl
e or
gra
phin
g on
a c
oord
inat
e pl
ane
and
obse
rvin
g w
heth
er th
e gr
aph
is a
stra
ight
line
thro
ugh
the
orig
in.
7-RP
.2b
Iden
tify
the
cons
tant
of p
ropo
rtion
ality
(uni
t rat
e) in
tabl
es, g
raph
s, e
quati
ons,
dia
gram
s, a
nd v
erba
l des
crip
tions
of p
ropo
rtion
al re
latio
nshi
ps.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…
MP2
- Re
ason
abs
trac
tly a
nd q
uanti
tativ
ely.
Reco
gnize
the
rela
tions
hips
bet
wee
n nu
mbe
rs/q
uanti
ties w
ithin
the
pro-
cess
to e
valu
ate
a pr
oble
m.
MP3
- Co
nstr
uct v
iabl
e ar
gum
ents
and
criti
que
the
reas
onin
g of
oth
ers.
Justi
fy (o
rally
and
in w
ritten
form
) the
app
roac
h us
ed, i
nclu
ding
how
it fi
ts
in th
e co
ntex
t fro
m w
hich
the
data
aro
se.
MP4
- M
odel
with
mat
hem
atics
.Si
mpl
ify a
com
plic
ated
pro
blem
by
mak
ing
assu
mpti
ons a
nd a
ppro
xim
a-tio
ns.
MP8
- Lo
ok fo
r and
exp
ress
regu
larit
y in
repe
ated
reas
onin
g.Re
cogn
ize si
mila
rities
and
patt
erns
in re
peat
ed tr
ials
with
a p
roce
ss.
Prio
r Kno
wle
dge
•In
terp
ret u
nit r
ate
as th
e slo
pe.
•U
nder
stan
d ra
tios.
•Co
mpu
te u
nit r
ates
ass
ocia
ted
with
ratio
s of f
racti
ons.
Stud
ents
will
be
able
to:
•Sh
ow u
nder
stan
ding
of t
he c
once
pts o
f spe
ed a
nd ra
te.
•Cr
eate
moti
on g
raph
s (di
stan
ce v
s. ti
me)
.
•De
scrib
e ho
w c
hang
es in
moti
on a
ffect
the
grap
h.
•U
se c
omm
on o
r firs
t diff
eren
ces t
o de
term
ine
if a
func
tion
is lin
ear.
•Id
entif
y re
latio
nshi
ps a
s lin
ear o
r non
linea
r usin
g a
tabl
e, g
raph
, or
equa
tion.
•Fi
nd ra
tes f
or d
ata
in ta
bles
, gra
phs a
nd p
robl
em si
tuati
ons.
Poss
ible
task
(s)*
Chicago Public SchoolsPage 60
Alge
bra
I Pla
nnin
g G
uide
– S
Y12-
13
Big
Idea
: Mod
elin
g w
ith L
inea
r Fun
ction
s
Topi
c: C
onst
ructi
ng L
inea
r Fun
ction
sCC
SS-M
Con
tent
Sta
ndar
ds
F-LE
.2
Cons
truc
t lin
ear …
func
tions
, inc
ludi
ng a
rithm
etic
… se
quen
ces,
giv
en a
gra
ph, a
des
crip
tion
of a
rela
tions
hip,
or t
wo
inpu
t-out
put p
airs
(inc
lude
read
ing
thes
e fr
om a
tabl
e).
A-CE
D.2
Crea
te e
quati
ons i
n tw
o or
mor
e va
riabl
es to
repr
esen
t rel
ation
ship
s bet
wee
n qu
antiti
es; g
raph
equ
ation
s on
a co
ordi
nate
axe
s with
labe
ls an
d sc
ales
.8-
F.4
Cons
truc
t a fu
nctio
n to
mod
el a
line
ar re
latio
nshi
p be
twee
n tw
o qu
antiti
es. D
eter
min
e th
e ra
te o
f cha
nge
and
initi
al v
alue
of t
he fu
nctio
n fr
om a
des
crip
tion
of a
rela
tions
hip
or
from
two
(x, y
) val
ues,
incl
udin
g re
adin
g th
ese
from
a ta
ble
or fr
om a
gra
ph. I
nter
pret
the
rate
of c
hang
e an
d in
itial
val
ue o
f a li
near
func
tion
in te
rms o
f the
situ
ation
it m
odel
s,
and
in te
rms o
f its
gra
ph o
r a ta
ble
of v
alue
s.F-
IF.7a
Gr
aph
linea
r … fu
nctio
ns a
nd sh
ow in
terc
epts
...
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…
MP2
- Re
ason
abs
trac
tly a
nd q
uanti
tativ
ely.
Tran
slate
giv
en in
form
ation
to c
reat
e a
mat
hem
atica
l rep
rese
ntati
on fo
r a
conc
ept.
MP7
- Lo
ok fo
r and
mak
e us
e of
stru
ctur
e.U
se p
atter
ns o
r str
uctu
re to
mak
e se
nse
of m
athe
mati
cs a
nd c
onne
ct
prio
r kno
wle
dge
to si
mila
r situ
ation
s and
ext
end
to n
ovel
situ
ation
s.
MP8
- Lo
ok fo
r and
exp
ress
regu
larit
y in
repe
ated
reas
onin
g.Ev
alua
te th
e re
ason
able
ness
of r
esul
ts th
roug
hout
the
mat
hem
atica
l pr
oces
s whi
le a
ttend
ing
to d
etai
l.
Prio
r Kno
wle
dge
•Ap
ply
prop
ertie
s of o
pera
tions
.
•Cr
eate
equ
ation
s and
ineq
ualiti
es in
one
var
iabl
e.
Stud
ents
will
be
able
to:
•W
rite
an e
quati
on b
ased
on
an a
rithm
etic
sequ
ence
.
•De
term
ine
a re
curs
ive
form
ula.
•Ex
plai
n ho
w to
det
erm
ine
the
next
step
in a
patt
ern.
•Kn
ow a
nd u
se th
e re
latio
nshi
p be
twee
n th
e y-
inte
rcep
t of t
he g
raph
of
a li
near
mod
el a
nd th
e sit
uatio
n be
ing
mod
eled
.
•U
se c
onst
ant r
ate
of c
hang
e an
d slo
pe to
ana
lyze
and
gra
ph li
near
fu
nctio
ns.
Poss
ible
task
(s)*
*The
Alg
ebra
I to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
CPS Mathematics Content Framework_Version 1.0 Page 61
Plan
ning
Gui
des
Alge
bra
I Pla
nnin
g G
uide
– S
Y12-
13
Big
Idea
: Mod
elin
g w
ith L
inea
r Fun
ction
s
*The
Alg
ebra
I to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
Topi
c: A
naly
zing
Lin
ear F
uncti
ons
CCSS
-M C
onte
nt S
tand
ards
F-IF.
4 Fo
r a fu
nctio
n th
at m
odel
s a re
latio
nshi
p be
twee
n tw
o qu
antiti
es, i
nter
pret
key
feat
ures
of g
raph
s and
tabl
es in
term
s of t
he q
uanti
ties,
and
sket
ch g
raph
s sho
win
g ke
y fe
atur
es
give
n a
verb
al d
escr
iptio
n of
the
rela
tions
hip.
Key
feat
ures
incl
ude:
inte
rcep
ts; i
nter
vals
whe
re th
e fu
nctio
n is
incr
easin
g, d
ecre
asin
g, p
ositi
ve, o
r neg
ative
; rel
ative
max
imum
s an
d m
inim
ums;
sym
met
ries;
end
beh
avio
r; an
d pe
riodi
city
.F-
BF.3
Id
entif
y th
e eff
ect o
n th
e gr
aph
of re
plac
ing
f(x) b
y f(x
) + k
, k f(
x), f
(kx)
, and
f(x
+ k)
for s
peci
fic v
alue
s of k
(bot
h po
sitive
and
neg
ative
); fin
d th
e va
lue
of k
giv
en th
e gr
aphs
. Ex
perim
ent w
ith c
ases
and
illu
stra
te a
n ex
plan
ation
of t
he e
ffect
s on
the
grap
h us
ing
tech
nolo
gy. I
nclu
de re
cogn
izing
eve
n an
d od
d fu
nctio
ns fr
om th
eir g
raph
s and
alg
ebra
ic
expr
essio
ns fo
r the
m.
F-LE
.5
Inte
rpre
t the
par
amet
ers i
n a
linea
r or e
xpon
entia
l fun
ction
in te
rms o
f a c
onte
xt.
8.-F
.3
Inte
rpre
t the
equ
ation
y =
mx
+ b
as d
efini
ng a
line
ar fu
nctio
n, w
hose
gra
ph is
a st
raig
ht li
ne; g
ive
exam
ples
of f
uncti
ons t
hat a
re n
ot li
near
.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…
MP2
- Re
ason
abs
trac
tly a
nd q
uanti
tativ
ely.
Tran
slate
giv
en in
form
ation
to c
reat
e a
mat
hem
atica
l rep
rese
ntati
on fo
r a
conc
ept.
MP3
- Co
nstr
uct v
iabl
e ar
gum
ents
and
criti
que
the
reas
onin
g of
oth
ers.
Justi
fy (o
rally
and
in w
ritten
form
) the
app
roac
h us
ed, i
nclu
ding
how
it fi
ts
in th
e co
ntex
t fro
m w
hich
the
data
aro
se.
MP4
- M
odel
with
mat
hem
atics
.Si
mpl
ify a
com
plic
ated
pro
blem
by
mak
ing
assu
mpti
ons a
nd
appr
oxim
ation
s.
MP8
- Lo
ok fo
r and
exp
ress
regu
larit
y in
repe
ated
reas
onin
g.Ev
alua
te th
e re
ason
able
ness
of r
esul
ts th
roug
hout
the
mat
hem
atica
l pr
oces
s whi
le a
ttend
ing
to d
etai
l.
Prio
r Kno
wle
dge
•In
terp
ret u
nit r
ate
as th
e slo
pe.
•U
nder
stan
d ra
tios.
•Co
mpu
te u
nit r
ates
ass
ocia
ted
with
ratio
s of f
racti
ons.
Stud
ents
will
be
able
to:
•W
rite
the
equa
tion
of a
line
in d
iffer
ent f
orm
s (slo
pe-in
terc
ept,
st
anda
rd, a
nd p
oint
-slo
pe fo
rms)
.
•Id
entif
y slo
pe a
nd y
-inte
rcep
t fro
m g
raph
s, ta
bles
and
pro
blem
sit
uatio
ns.
•Id
entif
y eq
uatio
ns a
s lin
ear o
r non
-line
ar
•U
nder
stan
d th
e eff
ects
of c
hang
ing
m o
r b o
n th
e gr
aph
of y
= m
x +
b.
•Tr
ansf
orm
the
pare
nt fu
nctio
n y
= x
to c
reat
e ot
her l
inea
r fun
ction
s.
•In
terp
ret t
he m
eani
ng o
f m a
nd b
from
tabl
es a
nd g
raph
s.
Poss
ible
task
(s)*
Fun
ction
s
Fro
m M
athe
mati
cs A
sses
smen
t Pro
ject
Chicago Public SchoolsPage 62
Big
Idea
Ass
essm
ent:
Solv
ing
Syst
ems o
f Equ
ation
s and
Ineq
ualiti
esSe
e Al
gebr
a I T
ools
et*:
Pas
seng
er Je
t
Topi
c: S
olvi
ng S
yste
ms o
f Equ
ation
s thr
ough
Tab
les,
Cha
rts,
and
Gra
phs
CCSS
-M C
onte
nt S
tand
ards
8-EE
.8a
Und
erst
and
that
solu
tions
to a
syst
em o
f tw
o lin
ear e
quati
ons i
n tw
o va
riabl
es c
orre
spon
d to
poi
nts o
f int
erse
ction
of t
heir
grap
hs, b
ecau
se p
oint
s of i
nter
secti
on sa
tisfy
bot
h eq
uatio
ns si
mul
tane
ously
.8-
EE.8
b So
lve
syst
ems o
f tw
o lin
ear e
quati
ons i
n tw
o va
riabl
es a
lgeb
raic
ally,
and
esti
mat
e so
lutio
ns b
y gr
aphi
ng th
e eq
uatio
ns. S
olve
sim
ple
case
s by
insp
ectio
n.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…
MP1
- M
ake
sens
e of
pro
blem
s and
per
seve
re in
solv
ing
them
.An
alyz
e gi
ven
info
rmati
on to
dev
elop
pos
sible
stra
tegi
es fo
r sol
ving
the
prob
lem
MP6
- Att
end
to p
reci
sion.
Form
ulat
e pr
ecise
exp
lana
tions
(ora
lly a
nd in
writt
en fo
rm) u
sing
both
mat
hem
atica
l rep
rese
ntati
ons a
nd w
ords
MP7
- Lo
ok fo
r and
mak
e us
e of
stru
ctur
e.U
se p
atter
ns o
r str
uctu
re to
mak
e se
nse
of m
athe
mati
cs a
nd c
onne
ct
prio
r kno
wle
dge
to si
mila
r situ
ation
s and
ext
end
to n
ovel
situ
ation
s
Prio
r Kno
wle
dge
Solv
e lin
ear e
quati
ons a
nd in
equa
lities
in o
ne v
aria
ble.
Stud
ents
will
be
able
to:
•Id
entif
y th
e tw
o va
riabl
es n
eede
d to
solv
e a
wor
d pr
oble
m a
nd w
rite
a sy
stem
of l
inea
r equ
ation
s in
thos
e tw
o va
riabl
es to
mod
el th
e sit
u-ati
on.
•So
lve
a sy
stem
of t
wo
linea
r equ
ation
s by
mak
ing
an a
ppro
pria
te ta
ble
of v
alue
s by
hand
and
usin
g te
chno
logy
.
•So
lve
a sy
stem
of t
wo
linea
r equ
ation
s by
grap
hing
the
equa
tions
and
fin
ding
thei
r poi
nt o
f int
erse
ction
, by
hand
and
usin
g te
chno
logy
.
•Ch
eck
solu
tions
to a
syst
em o
f tw
o lin
ear e
quati
ons.
Poss
ible
task
(s)*
Alge
bra
I Pla
nnin
g G
uide
– S
Y12-
13
Big
Idea
: Sol
ving
Sys
tem
s of E
quati
ons a
nd In
equa
lities
*The
Alg
ebra
I to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
CPS Mathematics Content Framework_Version 1.0 Page 63
Plan
ning
Gui
des
Alge
bra
I Pla
nnin
g G
uide
– S
Y12-
13
Big
Idea
: Sol
ving
Sys
tem
s of E
quati
ons a
nd In
equa
lities
*The
Alg
ebra
I to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
Topi
c: S
olvi
ng S
yste
ms u
sing
Sub
stitu
tion
and
Elim
inati
onCC
SS-M
Con
tent
Sta
ndar
dsA-
REI.6
So
lve
syst
ems o
f lin
ear e
quati
ons e
xact
ly a
nd a
ppro
xim
atel
y (e
.g.,
with
gra
phs)
, foc
usin
g on
pai
rs o
f lin
ear e
quati
ons i
n tw
o va
riabl
es.
A-RE
I.5
Prov
e th
at, g
iven
a sy
stem
of t
wo
equa
tions
in tw
o va
riabl
es, r
epla
cing
one
equ
ation
by
the
sum
of t
hat e
quati
on a
nd a
mul
tiple
of t
he o
ther
pro
duce
s a sy
stem
with
the
sam
e so
lutio
ns.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…
MP1
- M
ake
sens
e of
pro
blem
s and
per
seve
re in
solv
ing
them
.An
alyz
e gi
ven
info
rmati
on to
dev
elop
pos
sible
stra
tegi
es fo
r sol
ving
the
prob
lem
.
MP6
- Att
end
to p
reci
sion.
Form
ulat
e pr
ecise
exp
lana
tions
(ora
lly a
nd in
writt
en fo
rm) u
sing
both
m
athe
mati
cal r
epre
sent
ation
s and
wor
ds.
MP7
- Lo
ok fo
r and
mak
e us
e of
stru
ctur
e.U
se p
atter
ns o
r str
uctu
re to
mak
e se
nse
of m
athe
mati
cs a
nd c
onne
ct
prio
r kno
wle
dge
to si
mila
r situ
ation
s and
ext
end
to n
ovel
situ
ation
s.
Prio
r Kno
wle
dge
•So
lve
linea
r equ
ation
s and
ineq
ualiti
es in
one
var
iabl
e.
•Cr
eate
a sy
stem
of e
quati
ons.
•So
lve
syst
em o
f equ
ation
s by
grap
hing
.
Stud
ents
will
be
able
to:
•Be
abl
e to
solv
e sy
stem
s of l
inea
r equ
ation
s usin
g th
e su
bstit
ution
met
hod.
•Be
abl
e to
solv
e sy
stem
s of l
inea
r equ
ation
s usin
g th
e lin
ear c
ombi
natio
n m
etho
d (e
limin
ation
).
•Be
abl
e to
reco
gnize
dep
ende
nt a
nd in
cons
isten
t sys
tem
s and
writ
e th
e so
lutio
n se
t of e
ach.
Poss
ible
task
(s)*
Chicago Public SchoolsPage 64
Alge
bra
I Pla
nnin
g G
uide
– S
Y12-
13
Big
Idea
: Sol
ving
Sys
tem
s of E
quati
ons a
nd In
equa
lities
Topi
c: P
robl
em S
olvi
ng w
ith S
yste
ms o
f Equ
ation
sCC
SS-M
Con
tent
Sta
ndar
ds8-
EE.8
c So
lve
real
-wor
ld a
nd m
athe
mati
cal p
robl
ems l
eadi
ng to
two
linea
r equ
ation
s in
two
varia
bles
.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…
MP1
- M
ake
sens
e of
pro
blem
s and
per
seve
re in
solv
ing
them
.An
alyz
e gi
ven
info
rmati
on to
dev
elop
pos
sible
stra
tegi
es fo
r sol
ving
the
prob
lem
.
MP4
- M
odel
with
mat
hem
atics
.Si
mpl
ify a
com
plic
ated
pro
blem
by
mak
ing
assu
mpti
ons a
nd
appr
oxim
ation
s.
MP5
- U
se a
ppro
pria
te to
ols s
trat
egic
ally.
Use
a v
arie
ty o
f tec
hnol
ogie
s, in
clud
ing
digi
tal c
onte
nt, t
o ex
plor
e,
confi
rm, a
nd d
eepe
n co
ncep
tual
und
erst
andi
ng.
Prio
r Kno
wle
dge
•So
lve
linea
r equ
ation
s and
ineq
ualiti
es in
one
var
iabl
e.
•Cr
eate
a sy
stem
of e
quati
ons.
•So
lve
syst
em o
f equ
ation
s.
Stud
ents
will
be
able
to:
•W
rite
a sy
stem
of l
inea
r equ
ation
s in
two
varia
bles
to m
odel
a p
robl
em si
tuati
on.
•De
term
ine
whi
ch so
lutio
n m
etho
d m
ight
be
mos
t effi
cien
t for
a g
iven
syst
em o
f lin
ear e
quati
ons.
•So
lve
syst
em o
f lin
ear i
nequ
aliti
es g
raph
ical
ly.
Poss
ible
task
(s)*
*The
Alg
ebra
I to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
CPS Mathematics Content Framework_Version 1.0 Page 65
Plan
ning
Gui
des
Alge
bra
I Pla
nnin
g G
uide
– S
Y12-
13
Big
Idea
: Non
-line
ar F
uncti
ons a
nd E
quati
ons
*The
Alg
ebra
I to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
Big
Idea
Ass
essm
ent:
Non
-line
ar F
uncti
ons a
nd E
quati
ons
See
Alge
bra
I Too
lset
*: S
umm
er O
lym
pics
Topi
c: E
xpon
ents
and
Exp
onen
tial F
uncti
ons
CCSS
-M C
onte
nt S
tand
ards
N-R
N.1
Ex
plai
n ho
w th
e de
finiti
on o
f the
mea
ning
of r
ation
al e
xpon
ents
follo
ws f
rom
ext
endi
ng th
e pr
oper
ties o
f int
eger
exp
onen
ts to
thos
e va
lues
, allo
win
g fo
r a n
otati
on fo
r rad
ical
s in
term
s of r
ation
al e
xpon
ents
. For
exa
mpl
e, w
e de
fine
5 1/
3 to b
e th
e cu
be ro
ot o
f 5 b
ecau
se w
e w
ant (
5 1/
3 )3 = 5
(1/3 )^3
to h
old,
so (5
1/3 )3 m
ust e
qual
5.
N-R
N.2
Re
writ
e ex
pres
sions
invo
lvin
g ra
dica
ls an
d ra
tiona
l exp
onen
ts u
sing
the
prop
ertie
s of e
xpon
ents
.F-
IF.7e
Gr
aph
expo
nenti
al a
nd lo
garit
hmic
func
tions
, sho
win
g in
terc
epts
and
end
beh
avio
r.F-
IF.9
Com
pare
pro
perti
es o
f tw
o fu
nctio
ns e
ach
repr
esen
ted
in a
diff
eren
t way
(alg
ebra
ical
ly, g
raph
ical
ly, n
umer
ical
ly in
tabl
es, o
r by
verb
al d
escr
iptio
ns).
For e
xam
ple,
giv
en a
gra
ph
of o
ne q
uadr
atic
func
tion
and
an a
lgeb
raic
exp
ress
ion
for a
noth
er, s
ay w
hich
has
the
larg
er m
axim
um.
F.LE.
1a
Prov
e th
at li
near
func
tions
gro
w b
y eq
ual d
iffer
ence
s ove
r equ
al in
terv
als,
and
that
exp
onen
tial f
uncti
ons g
row
by
equa
l fac
tors
ove
r equ
al in
terv
als.
F.LE.
1c
Reco
gnize
situ
ation
s in
whi
ch a
qua
ntity
gro
ws o
r dec
ays b
y a
cons
tant
per
cent
rate
per
uni
t int
erva
l rel
ative
to a
noth
er.
F-LE
.2
Cons
truc
t lin
ear a
nd e
xpon
entia
l fun
ction
s, in
clud
ing
arith
meti
c an
d ge
omet
ric se
quen
ces,
giv
en a
gra
ph, a
des
crip
tion
of a
rela
tions
hip,
or t
wo
inpu
t-out
put p
airs
(inc
lude
re
adin
g th
ese
from
a ta
ble)
.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…
MP4
- M
odel
with
mat
hem
atics
.Si
mpl
ify a
com
plic
ated
pro
blem
by
mak
ing
assu
mpti
ons a
nd
appr
oxim
ation
s.
MP7
- Lo
ok fo
r and
mak
e us
e of
stru
ctur
e.U
se p
atter
ns o
r str
uctu
re to
mak
e se
nse
of m
athe
mati
cs a
nd c
onne
ct
prio
r kno
wle
dge
to si
mila
r situ
ation
s and
ext
end
to n
ovel
situ
ation
s.
MP8
- Lo
ok fo
r and
exp
ress
regu
larit
y in
repe
ated
reas
onin
g.Ev
alua
te th
e re
ason
able
ness
of r
esul
ts th
roug
hout
the
mat
hem
atica
l pr
oces
s whi
le a
ttend
ing
to d
etai
l.
Prio
r Kno
wle
dge
•W
rite
and
eval
uate
num
eric
al e
xpre
ssio
ns.
•Kn
ow a
nd a
pply
the
prop
ertie
s of e
xpon
ents
.•
Anal
yze
a gr
aph
whe
re th
e fu
nctio
n is
incr
easin
g or
dec
reas
ing.
Stud
ents
will
be
able
to:
•De
velo
p an
d un
ders
tand
the
law
s of e
xpon
ents
.•
Sim
plify
num
eric
al a
nd v
aria
ble
expr
essio
ns in
volv
ing
expo
nent
.•
Deve
lop
and
unde
rsta
nd th
e la
ws o
f exp
onen
ts.
•Si
mpl
ify n
umer
ical
and
var
iabl
e ex
pres
sions
invo
lvin
g ex
pone
nt.
•De
term
ine
if a
rela
tions
hip
repr
esen
ted
by a
tabl
e, ru
le, g
raph
, or
stat
emen
t can
be
repr
esen
ted
by a
n ex
pone
ntial
func
tion.
•Us
e fu
nctio
ns o
f the
form
y =
ab^
x to
repr
esen
t exp
onen
tial r
elati
onsh
ips.
•Ex
plai
n ho
w c
hang
es in
the
para
met
ers a
and
b fo
r y =
ab^
x aff
ect t
he
grap
h of
an
expo
nenti
al fu
nctio
n.
•An
alyz
e da
ta a
nd re
pres
ent s
ituati
ons i
nvol
ving
exp
onen
tial r
elati
on-
ship
s.•
Reas
on a
bout
the
rela
tive
mag
nitu
de o
f qua
ntitie
s•
Use
scie
ntific
not
ation
to re
pres
ent q
uanti
ties a
nd so
lve
prob
lem
s.•
Dete
rmin
e if
a ta
ble,
gra
ph, r
ule
or st
atem
ent c
an b
e re
pres
ente
d by
a
linea
r or e
xpon
entia
l fun
ction
.•
Reco
gnize
that
exp
onen
tial f
uncti
on v
alue
s are
gen
erat
ed b
y
com
mon
mul
tiplie
rs, n
ot a
dditi
ve c
onst
ants
.•
Und
erst
and
the
role
s tha
t the
par
amet
ers a
and
b in
the
gene
ral
form
the
expo
nenti
al fu
nctio
n y
= a
• b^
x pl
ay in
det
erm
inin
g th
e st
artin
g po
ints
and
the
grow
th o
r dec
ay o
f the
func
tion.
Poss
ible
task
(s)*
Chicago Public SchoolsPage 66
Alge
bra
I Pla
nnin
g G
uide
– S
Y12-
13
Big
Idea
: Non
-line
ar F
uncti
ons a
nd E
quati
ons
Topi
c: P
olyn
omia
l Add
ition
and
Mul
tiplic
ation
and
the
Appl
icati
on o
f Ope
ratio
ns o
n Po
lyno
mia
l Exp
ress
ions
CCSS
-M C
onte
nt S
tand
ards
A-SS
E.1a
In
terp
ret p
arts
of a
n ex
pres
sion,
such
as t
erm
s, fa
ctor
s, a
nd c
oeffi
cien
ts.
A-AP
R.1
Und
erst
and
that
pol
ynom
ials
form
a sy
stem
ana
logo
us to
the
inte
gers
, nam
ely,
they
are
clo
sed
unde
r the
ope
ratio
ns o
f add
ition
, sub
trac
tion,
and
mul
tiplic
ation
; add
, su
btra
ct, a
nd m
ultip
ly p
olyn
omia
ls.A-
SSE.
3a F
acto
r a q
uadr
atic
expr
essio
n to
reve
al th
e ze
ros o
f the
func
tion
it de
fines
.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…
MP2
- Re
ason
abs
trac
tly a
nd q
uanti
tativ
ely.
Reco
gnize
the
rela
tions
hips
bet
wee
n nu
mbe
rs/q
uanti
ties w
ithin
the
pr
oces
s to
eval
uate
a p
robl
em.
MP7
- Lo
ok fo
r and
mak
e us
e of
stru
ctur
e.U
se p
atter
ns o
r str
uctu
re to
mak
e se
nse
of m
athe
mati
cs a
nd c
onne
ct
prio
r kno
wle
dge
to si
mila
r situ
ation
s and
ext
end
to n
ovel
situ
ation
s.
MP8
- Lo
ok fo
r and
exp
ress
regu
larit
y in
repe
ated
reas
onin
g.Ev
alua
te th
e re
ason
able
ness
of r
esul
ts th
roug
hout
the
mat
hem
atica
l pr
oces
s whi
le a
ttend
ing
to d
etai
l
Prio
r Kno
wle
dge
•W
rite
an e
quati
on.
•U
nder
stan
d an
d kn
ow th
e di
ffere
nce
betw
een
depe
nden
t and
inde
pen-
dent
var
iabl
es.
•An
alyz
e th
e re
latio
nshi
p be
twee
n th
e de
pend
ent a
nd in
depe
nden
t va
riabl
es u
sing
a gr
aph.
•Fi
nd th
e gr
eate
st c
omm
on fa
ctor
.
•De
fine
ratio
nal a
nd ir
ratio
nal n
umbe
rs.
Stud
ents
will
be
able
to:
•Cl
assif
y po
lyno
mia
ls by
type
and
deg
ree.
•M
odel
a si
tuati
on w
ith a
pol
ynom
ial e
xpre
ssio
n.
•M
ultip
ly m
onom
ials,
bin
omia
ls, a
nd tr
inom
ials
with
a v
arie
ty o
f m
etho
ds, i
nclu
ding
(but
not
lim
ited
to) u
sing
conc
rete
mod
els a
nd
dire
ctly
app
lyin
g th
e di
strib
utive
pro
pert
y.
•Ad
d an
d su
btra
ct p
olyn
omia
ls, si
mpl
ifyin
g w
ith a
var
iety
of m
etho
ds,
incl
udin
g (b
ut n
ot li
mite
d to
) usin
g co
ncre
te m
odel
s and
alg
ebra
ical
ly
com
bini
ng li
ke te
rms.
•Fa
ctor
qua
drati
c ex
pres
sions
.
Poss
ible
task
(s)*
*The
Alg
ebra
I to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
CPS Mathematics Content Framework_Version 1.0 Page 67
Plan
ning
Gui
des
Alge
bra
I Pla
nnin
g G
uide
– S
Y12-
13
Big
Idea
: Non
-line
ar F
uncti
ons a
nd E
quati
ons
*The
Alg
ebra
I to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
Topi
c: M
odel
ing
with
Qua
drati
c Fu
nctio
ns a
nd S
olvi
ng Q
uadr
atic
Equa
tions
CCSS
-M C
onte
nt S
tand
ards
F-IF.
7a
Grap
h lin
ear a
nd q
uadr
atic
func
tions
and
show
inte
rcep
ts, m
axim
a, a
nd m
inim
a.F-
IF.9
Com
pare
pro
perti
es o
f tw
o fu
nctio
ns e
ach
repr
esen
ted
in a
diff
eren
t way
(alg
ebra
ical
ly, g
raph
ical
ly, n
umer
ical
ly in
tabl
es, o
r by
verb
al d
escr
iptio
ns).
For e
xam
ple,
giv
en a
gr
aph
of o
ne q
uadr
atic
func
tion
and
an a
lgeb
raic
exp
ress
ion
for a
noth
er, s
ay w
hich
has
the
larg
er m
axim
um.
A-RE
I.4b
Solv
e qu
adra
tic e
quati
ons b
y in
spec
tion
(e.g
., fo
r x 2 =
49)
, tak
ing
squa
re ro
ots,
com
pleti
ng th
e sq
uare
, the
qua
drati
c fo
rmul
a an
d fa
ctor
ing,
as a
ppro
pria
te to
the
initi
al fo
rm o
f th
e eq
uatio
n. R
ecog
nize
whe
n th
e qu
adra
tic fo
rmul
a gi
ves c
ompl
ex so
lutio
ns a
nd w
rite
them
as a
± b
i for
real
num
bers
a a
nd b
.A-
REI.1
1 Ex
plai
n w
hy th
e x-
coor
dina
tes o
f the
poi
nts w
here
the
grap
hs o
f the
equ
ation
s y =
f(x)
and
y =
g(x
) int
erse
ct a
re th
e so
lutio
ns o
f the
equ
ation
f(x)
= g
(x);
find
the
solu
tions
ap
prox
imat
ely,
e.g.
, usin
g te
chno
logy
to g
raph
the
func
tions
, mak
e ta
bles
of v
alue
s, o
r find
succ
essiv
e ap
prox
imati
ons.
Incl
ude
case
s whe
re f(
x) a
nd/o
r g(x
) are
line
ar,
poly
nom
ial,
ratio
nal,
abso
lute
val
ue, e
xpon
entia
l, an
d lo
garit
hmic
func
tions
.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…
MP1
- M
ake
sens
e of
pro
blem
s and
per
seve
re in
solv
ing
them
.An
alyz
e gi
ven
info
rmati
on to
dev
elop
pos
sible
stra
tegi
es fo
r sol
ving
the
prob
lem
.
MP4
- M
odel
with
mat
hem
atics
.U
se a
var
iety
of m
etho
ds to
mod
el, r
epre
sent
, and
solv
e re
al-w
orld
pr
oble
ms.
MP5
- U
se a
ppro
pria
te to
ols s
trat
egic
ally.
Sele
ct a
nd u
se a
ppro
pria
te to
ols t
o be
st m
odel
/sol
ve p
robl
ems.
Prio
r Kno
wle
dge
•Id
entif
y ra
tiona
l and
irra
tiona
l num
bers
•Ap
prox
imat
e irr
ation
al n
umbe
rs
•So
lve
equa
tions
and
ineq
ualiti
es
•Cr
eate
equ
ation
s and
ineq
ualiti
es
Stud
ents
will
be
able
to:
•De
term
ine
if a
rela
tions
hip
repr
esen
ted
by a
tabl
e, ru
le, g
raph
, or
stat
emen
t can
be
repr
esen
ted
by a
qua
drati
c fu
nctio
n.
•U
se fu
nctio
ns o
f the
form
y =
ax^
2 +
c to
repr
esen
t som
e qu
adra
tic
rela
tions
hips
.
•Ex
plai
n ho
w c
hang
es in
the
para
met
ers a
and
c fo
r y =
ax^
2 +
c aff
ect
the
grap
h of
the
pare
nt q
uadr
atic
func
tion
y =
x^2.
•Id
entif
y an
d m
ake
conn
ectio
ns b
etw
een
solu
tions
and
x-in
terc
epts
.
•Si
mpl
ify sq
uare
root
s alg
ebra
ical
ly a
nd c
onne
ct th
e sim
plifi
ed fo
rm to
th
e ge
omet
ric m
odel
s for
squa
re ro
ots.
•U
se th
e di
scrim
inan
t to
dete
rmin
e th
e nu
mbe
r of s
oluti
ons f
or a
qu
adra
tic e
quati
on.
•So
lve
quad
ratic
equ
ation
s by
fact
orin
g.
•Id
entif
y an
d m
ake
conn
ectio
ns a
mon
g fa
ctor
s, so
lutio
ns, x
-inte
rcep
ts,
and
zero
s.
•So
lve
quad
ratic
s by
grap
hing
.
•Ex
plai
n th
e m
eani
ng o
f sol
ution
s for
giv
en si
tuati
ons.
•So
lve
quad
ratic
equ
ation
s usin
g th
e qu
adra
tic fo
rmul
a.
•U
se th
e di
scrim
inan
t to
dete
rmin
e th
e nu
mbe
r of s
oluti
ons f
or a
qu
adra
tic e
quati
on.
•Ex
plai
n th
e m
eani
ng o
f sol
ution
s for
giv
en si
tuati
on.
Poss
ible
task
(s)*
Chicago Public SchoolsPage 68
Geometry Planning Guide – SY12-13 Introduction
The purpose of this Geometry Planning Guide is to define the scope of the Common Core State Standards for Mathematics (CCSS-M) Content Standards to be taught in the 2012-2013 school year. It has been designed by CPS teachers to be useful to CPS teachers during the three-year transition to full implementation of the CCSS-M.
In general, a Geometry course formalizes and extends students’ geometric experiences from the middle grades. In Year 1 of our transition to CCSS-M, 2012-2013, the Geometry Planning Guide focuses student learning on defining geometric shapes, points, lines, and planes, concepts found in the CCSS-M for Grades 7 and 8. Because this focus on geometric construction is new for many CPS students, we have allocated more time than is provided for in the traditional Geometry course outlined in CCSS-M Appendix A: Designing High School Mathematics Courses on the Common Core State Standards (2011). To accommodate this, the Applications of Probability unit (found in the Appendix A course outline) will not be included in Year 1, but will be included in Year 2 (for 2013-2014).
This Planning Guide includes five Big Ideas (below). For each Big Idea, a summative assessment is referenced, and topics provide the detailed components to support instruction. Topic components include a targeted set of content standards, target mathematical practices (and how they apply to the specific topic), and sample instructional tasks. The sample tasks have been selected for their rigor and their potential usefulness as formative performance assessments for the topic. A toolset to support instruction includes: samples of lesson planning templates; samples of rigorous tasks; samples of formative assessment options based on MARS (Mathematics Assessment Resource Service)tasks; tools for examining and modifying lessons/tasks to increase rigor; samples of modified lessons/tasks; and a professional resource list. These tools are available at https://ocs.cps.k12.il.us/sites/IKMC/default.aspx and on the Department of Mathematics and Science website (cmsi.cps.k12.il.us).
The guide assumes 156 days for instruction, including time for formative and summative assessments. The Big Ideas are sequenced in a way that we believe best develops and connects the mathematical content of the CCSS-M. However, teachers should review the Big Ideas and decide the order and time allocation appropriate for their classrooms, given their students, instructional materials, and other considerations. The order of the standards presented in a topic does not imply a sequence of the content. Some standards may be revisited several times while addressing the topic, while others may be only partially addressed, depending on the topic’s mathematical focus.
Throughout the Geometry course, students should continue to develop proficiency in the Common Core’s Standards for Mathematical Practice (below). Teachers should integrate the instruction of content standards with mathematical practices. This Planning Guide includes “how to” guidance to support this approach to integrated instruction. When the mathematical practices are taught alongside the indicated content standards, students will be more likely to achieve the depth of conceptual understanding and procedural fluency that are expected by the CCSS-M.
Page 69
Plan
ning
Gui
des
CPS Mathematics Content Framework_Version 1.0
The CCSS-M Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning
Finally, this document reflects our current thinking about the intent of CCSS-M, but we recognize that we will learn more as we begin the transition. This learning will be reflected in refinements to the materials. Please share your implementation insights with us, so that the District can benefit from your experience. Network Instructional Support Leaders (ISLs) provide the interface between you, the teacher, and the Department of Mathematics and Science.
BIG IDEAS in GEOMETRY, YEAR 1 PAGE
Congruence, Proof, and Constructions.................................................................................................................70
Similarity & Congruence ..................................................................................................................................... 74
Extending to Three Dimensions .......................................................................................................................... 77
Connecting Algebra and Geometry through Coordinates .................................................................................... 80
Circles With and Without Coordinates .................................................................................................................82
Chicago Public SchoolsPage 70
Geo
met
ry P
lann
ing
Gui
de –
SY1
2-13
Bi
g Id
ea: C
ongr
uenc
e, P
roof
, and
Con
stru
ction
s
*The
Geo
met
ry to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
Big
Idea
Ass
essm
ent:
Cong
ruen
ce, P
roof
, and
Con
stru
ction
sSe
e G
eom
etry
Too
lset
*: U
sing
Too
ls o
f Geo
met
ry
Topi
c: In
trod
uctio
n to
Geo
met
ryCC
SS-M
Con
tent
Sta
ndar
ds
G-CO
.1
Know
pre
cise
defi
nitio
ns o
f ang
le, c
ircle
, per
pend
icul
ar li
ne, p
aral
lel l
ine,
and
line
segm
ent,
base
d on
the
unde
fined
noti
ons o
f poi
nt, l
ine,
dist
ance
alo
ng a
line
, and
dist
ance
aro
und
a ci
rcul
ar a
rc.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…
MP5
- U
se a
ppro
pria
te to
ols s
trat
egic
ally.
Iden
tify
mat
hem
atica
l too
ls an
d re
cogn
ize th
eir s
tren
gths
and
wea
knes
ses
MP6
- Att
end
to p
reci
sion.
Com
mun
icat
e us
ing
clea
r mat
hem
atica
l defi
nitio
ns, v
ocab
ular
y, an
d sy
mbo
ls
Prio
r Kno
wle
dge
•De
finiti
ons o
f: Ac
ute
Angl
es, O
btus
e An
gles
, Rig
ht A
ngle
s, A
cute
Tria
ngle
, Obt
use
Tria
ngle
, Rig
ht T
riang
le, I
sosc
eles
Tria
ngle
s, S
cale
ne T
riang
les,
Eq
uila
tera
l Tria
ngle
, Pol
ygon
s, R
egul
ar P
olyg
ons,
Tra
pezo
ids,
Isos
cele
s Tra
pezo
ids
Stud
ents
will
be
able
to:
•N
ame
and/
or u
se th
e co
rrec
t sym
bolic
not
ation
al c
onve
ntion
s for
: Ang
le, M
easu
re o
f an
Angl
e, C
ircle
, Per
pend
icul
ar, P
aral
lel,
Segm
ent,
Mea
sure
of
a Se
gmen
t, Po
int,
Line
, Pla
ne, E
ndpo
int,
Ray,
Plan
e, A
ngle
Bise
ctor
, Mid
poin
t, Se
gmen
t Bise
ctor
s, P
erpe
ndic
ular
Bise
ctor
s, B
etw
eene
ss, C
ongr
uent
Se
gmen
ts, C
ongr
uent
Ang
les,
Tic
k M
arks
, Len
gth
of S
egm
ent,
Mea
sure
of a
n An
gle,
Col
linea
r, An
gle
Addi
tion,
Seg
men
t Add
ition
Poss
ible
task
s*
CPS Mathematics Content Framework_Version 1.0 Page 71
Plan
ning
Gui
des
Geo
met
ry P
lann
ing
Gui
de –
SY1
2-13
Bi
g Id
ea: C
ongr
uenc
e, P
roof
, and
Con
stru
ction
s
*The
Geo
met
ry to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
Topi
c: T
ools
of G
eom
etry
CCSS
-M C
onte
nt S
tand
ards
G-CO
.12
Mak
e fo
rmal
geo
met
ric c
onst
ructi
ons w
ith a
var
iety
of t
ools
and
met
hods
(com
pass
and
stra
ight
edge
, str
ing,
refle
ctive
dev
ices
, pap
er fo
ldin
g, d
ynam
ic g
eom
etric
softw
are,
etc
.). C
opyi
ng
a se
gmen
t; co
pyin
g an
ang
le; b
isecti
ng a
segm
ent;
bise
cting
an
angl
e; c
onst
ructi
ng p
erpe
ndic
ular
line
s, in
clud
ing
the
perp
endi
cula
r bise
ctor
of a
line
segm
ent;
and
cons
truc
ting
a lin
e pa
ralle
l to
a gi
ven
line
thro
ugh
a po
int n
ot o
n th
e lin
e.G-
CO.1
3 Co
nstr
uct a
n eq
uila
tera
l tria
ngle
, a sq
uare
, and
a re
gula
r hex
agon
insc
ribed
in a
circ
le.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…
MP1
-Mak
e se
nse
of p
robl
ems a
nd p
erse
vere
in so
lvin
g th
em.
Chec
k fo
r acc
urac
y an
d re
ason
able
ness
of w
ork,
stra
tegy
and
solu
tion
MP5
- U
se a
ppro
pria
te to
ols s
trat
egic
ally.
Sele
ct a
nd u
se a
ppro
pria
te to
ols t
o be
st m
odel
/sol
ve p
robl
ems
MP6
- Att
end
to p
reci
sion.
Calc
ulat
e an
swer
s effi
cien
tly a
nd a
ccur
atel
y an
d la
bel t
hem
app
ropr
iate
ly
Prio
r Kno
wle
dge
•De
finiti
ons o
f Con
grue
nt S
egm
ents
, Con
grue
nt A
ngle
s, R
adiu
s, A
rc, M
idpo
int,
Bise
ctor
, Per
pend
icul
ar L
ines
, Par
alle
l Lin
es, P
erpe
ndic
ular
Bise
ctor
, Eq
uila
tera
l Tria
ngle
, Squ
are,
Per
pend
icul
ar, E
quid
istan
t
Stud
ents
will
be
able
to:
•Ac
cura
tely
con
stru
ct: a
) a se
gmen
t con
grue
nt to
a g
iven
segm
ent;
b) a
n an
gle
to c
ongr
uent
to a
giv
en a
ngle
; c) a
n an
gle
bise
ctor
of a
giv
en a
ngle
; d)
the
segm
ent b
isect
or a
nd p
erpe
ndic
ular
bise
ctor
of a
giv
en se
gmen
t; e)
a li
ne p
erpe
ndic
ular
to a
giv
en li
ne th
roug
h a
poin
t on
the
give
n lin
e; f)
a
line
perp
endi
cula
r to
a gi
ven
line
thro
ugh
a po
int n
ot o
n th
e gi
ven
line;
g) a
line
par
alle
l to
a gi
ven
line
thro
ugh
a gi
ven
poin
t
•Ac
cura
tely
con
stru
ct: a
) an
equ
ilate
ral t
riang
le g
iven
a si
de le
ngth
; b) a
squa
re g
iven
a si
de le
ngth
Poss
ible
task
(s)*
See
CME
Proj
ect G
eom
etry
: P
age
7 M
odel
the
Prob
lem
Chicago Public SchoolsPage 72
Topi
c: T
rans
form
ation
sCC
SS-M
Con
tent
Sta
ndar
ds
G-CO
.2
Repr
esen
t tra
nsfo
rmati
ons i
n th
e pl
ane
usin
g, e
.g.,
tran
spar
enci
es a
nd g
eom
etry
softw
are;
des
crib
e tr
ansf
orm
ation
s as f
uncti
ons t
hat t
ake
poin
ts in
the
plan
e as
inpu
ts a
nd
give
oth
er p
oint
s as o
utpu
ts. C
ompa
re tr
ansf
orm
ation
s tha
t pre
serv
e di
stan
ce a
nd a
ngle
to th
ose
that
do
not (
e.g.
, tra
nsla
tion
vers
us h
orizo
ntal
stre
tch)
.G-
CO.3
Gi
ven
a re
ctan
gle,
par
alle
logr
am, t
rape
zoid
, or r
egul
ar p
olyg
on, d
escr
ibe
the
rota
tions
and
refle
ction
s tha
t car
ry it
ont
o its
elf.
G-CO
.4
Deve
lop
defin
ition
s of r
otati
ons,
refle
ction
s, a
nd tr
ansla
tions
in te
rms o
f ang
les,
circ
les,
per
pend
icul
ar li
nes,
par
alle
l lin
es, a
nd li
ne se
gmen
ts.
G-CO
.5
Give
n a
geom
etric
figu
re a
nd a
rota
tion,
refle
ction
, or t
rans
latio
n, d
raw
the
tran
sfor
med
figu
re u
sing,
e.g
., gr
aph
pape
r, tr
acin
g pa
per,
or g
eom
etry
softw
are.
Spe
cify
a se
quen
ce
of tr
ansf
orm
ation
s tha
t will
car
ry a
giv
en fi
gure
ont
o an
othe
r.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…
MP4
- M
odel
with
mat
hem
atics
.Ch
oose
a m
odel
that
is b
oth
appr
opria
te a
nd e
ffici
ent t
o ar
rive
at o
ne o
r m
ore
desir
ed so
lutio
ns.
MP5
- U
se a
ppro
pria
te to
ols s
trat
egic
ally.
Iden
tify
and
succ
essf
ully
use
ext
erna
l mat
hem
atica
l res
ourc
es to
pos
e or
so
lve
prob
lem
s.
MP6
- Att
end
to p
reci
sion.
Calc
ulat
e an
swer
s effi
cien
tly a
nd a
ccur
atel
y an
d la
bel t
hem
app
ropr
iate
ly
MP7
- Lo
ok fo
r and
mak
e us
e of
stru
ctur
e.Lo
ok fo
r, id
entif
y, an
d ac
cept
patt
erns
or s
truc
ture
with
in re
latio
nshi
ps
MP8
- Lo
ok fo
r and
exp
ress
regu
larit
y in
repe
ated
reas
onin
g.Ge
nera
lize
the
proc
ess t
o cr
eate
a sh
ortc
ut w
hich
may
lead
to d
evel
opin
g ru
les o
r cre
ating
a fo
rmul
a.
Prio
r Kno
wle
dge
•Se
gmen
t len
gth
and
angl
e m
easu
re
•De
finiti
ons o
f par
alle
logr
am, r
ecta
ngle
, tra
pezo
id, a
nd re
gula
r pol
ygon
•St
uden
ts sh
ould
be
able
to a
ccur
atel
y co
nstr
uct p
erpe
ndic
ular
line
s th
roug
h a
poin
t not
on
the
line,
con
grue
nt a
ngle
s, a
nd c
ongr
uent
se
gmen
ts
Stud
ents
will
be
able
to:
•De
scrib
e th
e di
ffere
nt ty
pes o
f tra
nsfo
rmati
ons a
nd b
e ab
le to
diff
eren
-tia
te b
etw
een
them
.
•U
nder
stan
d th
e di
ffere
nce
betw
een
tran
sfor
mati
ons t
hat a
re is
om-
etrie
s and
thos
e th
at a
re n
ot.
•Id
entif
y an
d co
nstr
uct t
he li
ne(s
) of s
ymm
etry
, if t
hey
exist
, for
any
po
lygo
n.
•Id
entif
y an
d lo
cate
the
cent
er o
f rot
ation
for a
ny p
olyg
on, i
f it e
xist
s,
and
dete
rmin
e th
e de
gree
of r
otati
onal
sym
met
ry.
•Cl
assif
y ob
ject
s as n
-fold
refle
ction
al a
nd/o
r n-r
otati
onal
and
be
able
to
dete
rmin
e th
e va
lue
of n
for e
ach.
•Re
flect
an
obje
ct o
ver a
giv
en li
ne b
y co
nstr
uctin
g pe
rpen
dicu
lar l
ines
an
d co
ngru
ent s
egm
ents
or b
y us
ing
tech
nolo
gy.
•Ro
tate
an
obje
ct b
y re
flecti
ng it
ove
r tw
o in
ters
ectin
g lin
es w
ith a
nd
with
out t
echn
olog
y.
•Tr
ansla
te a
n ob
ject
by
refle
cting
it o
ver t
wo
para
llel l
ines
with
and
w
ithou
t tec
hnol
ogy.
•Ro
tate
an
obje
ct a
bout
a g
iven
cen
ter w
ith a
giv
en a
ngle
and
dire
c-tio
n w
ith a
nd w
ithou
t tec
hnol
ogy.
•Tr
ansla
te a
n ob
ject
a g
iven
dist
ance
and
giv
en d
irecti
on o
n th
e co
or-
dina
te p
lane
and
geo
met
ric p
lane
with
and
with
out t
echn
olog
y.
•De
scrib
e an
d pe
rfor
m th
e co
mpo
sition
of t
rans
form
ation
s tha
t will
m
ap a
giv
en o
bjec
t ont
o a
cong
ruen
t obj
ect i
n th
e pl
ane
with
or
with
out t
echn
olog
y.
Poss
ible
task
(s)*
CME
Proj
ect G
eom
etry
:
P
ages
537
and
538
In C
lass
Exp
erim
ent a
nd F
or Y
ou to
Exp
lore
Geo
met
ry P
lann
ing
Gui
de –
SY1
2-13
Bi
g Id
ea: C
ongr
uenc
e, P
roof
, and
Con
stru
ction
s
*The
Geo
met
ry to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
CPS Mathematics Content Framework_Version 1.0 Page 73
Plan
ning
Gui
des
Geo
met
ry P
lann
ing
Gui
de –
SY1
2-13
Bi
g Id
ea: C
ongr
uenc
e, P
roof
, and
Con
stru
ction
s
*The
Geo
met
ry to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
Topi
c: P
roof
CCSS
-M C
onte
nt S
tand
ards
G-CO
.9
Prov
e th
eore
ms a
bout
line
s and
ang
les.
The
orem
s inc
lude
: ver
tical
ang
les a
re c
ongr
uent
; whe
n a
tran
sver
sal c
ross
es p
aral
lel l
ines
, alte
rnat
e in
terio
r ang
les a
re c
ongr
uent
and
co
rres
pond
ing
angl
es a
re c
ongr
uent
; poi
nts o
n a
perp
endi
cula
r bise
ctor
of a
line
segm
ent a
re e
xact
ly th
ose
equi
dist
ant f
rom
the
segm
ent’s
end
poin
ts.
G-CO
.1
Prov
e th
eore
ms a
bout
tria
ngle
s. T
heor
ems i
nclu
de: m
easu
res o
f int
erio
r ang
les o
f a tr
iang
le su
m to
180
°; ba
se a
ngle
s of i
sosc
eles
tria
ngle
s are
con
grue
nt; t
he se
gmen
t joi
ning
m
idpo
ints
of t
wo
sides
of a
tria
ngle
is p
aral
lel t
o th
e th
ird si
de a
nd h
alf t
he le
ngth
; the
med
ians
of a
tria
ngle
mee
t at a
poi
nt.
G-CO
.11
Prov
e th
eore
ms a
bout
par
alle
logr
ams.
The
orem
s inc
lude
: opp
osite
side
s are
con
grue
nt, o
ppos
ite a
ngle
s are
con
grue
nt, t
he d
iago
nals
of a
par
alle
logr
am b
isect
eac
h ot
her,
and
conv
erse
ly, re
ctan
gles
are
par
alle
logr
ams w
ith c
ongr
uent
dia
gona
ls.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…
MP1
-Mak
e se
nse
of p
robl
ems a
nd p
erse
vere
in so
lvin
g th
em.
Anal
yze
give
n in
form
ation
to d
evel
op p
ossib
le st
rate
gies
for s
olvi
ng th
e pr
oble
m
MP2
- Re
ason
abs
trac
tly a
nd q
uanti
tativ
ely.
Tran
slate
giv
en in
form
ation
to c
reat
e a
mat
hem
atica
l rep
rese
ntati
on fo
r a
conc
ept.
MP3
- Co
nstr
uct v
iabl
e ar
gum
ents
and
criti
que
the
reas
onin
g of
oth
ers.
Justi
fy (o
rally
and
in w
ritten
form
) the
app
roac
h us
ed, i
nclu
ding
how
it fi
ts
in th
e co
ntex
t fro
m w
hich
the
data
aro
se
Prio
r Kno
wle
dge
•U
nder
stan
d an
d ap
ply
the
defin
ition
s of:
Line
ar P
air o
f Ang
les,
Pa
ralle
l Lin
es a
nd A
ngle
s For
med
by
Tran
sver
sals,
Cor
resp
ondi
ng
Angl
es, C
orre
spon
ding
Sid
es, P
erpe
ndic
ular
Lin
es, P
erpe
ndic
ular
Bi
sect
or, a
nd E
quid
istan
ce
•U
nder
stan
d an
d ap
ply
the
defin
ition
s of:
Para
llelo
gram
s, D
iago
nals,
O
ppos
ite S
ides
, Opp
osite
Ang
les,
Con
secu
tive
Angl
es, R
ecta
ngle
s,
Bise
ct
•U
nder
stan
d an
d ap
ply
the
defin
ition
s of:
Inte
rior A
ngle
s, Is
osce
les
Tria
ngle
s, P
arts
of I
sosc
eles
Tria
ngle
s, M
idse
gmen
t of a
Tria
ngle
, M
edia
ns o
f a T
riang
le
Stud
ents
will
be
able
to:
Prov
e th
e fo
llow
ing
theo
rem
s:
•Ve
rtica
l ang
le th
eore
m
•Al
tern
ate
inte
rior a
ngle
theo
rem
•Co
rres
pond
ing
angl
e th
eore
m
•Pe
rpen
dicu
lar b
isect
or th
eore
m
•Tr
iang
le su
m th
eore
m
•Is
osce
les t
riang
le th
eore
m
•M
id-s
egm
ent o
f a tr
iang
le p
aral
lel t
o a
side
and
half
the
leng
th
•M
edia
ns o
f a tr
iang
le a
re c
oncu
rren
t
•O
ppos
ite si
des o
f a p
aral
lelo
gram
are
con
grue
nt
•O
ppos
ite a
ngle
s of a
par
alle
logr
am a
re c
ongr
uent
•Di
agon
als o
f a p
aral
lelo
gram
bise
ct e
ach
othe
r
•Re
ctan
gles
are
par
alle
logr
ams w
ith c
ongr
uent
dia
gona
ls
Poss
ible
task
(s)*
See
Carn
egie
Lea
rnin
g, In
c. G
eom
etry
: P
ages
91
and
92 W
hat’s
You
r Pro
of?
Chicago Public SchoolsPage 74
Geo
met
ry P
lann
ing
Gui
de –
SY1
2-13
Bi
g Id
ea: S
imila
rity
& C
ongr
uenc
e
Big
Idea
Ass
essm
ent:
Sim
ilarit
y &
Con
grue
nce
See
Geo
met
ry T
ools
et*:
Mon
tros
e Hi
gh R
ocke
t Clu
b
Topi
c: S
imila
rity
CCSS
-M C
onte
nt S
tand
ards
G-SR
T.2
Give
n tw
o fig
ures
, use
the
defin
ition
of s
imila
rity
in te
rms o
f sim
ilarit
y tr
ansf
orm
ation
s to
deci
de if
they
are
sim
ilar;
expl
ain
usin
g sim
ilarit
y tr
ansf
orm
ation
s the
mea
ning
of
simila
rity
for t
riang
les a
s the
equ
ality
of a
ll co
rres
pond
ing
pairs
of a
ngle
s and
the
prop
ortio
nalit
y of
all
corr
espo
ndin
g pa
irs o
f sid
es.
G-SR
T.3
Use
the
prop
ertie
s of s
imila
rity
tran
sfor
mati
ons t
o es
tabl
ish th
e AA
crit
erio
n fo
r tw
o tr
iang
les t
o be
sim
ilar.
G-SR
T.5
Use
con
grue
nce
and
simila
rity
crite
ria fo
r tria
ngle
s to
solv
e pr
oble
ms a
nd to
pro
ve re
latio
nshi
ps in
geo
met
ric fi
gure
s.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…
MP3
- Co
nstr
uct v
iabl
e ar
gum
ents
and
criti
que
the
reas
onin
g of
oth
ers.
Use
obs
erva
tions
and
prio
r kno
wle
dge
(sta
ted
assu
mpti
ons,
defi
nitio
ns,
and
prev
ious
est
ablis
hed
resu
lts) t
o m
ake
conj
ectu
res a
nd c
onst
ruct
ar
gum
ents
MP4
- M
odel
with
mat
hem
atics
.Si
mpl
ify a
com
plic
ated
pro
blem
by
mak
ing
assu
mpti
ons a
nd
appr
oxim
ation
s
MP5
- U
se a
ppro
pria
te to
ols s
trat
egic
ally.
Sele
ct a
nd u
se a
ppro
pria
te to
ols t
o be
st m
odel
/sol
ve p
robl
ems
MP7
- Lo
ok fo
r and
mak
e us
e of
stru
ctur
e.Lo
ok fo
r, id
entif
y, an
d ac
cept
patt
erns
or s
truc
ture
with
in re
latio
nshi
ps
Prio
r Kno
wle
dge
•Pr
opor
tiona
l rea
soni
ng a
nd u
sing
tran
sfor
mati
ons
•Ba
sic a
bilit
y to
mat
hem
atica
lly su
ppor
t a p
redi
ction
or h
ypot
hesis
•Re
cogn
ize a
situ
ation
’s co
nnec
tion
to m
athe
mati
cs m
odel
Stud
ents
will
be
able
to:
•So
lve
linea
r equ
ation
s in
one
varia
ble.
•Id
entif
y lin
ear e
quati
ons w
ith o
ne, i
nfini
tely
man
y, or
no
solu
tion.
•Tr
ansf
orm
an
expr
essio
n or
equ
ation
into
alte
rnat
e fo
rms.
•Fi
nd th
e so
lutio
n to
a sy
stem
of e
quati
ons f
rom
a g
raph
.
•So
lve
syst
ems a
lgeb
raic
ally
usin
g su
bstit
ution
and
elim
inati
on/
com
bina
tion.
•U
nder
stan
d th
e so
lutio
n of
a sy
stem
of e
quati
ons i
n th
e co
ntex
t of
the
prob
lem
and
writ
e as
an
orde
red
pair.
Poss
ible
task
(s)*
*The
Geo
met
ry to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
CPS Mathematics Content Framework_Version 1.0 Page 75
Plan
ning
Gui
des
Geo
met
ry P
lann
ing
Gui
de –
SY1
2-13
Bi
g Id
ea: S
imila
rity
& C
ongr
uenc
e
*The
Geo
met
ry to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
Topi
c: S
imila
rity
in D
ilatio
nsCC
SS-M
Con
tent
Sta
ndar
ds
G-SR
T.1
Verif
y ex
perim
enta
lly th
e pr
oper
ties o
f dila
tions
giv
en b
y a
cent
er a
nd a
scal
e fa
ctor
:a.
A d
ilatio
n ta
kes a
line
not
pas
sing
thro
ugh
the
cent
er o
f the
dila
tion
to a
par
alle
l lin
e, a
nd le
aves
a li
ne p
assin
g th
roug
h th
e ce
nter
unc
hang
ed.
b. T
he d
ilatio
n of
a li
ne se
gmen
t is l
onge
r or s
hort
er in
the
ratio
giv
en b
y th
e sc
ale
fact
or
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…
MP1
-Mak
e se
nse
of p
robl
ems a
nd p
erse
vere
in so
lvin
g th
em.
Iden
tify
and
exec
ute
appr
opria
te st
rate
gies
to so
lve
the
prob
lem
MP3
- Co
nstr
uct v
iabl
e ar
gum
ents
and
criti
que
the
reas
onin
g of
oth
ers.
Justi
fy (o
rally
and
in w
ritten
form
) the
app
roac
h us
ed, i
nclu
ding
how
it fi
ts
in th
e co
ntex
t fro
m w
hich
the
data
aro
se
MP8
- Lo
ok fo
r and
exp
ress
regu
larit
y in
repe
ated
reas
onin
g.Ge
nera
lize
the
proc
ess t
o cr
eate
a sh
ortc
ut w
hich
may
lead
to d
evel
op-
ing
rule
s or c
reati
ng a
form
ula
Prio
r Kno
wle
dge
•Fl
uenc
y w
ith si
mila
rity
of tw
o-di
men
siona
l figu
res.
(im
plie
s flue
ncy
with
ratio
s and
pro
porti
onal
reas
onin
g)
•Pa
ralle
l lin
es c
ut b
y tr
aver
sals
prop
ertie
s or e
stab
lish
the
prop
ertie
s
Stud
ents
will
be
able
to:
•Gi
ven
a on
e- o
r tw
o-di
men
siona
l figu
re in
a p
lane
and
a p
oint
on
or n
ot
on th
e pl
ane,
cre
ate
spec
ific
dila
tions
.
•Be
abl
e to
exp
lain
(jus
tify)
the
rela
tion
betw
een
area
s of a
figu
re a
nd
its d
ilatio
n (fo
r sim
ple
figur
es: t
riang
les,
qua
drila
tera
ls, a
nd h
exag
ons
or o
ther
sim
ple
figur
es).
•So
me
stud
ents
mig
ht e
njoy
ext
endi
ng d
ilatio
ns to
thre
e-di
men
siona
l so
lids a
nd e
xpla
inin
g th
e re
latio
n be
twee
n th
e or
igin
al so
lid a
nd th
e di
late
d so
lid.
Poss
ible
task
(s)*
Chicago Public SchoolsPage 76
Geo
met
ry P
lann
ing
Gui
de –
SY1
2-13
Bi
g Id
ea: S
imila
rity
& C
ongr
uenc
e
Topi
c: S
imila
rity
and
Trig
onom
etric
Rati
osCC
SS-M
Con
tent
Sta
ndar
ds
G-SR
T.6
Und
erst
and
that
by
simila
rity,
side
ratio
s in
right
tria
ngle
s are
pro
perti
es o
f the
ang
les i
n th
e tr
iang
le, l
eadi
ng to
defi
nitio
ns o
f trig
onom
etric
ratio
s for
acu
te a
ngle
s.G-
SRT.
8 U
se tr
igon
omet
ric ra
tios a
nd th
e Py
thag
orea
n Th
eore
m to
solv
e rig
ht tr
iang
les i
n ap
plie
d pr
oble
ms.★
G-SR
T.7
Expl
ain
and
use
the
rela
tions
hip
betw
een
the
sine
and
cosin
e of
com
plem
enta
ry a
ngle
s.
★Sp
ecifi
c m
odel
ing
stan
dard
(ver
sus a
n ex
ampl
e of
the
mod
elin
g St
anda
rd fo
r Mat
hem
atica
l Pra
ctice
).
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…
MP1
-Mak
e se
nse
of p
robl
ems a
nd p
erse
vere
in so
lvin
g th
em.
Anal
yze
give
n in
form
ation
to d
evel
op p
ossib
le st
rate
gies
for s
olvi
ng th
e pr
oble
m
MP3
- Co
nstr
uct v
iabl
e ar
gum
ents
and
criti
que
the
reas
onin
g of
oth
ers.
Justi
fy (o
rally
and
in w
ritten
form
) the
app
roac
h us
ed, i
nclu
ding
how
it fi
ts
in th
e co
ntex
t fro
m w
hich
the
data
aro
se
MP4
- M
odel
with
mat
hem
atics
.U
se a
var
iety
of m
etho
ds to
mod
el, r
epre
sent
, and
solv
e re
al-w
orld
pr
oble
ms
MP8
- Lo
ok fo
r and
expr
ess r
egul
arity
in re
peat
ed re
ason
ing.
Reco
gnize
sim
ilariti
es a
nd p
atter
ns in
repe
ated
tria
ls w
ith a
pro
cess
Prio
r Kno
wle
dge
•Fl
uenc
y w
ith ra
tios a
nd p
ropo
rtion
al re
ason
ing
•Fl
uenc
y w
ith d
ilatio
ns
•Re
cogn
ize a
situ
ation
’s co
nnec
tion
to m
athe
mati
cs m
odel
•Ba
sic a
bilit
y to
mat
hem
atica
lly su
ppor
t a p
redi
ction
or h
ypot
hesis
Stud
ents
will
be
able
to:
•Ex
plai
n/ju
stify
that
the
sine,
cos
ine,
and
tang
ent o
f any
giv
en a
cute
an
gle
in a
righ
t tria
ngle
are
con
stan
t for
any
pai
r of d
ilate
d tr
iang
les.
•Id
entif
y th
e op
posit
e sid
e of
an
angl
e, th
e ad
jace
nt si
de o
f an
angl
e,
and
the
hypo
tenu
se in
any
righ
t tria
ngle
.
•De
term
ine
the
spec
ific
ratio
s for
sine
, cos
ine,
and
tang
ent f
or a
spec
i-fie
d an
gle
in a
righ
t tria
ngle
if a
ll th
e sid
es a
re k
now
n or
if o
nly
two
sides
are
kno
wn.
•Ap
ply
trig
onom
etry
to tr
iang
les o
r rea
l wor
ld si
tuati
ons.
•U
nder
stan
d an
d be
abl
e to
exp
lain
whe
n th
e sin
e an
d co
sine
have
the
sam
e va
lue
for d
iffer
ent a
ngle
s and
exp
lain
the
rela
tions
hip
betw
een
thes
e an
gles
.
Poss
ible
task
(s)*
Hope
wel
l Tria
ngle
s
Fr
om: M
athe
mati
cs A
sses
smen
t Pro
ject
*The
Geo
met
ry to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
CPS Mathematics Content Framework_Version 1.0 Page 77
Plan
ning
Gui
des
Geo
met
ry P
lann
ing
Gui
de –
SY1
2-13
Bi
g Id
ea: E
xten
ding
to T
hree
Dim
ensi
ons
*The
Geo
met
ry to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
Big
Idea
Ass
essm
ent:
Exte
ndin
g to
Thr
ee D
imen
sion
sSe
e G
eom
etry
Too
lset
*: D
ream
Hou
se
Topi
c: V
olum
e Fo
rmul
as a
nd P
robl
em S
olvi
ngCC
SS-M
Con
tent
Sta
ndar
ds
G-GM
D.1
Give
an
info
rmal
arg
umen
t for
the
form
ulas
for t
he c
ircum
fere
nce
of a
circ
le, a
rea
of a
circ
le, v
olum
e of
a c
ylin
der,
pyra
mid
, and
con
e. U
se d
issec
tion
argu
men
ts, C
aval
ieri’
s pr
inci
ple,
and
info
rmal
lim
it ar
gum
ents
.G-
GMD.
2 (+
) Giv
e an
info
rmal
arg
umen
t usin
g Ca
valie
ri’s p
rinci
ple
for t
he fo
rmul
as fo
r the
vol
ume
of a
sphe
re a
nd o
ther
solid
figu
res.
G-GM
D.3
Use
vol
ume
form
ulas
for c
ylin
ders
, pyr
amid
s, c
ones
, and
sphe
res t
o so
lve
prob
lem
s.★
+ Ad
ditio
nal c
onte
nt th
at st
uden
ts sh
ould
lear
n in
ord
er to
take
adv
ance
d co
urse
s suc
h as
cal
culu
s, a
dvan
ced
stati
stics
, or d
iscre
te m
athe
mati
cs★
Spec
ific
mod
elin
g st
anda
rd (v
ersu
s an
exam
ple
of th
e m
odel
ing
Stan
dard
for M
athe
mati
cal P
racti
ce)
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…
MP3
- Co
nstr
uct v
iabl
e ar
gum
ents
and
criti
que
the
reas
onin
g of
oth
ers.
Use
obs
erva
tions
and
prio
r kno
wle
dge
(sta
ted
assu
mpti
ons,
defi
nitio
ns,
and
prev
ious
est
ablis
hed
resu
lts) t
o m
ake
conj
ectu
res a
nd c
onst
ruct
ar
gum
ents
MP4
- M
odel
with
mat
hem
atics
.Si
mpl
ify a
com
plic
ated
pro
blem
by
mak
ing
assu
mpti
ons a
nd
appr
oxim
ation
s
MP7
- Lo
ok fo
r and
mak
e us
e of
stru
ctur
e.U
se p
atter
ns o
r str
uctu
re to
mak
e se
nse
of m
athe
mati
cs a
nd c
onne
ct
prio
r kno
wle
dge
to si
mila
r situ
ation
s and
ext
end
to n
ovel
situ
ation
s
MP8
- Lo
ok fo
r and
exp
ress
regu
larit
y in
repe
ated
reas
onin
g.Ev
alua
te th
e re
ason
able
ness
of r
esul
ts th
roug
hout
the
mat
hem
atica
l pr
oces
s whi
le a
ttend
ing
to d
etai
l
Prio
r Kno
wle
dge
•Th
is un
it bu
ilds o
n th
e 7t
h gr
ade
geom
etry
stan
dard
: 7G-
G So
lve
real
-lif
e an
d m
athe
mati
cal p
robl
ems i
nvol
ving
ang
le m
easu
re, a
rea,
surf
ace
area
, and
vol
ume
•Al
gebr
a m
anip
ulati
ons a
nd re
ason
ing
skill
s
•Ba
sic u
nder
stan
ding
of a
rea
as th
e sq
uare
uni
ts o
f cov
erin
g a
give
n fig
ure
or sh
ape
and
be a
ble
to c
alcu
late
are
as a
nd p
erim
eter
s of
simpl
e fig
ures
; rec
tang
les,
tria
ngle
s, a
nd p
ossib
ly c
ircle
s
Stud
ents
will
be
able
to:
•De
rive
the
form
ulas
for c
ircum
fere
nce,
per
imet
er, a
rea,
and
vol
ume
for v
ario
us g
eom
etric
figu
res w
ith a
nd w
ithou
t the
use
of t
echn
olog
y.
•Ap
ply
the
deriv
ed fo
rmul
as.
Poss
ible
task
(s)*
Glas
ses
Fr
om: M
athe
mati
cs A
sses
smen
t Pro
ject
Chicago Public SchoolsPage 78
Geo
met
ry P
lann
ing
Gui
de –
SY1
2-13
Bi
g Id
ea: E
xten
ding
to T
hree
Dim
ensi
ons
Topi
c: V
isua
lize
Sim
ilariti
es B
etw
een
Two-
and
Thr
ee-D
imen
sion
al O
bjec
tsCC
SS-M
Con
tent
Sta
ndar
ds
G-GM
D.4
Iden
tify
the
shap
es o
f tw
o-di
men
siona
l cro
ss-s
ectio
ns o
f thr
ee-d
imen
siona
l obj
ects
, and
iden
tify
thre
e-di
men
siona
l obj
ects
gen
erat
ed b
y ro
tatio
ns o
f tw
o-di
men
siona
l ob
ject
s.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…
MP1
-Mak
e se
nse
of p
robl
ems a
nd p
erse
vere
in so
lvin
g th
em.
Anal
yze
give
n in
form
ation
to d
evel
op p
ossib
le st
rate
gies
for s
olvi
ng th
e pr
oble
m
MP3
- Co
nstr
uct v
iabl
e ar
gum
ents
and
criti
que
the
reas
onin
g of
oth
ers.
Use
obs
erva
tions
and
prio
r kno
wle
dge
(sta
ted
assu
mpti
ons,
defi
nitio
ns,
and
prev
ious
est
ablis
hed
resu
lts) t
o m
ake
conj
ectu
res a
nd c
onst
ruct
ar
gum
ents
Prio
r Kno
wle
dge
•Kn
ow th
e di
ffere
nce
betw
een
two-
and
thre
e-di
men
siona
l obj
ects
•Vi
sual
izatio
n sk
ills a
nd/o
r too
ls to
hel
p vi
sual
ize. F
or e
xam
ple,
cla
y m
odel
s to
be c
ut b
y a
strin
g to
“see
” w
hat t
he c
ross
secti
onal
are
a lo
oks l
ike
Stud
ents
will
be
able
to:
•Gi
ven
a sp
ecifi
c th
ree-
dim
ensio
nal s
hape
, ide
ntify
and
exp
lain
pla
nar c
ross
secti
ons o
f tha
t sha
pe.
•De
scrib
e th
e th
ree-
dim
ensio
nal o
bjec
t gen
erat
ed w
hen
any
two-
dim
ensio
nal fi
gure
is ro
tate
d ab
out:
a)
eith
er a
xis
b)
any
line
para
llel t
o ei
ther
axi
s
Poss
ible
task
(s)*
Prop
ane
Tank
Fro
m: M
athe
mati
cs A
sses
smen
t Pro
ject
*The
Geo
met
ry to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
CPS Mathematics Content Framework_Version 1.0 Page 79
Plan
ning
Gui
des
Geo
met
ry P
lann
ing
Gui
de –
SY1
2-13
Bi
g Id
ea: E
xten
ding
to T
hree
Dim
ensi
ons
Topi
c: A
pply
ing
Geo
met
ric C
once
pts i
n M
odel
ing
Situ
ation
sCC
SS-M
Con
tent
Sta
ndar
ds
G-M
G.1
Use
geo
met
ric sh
apes
, the
ir m
easu
res,
and
thei
r pro
perti
es to
des
crib
e ob
ject
s (e.
g., m
odel
ing
a tr
ee tr
unk
or a
hum
an to
rso
as a
cyl
inde
r). ★
★Sp
ecifi
c m
odel
ing
stan
dard
(ver
sus a
n ex
ampl
e of
the
mod
elin
g St
anda
rd fo
r Mat
hem
atica
l Pra
ctice
)
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…
MP1
-Mak
e se
nse
of p
robl
ems a
nd p
erse
vere
in so
lvin
g th
em.
Iden
tify
and
exec
ute
appr
opria
te st
rate
gies
to so
lve
the
prob
lem
MP4
- M
odel
with
mat
hem
atics
.Ch
oose
a m
odel
that
is b
oth
appr
opria
te a
nd e
ffici
ent t
o ar
rive
at o
ne o
r m
ore
desir
ed so
lutio
ns
MP6
- Att
end
to p
reci
sion.
Com
mun
icat
e us
ing
clea
r mat
hem
atica
l defi
nitio
ns, v
ocab
ular
y, an
d sy
mbo
ls
MP7
- Lo
ok fo
r and
mak
ing
use
of st
ruct
ure.
Use
patt
erns
or s
truc
ture
to m
ake
sens
e of
mat
hem
atics
and
con
nect
pr
ior k
now
ledg
e to
sim
ilar s
ituati
ons a
nd e
xten
d to
nov
el si
tuati
ons
Prio
r Kno
wle
dge
•Kn
ow th
e di
ffere
nce
betw
een
two-
and
thre
e-di
men
siona
l obj
ects
•Vi
sual
izatio
n sk
ills a
nd/o
r too
ls to
hel
p vi
sual
ize. F
or e
xam
ple,
cla
y m
odel
s to
be c
ut b
y a
strin
g to
“see
” w
hat t
he c
ross
secti
onal
are
a lo
oks l
ike
Stud
ents
will
be
able
to:
•Dr
aw a
ny th
ree-
dim
ensio
nal o
bjec
t usin
g on
ly g
eom
etric
shap
es.
Poss
ible
task
(s)*
*The
Geo
met
ry to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
Chicago Public SchoolsPage 80
Geo
met
ry P
lann
ing
Gui
de –
SY1
2-13
Bi
g Id
ea: C
onne
cting
Alg
ebra
and
Geo
met
ry th
roug
h Co
ordi
nate
s
Big
Idea
Ass
essm
ent:C
onne
cting
Alg
ebra
and
Geo
met
ry T
hrou
gh C
oord
inat
esSe
e G
eom
etry
Too
lset
*: N
eigh
borh
ood
Map
Topi
c: S
lope
For
mul
aCC
SS-M
Con
tent
Sta
ndar
d
G-GP
E.5
Prov
e th
e slo
pe c
riter
ia fo
r par
alle
l and
per
pend
icul
ar li
nes a
nd u
se th
em to
solv
e ge
omet
ric p
robl
ems (
e.g.
, find
the
equa
tion
of a
line
par
alle
l or p
erpe
ndic
ular
to a
giv
en li
ne
that
pas
ses t
hrou
gh a
giv
en p
oint
.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…
MP3
- Co
nstr
uct v
iabl
e ar
gum
ents
and
criti
que
the
reas
onin
g of
oth
ers.
Use
obs
erva
tions
and
prio
r kno
wle
dge
(sta
ted
assu
mpti
ons,
defi
nitio
ns,
and
prev
ious
est
ablis
hed
resu
lts) t
o m
ake
conj
ectu
res a
nd c
onst
ruct
ar
gum
ents
MP5
- U
se a
ppro
pria
te to
ols s
trat
egic
ally.
Sele
ct a
nd u
se a
ppro
pria
te to
ols t
o be
st m
odel
/sol
ve p
robl
ems
MP6
- Att
end
to p
reci
sion.
Calc
ulat
e an
swer
s effi
cien
tly a
nd a
ccur
atel
y an
d la
bel t
hem
app
ropr
iate
ly
Prio
r Kno
wle
dge
•Sl
ope
(con
cept
ually
, gra
phic
ally,
alg
ebra
ical
ly)
•Ro
tatio
ns a
nd tr
ansla
tions
of c
oord
inat
e po
ints
•Sl
ope-
inte
rcep
t for
m
Stud
ents
will
be
able
to:
•Fi
nd th
e slo
pe b
etw
een
two
give
n po
ints
.
•Pr
ove
the
slope
crit
eria
for p
aral
lel a
nd p
erpe
ndic
ular
line
s.
•Fi
nd th
e eq
uatio
n of
a li
ne p
aral
lel o
r per
pend
icul
ar to
a g
iven
line
th
at p
asse
s thr
ough
a g
iven
poi
nt.
Poss
ible
task
(s)*
*The
Geo
met
ry to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
CPS Mathematics Content Framework_Version 1.0 Page 81
Plan
ning
Gui
des
Geo
met
ry P
lann
ing
Gui
de –
SY1
2-13
Bi
g Id
ea: C
onne
cting
Alg
ebra
and
Geo
met
ry th
roug
h Co
ordi
nate
s
*The
Geo
met
ry to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
Topi
c: D
ista
nce
Form
ula
CCSS
-M C
onte
nt S
tand
ards
G-GP
E.6
Find
the
poin
t on
a di
rect
ed li
ne se
gmen
t bet
wee
n tw
o gi
ven
poin
ts th
at p
artiti
on th
e se
gmen
t in
a gi
ven
ratio
.G-
GPE.
7 U
se c
oord
inat
es to
com
pute
per
imet
ers o
f pol
ygon
s and
are
as o
f tria
ngle
s and
rect
angl
es, e
.g.,
usin
g th
e di
stan
ce fo
rmul
a.G-
GPE.
4 U
se c
oord
inat
es to
pro
ve si
mpl
e ge
omet
ric th
eore
ms a
lgeb
raic
ally.
For
exa
mpl
e, p
rove
or d
ispro
ve th
at a
figu
re d
efine
d by
four
giv
en p
oint
s in
the
coor
dina
te p
lane
is a
re
ctan
gle.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…..
MP3
- Co
nstr
uct v
iabl
e ar
gum
ents
and
criti
que
the
reas
onin
g of
oth
ers.
Use
obs
erva
tions
and
prio
r kno
wle
dge
(sta
ted
assu
mpti
ons,
defi
nitio
ns,
and
prev
ious
est
ablis
hed
resu
lts) t
o m
ake
conj
ectu
res a
nd c
onst
ruct
ar
gum
ents
MP4
- M
odel
with
mat
hem
atics
.U
se a
var
iety
of m
etho
ds to
mod
el, r
epre
sent
, and
solv
e re
al-w
orld
pr
oble
ms
MP6
- Att
end
to p
reci
sion.
Calc
ulat
e an
swer
s effi
cien
tly a
nd a
ccur
atel
y an
d la
bel t
hem
app
ropr
iate
ly
Prio
r Kno
wle
dge
•Di
stan
ce o
n a
num
ber l
ine
•Py
thag
orea
n Th
eore
m
•Ra
tio
•Pe
rimet
er
•Ar
ea o
f pol
ygon
s
•Pr
oper
ties o
f qua
drila
tera
ls
Stud
ents
will
be
able
to:
•Fi
nd th
e di
stan
ce b
etw
een
two
poin
ts in
the
coor
dina
te p
lane
. •
Find
the
poin
t on
a lin
e se
gmen
t tha
t par
tition
s the
segm
ent i
n a
give
n ra
tio.
•U
se th
e di
stan
ce fo
rmul
a to
com
pute
per
imet
ers o
f pol
ygon
s and
ar
eas o
f tria
ngle
s and
rect
angl
es.
•U
se th
e di
stan
ce a
nd sl
ope
form
ulas
to p
rove
that
a fi
gure
is a
re
ctan
gle,
par
alle
logr
am, r
hom
bus,
trap
ezoi
d, k
ite, o
r squ
are.
Poss
ible
task
(s)
Chicago Public SchoolsPage 82
Geo
met
ry P
lann
ing
Gui
de –
SY1
2-13
Bi
g Id
ea: C
ircle
s With
and
With
out C
oord
inat
es
Big
Idea
Ass
essm
ent:
Circ
les W
ith a
nd W
ithou
t Coo
rdin
ates
See
Geo
met
ry T
ools
et*:
Circ
ular
Rea
soni
ng
Topi
c: D
efini
ng a
Circ
le a
nd P
arts
of C
ircle
sCC
SS-M
Con
tent
Sta
ndar
ds
G-C.
1 Pr
ove
that
all
circ
les a
re si
mila
r.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…
MP3
- Co
nstr
uct v
iabl
e ar
gum
ents
and
criti
que
the
reas
onin
g of
oth
ers.
Use
obs
erva
tions
and
prio
r kno
wle
dge
(sta
ted
assu
mpti
ons,
defi
nitio
ns,
and
prev
ious
est
ablis
hed
resu
lts) t
o m
ake
conj
ectu
res a
nd c
onst
ruct
ar
gum
ents
MP7
- Lo
ok fo
r and
mak
e us
e of
stru
ctur
e.U
se p
atter
ns o
r str
uctu
re to
mak
e se
nse
of m
athe
mati
cs a
nd c
onne
ct
prio
r kno
wle
dge
to si
mila
r situ
ation
s and
ext
end
to n
ovel
situ
ation
s
Prio
r Kno
wle
dge
•De
finiti
ons o
f: Ci
rcle
, Rad
ius,
Dia
met
er ,
Circ
umfe
renc
e, S
imila
rity
Stud
ents
will
be
able
to:
•Id
entif
y (a
nd n
ame)
radi
i, di
amet
ers,
and
cho
rds.
•Pr
ove
that
all
circ
les a
re si
mila
r (by
com
parin
g ra
tios o
f rad
ii an
d ci
rcum
fere
nce
of d
iffer
ent c
ircle
s and
pro
ving
that
a c
onst
ant s
cale
fact
or e
xist
s).
Poss
ible
task
(s)
*The
Geo
met
ry to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
CPS Mathematics Content Framework_Version 1.0 Page 83
Plan
ning
Gui
des
Geo
met
ry P
lann
ing
Gui
de –
SY1
2-13
Bi
g Id
ea: C
ircle
s With
and
With
out C
oord
inat
es
*The
Geo
met
ry to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
Topi
c: R
elati
onsh
ips A
mon
g In
scrib
ed A
ngle
s, R
adii,
and
Cho
rds
CCSS
-M C
onte
nt S
tand
ards
G-C.
2 Id
entif
y an
d de
scrib
e re
latio
nshi
ps a
mon
g in
scrib
ed a
ngle
s, ra
dii,
and
chor
ds. I
nclu
de th
e re
latio
nshi
p be
twee
n ce
ntra
l, in
scrib
ed, a
nd c
ircum
scrib
ed a
ngle
s; in
scrib
ed a
ngle
s on
a di
amet
er a
re ri
ght a
ngle
s; th
e ra
dius
of a
circ
le is
per
pend
icul
ar to
the
tang
ent l
ine
whe
re th
e ra
dius
inte
rsec
ts th
e ci
rcle
.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…
MP3
- Co
nstr
uct v
iabl
e ar
gum
ents
and
criti
que
the
reas
onin
g of
oth
ers.
Use
obs
erva
tions
and
prio
r kno
wle
dge
(sta
ted
assu
mpti
ons,
defi
nitio
ns,
and
prev
ious
est
ablis
hed
resu
lts) t
o m
ake
conj
ectu
res a
nd c
onst
ruct
ar
gum
ents
MP7
- Lo
ok fo
r and
mak
e us
e of
stru
ctur
e.Lo
ok fo
r, id
entif
y, an
d ac
cept
patt
erns
or s
truc
ture
with
in re
latio
nshi
ps
MP8
- Lo
ok fo
r and
exp
ress
regu
larit
y in
repe
ated
reas
onin
g.Ge
nera
lize
the
proc
ess t
o cr
eate
a sh
ortc
ut w
hich
may
lead
to d
evel
op-
ing
rule
s or c
reati
ng a
form
ula
Prio
r Kno
wle
dge
Stud
ents
will
be
able
to:
•Id
entif
y ta
ngen
t lin
es, i
nscr
ibed
ang
les,
and
cen
tral
ang
les.
•Ap
ply
the
rela
tions
hip
betw
een
two
cong
ruen
t ins
crib
ed a
ngle
s and
th
eir i
nter
cept
ed a
rcs.
•Id
entif
y (a
nd a
pply
) tha
t an
insc
ribed
ang
le is
one
hal
f of t
he m
easu
re
of th
e in
terc
epte
d ar
c or
cen
tral
ang
le th
at in
ters
ects
the
sam
e ar
c.
•Id
entif
y (a
nd a
pply
) tha
t a c
ircum
scrib
ed a
ngle
is th
e su
pple
men
t of
the
cent
ral a
ngle
with
the
sam
e in
terc
epte
d ar
c.
•Pr
ove
that
an
insc
ribed
ang
le in
a se
mi-c
ircle
is a
righ
t ang
le.
•De
scrib
e (a
nd a
pply
) how
the
radi
us o
f a c
ircle
is p
erpe
ndic
ular
to th
e ta
ngen
t lin
e at
the
poin
t of t
ange
ncy.
Poss
ible
task
(s)*
Chicago Public SchoolsPage 84
Geo
met
ry P
lann
ing
Gui
de –
SY1
2-13
Bi
g Id
ea: C
ircle
s With
and
With
out C
oord
inat
es
Topi
c: C
onst
ructi
ons
CCSS
-M C
onte
nt S
tand
ards
G-C.
3 C
onst
ruct
the
insc
ribed
and
circ
umsc
ribed
circ
les o
f a tr
iang
le a
nd p
rove
pro
perti
es o
f ang
les f
or a
qua
drila
tera
l ins
crib
ed in
a c
ircle
.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…
MP3
- Co
nstr
uct v
iabl
e ar
gum
ents
and
criti
que
the
reas
onin
g of
oth
ers.
Use
obs
erva
tions
and
prio
r kno
wle
dge
(sta
ted
assu
mpti
ons,
defi
nitio
ns,
and
prev
ious
est
ablis
hed
resu
lts) t
o m
ake
conj
ectu
res a
nd c
onst
ruct
ar
gum
ents
MP5
- U
se a
ppro
pria
te to
ols s
trat
egic
ally.
Sele
ct a
nd u
se a
ppro
pria
te to
ols t
o be
st m
odel
/sol
ve p
robl
ems
MP6
- Att
end
to p
reci
sion.
Form
ulat
e pr
ecise
exp
lana
tions
(ora
lly a
nd in
writt
en fo
rm) u
sing
both
mat
hem
atica
l rep
rese
ntati
ons a
nd w
ords
Prio
r Kno
wle
dge
•Co
nstr
uct a
per
pend
icul
ar b
isect
or
•Bi
sect
an
angl
e
•Co
nstr
uct a
per
pend
icul
ar
Stud
ents
will
be
able
to:
•Co
nstr
uct t
he in
scrib
ed a
nd c
ircum
scrib
ed c
ircle
s of a
tria
ngle
.
•Pr
ove
that
the
oppo
site
angl
es o
f a q
uadr
ilate
ral i
nscr
ibed
in a
circ
le a
re su
pple
men
tary
.
Poss
ible
task
(s)*
*The
Geo
met
ry to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
CPS Mathematics Content Framework_Version 1.0 Page 85
Plan
ning
Gui
des
Geo
met
ry P
lann
ing
Gui
de –
SY1
2-13
Bi
g Id
ea: C
ircle
s With
and
With
out C
oord
inat
es
*The
Geo
met
ry to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
Topi
c: A
rc L
engt
hs a
nd A
reas
of S
ecto
rsCC
SS-M
Con
tent
Sta
ndar
ds
G-C.
5 De
rive
usin
g sim
ilarit
y th
e fa
ct th
at th
e le
ngth
of t
he a
rc in
terc
epte
d by
an
angl
e is
prop
ortio
nal t
o th
e ra
dius
and
defi
ne th
e ra
dian
mea
sure
of t
he a
ngle
as t
he c
onst
ant o
f pr
opor
tiona
lity;
der
ive
the
form
ula
for t
he a
rea
of a
sect
or.
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…
MP3
- Co
nstr
uct v
iabl
e ar
gum
ents
and
criti
que
the
reas
onin
g of
oth
ers.
Use
obs
erva
tions
and
prio
r kno
wle
dge
(sta
ted
assu
mpti
ons,
defi
nitio
ns,
and
prev
ious
est
ablis
hed
resu
lts) t
o m
ake
conj
ectu
res a
nd c
onst
ruct
ar
gum
ents
MP7
- Lo
ok fo
r and
mak
e us
e of
stru
ctur
e.U
se p
atter
ns o
r str
uctu
re to
mak
e se
nse
of m
athe
mati
cs a
nd c
onne
ct
prio
r kno
wle
dge
to si
mila
r situ
ation
s and
ext
end
to n
ovel
situ
ation
s
MP8
- Lo
ok fo
r and
exp
ress
regu
larit
y in
repe
ated
reas
onin
g.Ge
nera
lize
the
proc
ess t
o cr
eate
a sh
ortc
ut w
hich
may
lead
to
deve
lopi
ng ru
les o
r cre
ating
a fo
rmul
a
Prio
r Kno
wle
dge
•Ci
rcum
fere
nce
and
area
of a
circ
le•
Set u
p pr
opor
tions
Stud
ents
will
be
able
to:
•Fi
nd th
e ci
rcum
fere
nce
and
area
of a
circ
le.
•De
rive,
usin
g sim
ilarit
y, th
e fa
ct th
at th
e le
ngth
of t
he a
rc in
terc
epte
d by
an
angl
e is
prop
ortio
nal t
o th
e ra
dius
.
•De
fine
the
radi
an m
easu
re o
f the
ang
le a
s the
con
stan
t of
prop
ortio
nalit
y.
•De
rive
the
form
ula
for t
he a
rea
of a
sect
or.
•Fi
nd th
e ar
c le
ngth
and
are
a of
sect
ors.
Poss
ible
task
(s)*
Chicago Public SchoolsPage 86
Topi
c: C
oord
inat
e G
eom
etry
of C
ircle
sCC
SS-M
Con
tent
Sta
ndar
ds
G-GP
E.1
Deriv
e th
e eq
uatio
n of
a c
ircle
giv
en c
ente
r and
radi
us u
sing
the
Pyth
agor
ean
Theo
rem
; com
plet
e th
e sq
uare
to fi
nd th
e ce
nter
and
radi
us o
f a c
ircle
giv
en b
y an
equ
ation
.G-
GPE.
4 U
se c
oord
inat
es to
pro
ve si
mpl
e ge
omet
ric th
eore
ms a
lgeb
raic
ally.
(For
exa
mpl
e, p
rove
or d
ispro
ve th
at th
e po
int (
1, )
lies o
n th
e ci
rcle
cen
tere
d at
the
orig
in a
nd c
onta
inin
g th
e po
int (
0, 2
).
Conn
ectio
ns to
Sta
ndar
ds
for M
athe
mati
cal P
racti
ceSt
anda
rds o
f Mat
hem
atica
l Pra
ctice
How
it a
pplie
s…
MP3
- Co
nstr
uct v
iabl
e ar
gum
ents
and
criti
que
the
reas
onin
g of
oth
ers.
Use
obs
erva
tions
and
prio
r kno
wle
dge
(sta
ted
assu
mpti
ons,
defi
nitio
ns,
and
prev
ious
est
ablis
hed
resu
lts) t
o m
ake
conj
ectu
res a
nd c
onst
ruct
ar
gum
ents
MP6
- Att
end
to p
reci
sion.
Form
ulat
e pr
ecise
exp
lana
tions
(ora
lly a
nd in
writt
en fo
rm) u
sing
both
mat
hem
atica
l rep
rese
ntati
ons a
nd w
ords
Prio
r Kno
wle
dge
•Py
thag
orea
n Th
eore
m (a
nd/o
r dist
ance
form
ula)
•Co
mpl
eting
the
squa
re
Stud
ents
will
be
able
to:
•De
rive
the
equa
tion
of a
circ
le g
iven
the
cent
er a
nd ra
dius
usin
g th
e Py
thag
orea
n Th
eore
m.
•Co
mpl
ete
the
squa
re to
find
the
cent
er a
nd ra
dius
of a
circ
le g
iven
by
an e
quati
on.
Poss
ible
task
(s)*
Geo
met
ry P
lann
ing
Gui
de –
SY1
2-13
Bi
g Id
ea: C
ircle
s With
and
With
out C
oord
inat
es
*The
Geo
met
ry to
olse
t inc
lude
s thi
s and
oth
er re
sour
ces a
nd c
an b
e fo
und
onlin
e at
http
s://
ocs.
cps.
k12.
il.us
/site
s/IK
MC/
defa
ult.a
spx
and
on th
e De
part
men
t of M
athe
mati
cs a
nd S
cien
ce w
ebsit
e (c
msi.
cps.
k12.
il.us
).
CPS Mathematics Content Framework_Version 1.0 Page 87
Sample TasksThe following sample tasks provide examples of rigorous tasks that support the expectations of the CCSS-M. Complete Toolsets for grades 6-8 as well as Algebra I and Geometry will be available on our Knowledge Management site: https://ocs.cps.k12.il.us/sites/IKMC and on the Department of Mathematics and Science website: http://cmsi.cps.k12.il.us/.
Chicago Public SchoolsPage 88
Grade 8 Sample Task
Algebra – 2008 Copyright © 2008 by Mathematics Assessment Resource Service
All rights reserved.
40
Sorting Functions
This problem gives you the chance to: ¥ Find relationships between graphs, equations, tables and rules ¥ Explain your reasons On the next page are four graphs, four equations, four tables, and four rules.
Your task is to match each graph with an equation, a table and a rule.
1. Write your answers in the following table.
Graph Equation Table Rule
A
B
C
D
2. Explain how you matched each of the four graphs to its equation.
Graph A ___________________________________________________________
___________________________________________________________________
___________________________________________________________________
Graph B ___________________________________________________________
___________________________________________________________________
_________________________________________________________
Graph C __________________________________________________________
___________________________________________________________________
___________________________________________________________________
Graph D ________________________________________________
___________________________________________________________________
____________________________________________________________________
CPS Mathematics Content Framework_Version 1.0 Page 89
Sam
ple
Task
s
Grade 8 Sample Task
Algebra – 2008 Copyright © 2008 by Mathematics Assessment Resource Service
All rights reserved.
41
Graph A
Equation A
xy = 2
Table A
Rule A
y is the same as x
multiplied by x
Graph B Equation B
y2 = x
Table B
Rule B
x multiplied
by y is equal to 2
Graph C
Equation C
y = x2
Table C
Rule C
y is 2 less than x
Graph D
Equation D
y = x - 2
Table D
Rule D
x is the same as y
multiplied by y
10
Chicago Public SchoolsPage 90
Algebra I Sample Task
Copyright © 2011 by Mathematics Assessment Page 1 Best Buy TicktsResource Service. All rights reserved.
Best Buy Tickets
Susie is organizing the printing of tickets for a show her friends are producing.
She has collected prices from several printers and these two seem to be the best.
Susie wants to go for the best buy
She doesn’t yet know how many people are going to come.
Show Susie a couple of ways in which she could make the right decision, whatever the number.
Illustrate your advice with a couple of examples.
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
Please continue your work on the page opposite
SURE PRINTTicket printing
25 tickets for $2
BEST PRINT
Tickets printed
$10 setting up
plus$1 for 25 tickets
Copyright © 2011 by Mathematics AssessmentResource Service. All rights reserved.
CPS Mathematics Content Framework_Version 1.0 Page 91
Sam
ple
Task
s
Algebra I Sample Task
Copyright © 2011 by Mathematics Assessment Page 2 Best Buy TicktsResource Service. All rights reserved.
Best Buy Tickets (continued)
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
________________________________________________________________________________
Copyright © 2011 by Mathematics AssessmentResource Service. All rights reserved.
Chicago Public SchoolsPage 92
CPS Mathematics Content Framework_Version 1.0 Page 93
Appe
ndix
AppendixResources to Support the Learning Expected by the CCSS-M
LESSON ANALYSIS AND MODIFICATION TOOL
This rubric is endorsed by the Illinois State Board of Education (ISBE).
http://engageny.org/resource/tri-state-quality-review-rubric-and-rating-process/ :: The Tri-State Collaborative (composed of educational leaders from Massachusetts, New York, and Rhode Island and facilitated by Achieve) has developed criterion-based rubrics and review processes to evaluate the quality of lessons and units intended to address the Common Core State Standards for Mathematics and ELA/Literacy.
SITES FOR SAMPLE TASKS THAT SUPPORT THE LEARNING EXPECTED BY THE CCSS-M
http://illustrativemathematics.org : : Site run by William McCallum with the goal of having a task to illustrate every CCSS-M Standard.
http://www.map.mathshell.org.uk : : Mathematics Assessment Project site, with MARS tasks associated with different content and practice standards.
http://www.turnonccmath.com/ : : Site designed to elaborate the "scientific basis" of learning trajectories research and links to the Common Core State Standards. Unpacked descriptors describe students' movement from naive to sophisticated ideas. Identifies: bridging standards; underlying cognitive principles; student misconceptions; strategies and inscriptions; and models, related representations and contexts. Also, each learning trajectory includes a diagram that gives a structural overview of the standards and descriptors within.
PERFORMANCE TASKS THAT INFORM INSTRUCTION: A GUIDE
Source: http://www.noycefdn.org/documents/math/howperformancetasks.pdf; accessed 23 April, 2012
How Performance Tasks are Used to Inform Instruction
Performance assessment opportunities are scheduled periodically throughout the school year to provide formative information to guide instruction. Most often performance tasks are administered the first week that school is in session, a second time at the end of the first semester, a third time at the end of the third quarter and then at the end of the year. The tasks are carefully selected to measure student growth from a pre-determined perspective. The most common perspectives are listed as follows:
1. Identify a big mathematical idea linked to a standard at a grade level. The students are assessed as to whether they understand and utilize that mathematical idea with tasks throughout the year.
2. Some examples might be multiplication at third grade, rational numbers at fifth grade, proportional reasoning at seventh grade or exponential growth at ninth grade.
3. Select specific tasks that measure the learning of students after a specific unit of instruction. These would be selected according to the mathematics of the curriculum taught each quarter. A unit of instruction might involve spatial visualization. The task administered following the teaching of that unit would involve spatial visualization and be tied to the geometric standard on spatial visualization at that grade span.
Chicago Public SchoolsPage 94
4. Select different types of tasks that would measure students' problem solving abilities with non routine, unrelated problems. A set of tasks that focus on different math strands as well as elicit different types of mathematical thinking and analysis are selected. Comparing the success of students in attacking, analyzing, solving and communicating their results as the year progresses is informative.
5. Select a specific strand or mathematical idea that is taught in more depth as a student proceeds through the grades. Mathematically related tasks, appropriate to a grade level, are administered to students at three or four grade levels to see comparison over a vertical slice.
6. The same mathematical task is given to two or three grades in a grade span to assess growth as students proceed through school and become more sophisticated mathematicians. These assessments can chart and compare depth in mathematical understanding.
CPS Mathematics Content Framework_Version 1.0 Page 95
Chicago Public SchoolsPage 96
Department of Math & ScienceOffice of Curriculum and Instruction125 South Clark StreetChicago, Illinois 60603
cps.edu