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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 8 Mathematics Planning Guide - SY12-13 Introduction ...............................................................................................44


TOC continued on next page


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


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


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


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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


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.


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.


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.


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.


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


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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.


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.


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).


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


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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.


� 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.


DRAFT05.22.12

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).


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**


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.


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


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.


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


Intr

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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


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.


� 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.











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.


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).


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).


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).


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).


Plan

ning

Gui

des





� 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).














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.




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).


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).


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).






� 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).





Plan

ning

Gui

des










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.




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).


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).


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).


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.



1. Make sense of problems and persevere in solving them







8. Look for and express regularity in repeated reasoning


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


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


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

).


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


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

).


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


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

).


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


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

).


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)*


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

).


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


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

).


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)*


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

).


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)*


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

).


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)*


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.

Plan

ning

Gui

des

CPS Mathematics Content Framework_Version 1.0









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


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*


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


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

).


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?


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

).


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)*


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

).


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


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

).


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

).


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

).


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)


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

).


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)*


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

).


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

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an

angl

e is

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ortio

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o th

e ra

dius

.

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fine

the

radi

an m

easu

re o

f the

ang

le a

s the

con

stan

t of

prop

ortio

nalit

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rive

the

form

ula

for t

he a

rea

of a

sect

or.

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nd th

e ar

c le

ngth

and

are

a of

sect

ors.

Poss

ible

task

(s)*


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

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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.

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exa

mpl

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rove

or d

ispro

ve th

at th

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int (

1, )

lies o

n th

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rcle

cen

tere

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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

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anda

rds o

f Mat

hem

atica

l Pra

ctice

How

it a

pplie

s…

MP3

- Co

nstr

uct v

iabl

e ar

gum

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criti

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reas

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oth

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obs

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and

prio

r kno

wle

dge

(sta

ted

assu

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defi

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and

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thag

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•Co

mpl

eting

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cent

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nd ra

dius

usin

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e Py

thag

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iven

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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

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*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

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athe

mati

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cien

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ebsit

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msi.

cps.

k12.

il.us

).


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/.


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 ________________________________________________

___________________________________________________________________

____________________________________________________________________


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


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.


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.


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.


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.

Department of Math & ScienceOffice of Curriculum and Instruction125 South Clark StreetChicago, Illinois 60603

cps.edu

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