## CPM Glossary

Special | A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X | Y | Z |

**ALL**

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## 5 |
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## 5 Practices for Orchestrating Productive Mathematics DiscussionsThe 5 Practices for Orchestrating Productive Mathematics Discussions provides teachers with concrete guidance for engaging students in discussions that make the mathematics in classroom lessons transparent to all. | ||

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## 6 Word Synthesis1. Read and mark up the text to gain an understanding of the ideas and applications. 2. Synthesize your ideas about the reading into only six words. Your six words could be a sentence, phrase, connection, personal learning, or an Aha. 3. Record your six words for presentation to the group. 4. Be prepared to connect your six words to content in the text. | |

## A |
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## Additional StrategiesAttached is a list of STTS. Some teachers make a poster of these for their classrooms and place a checkmark next to those that they have tried so that their students become familiar with these strategies. | |

## Ambassador
Students are eligible to be Ambassadors once the team has finished problem solving and the teacher has assessed for understanding. An Ambassador is sent to work with other teams to support productive struggle. The Ambassador asks the team questions to guide understanding during problem solving. Teacher appoints Ambassador Ambassadors help other teams
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## Anticipating- Do the problem yourself.
- What are students likely to produce?
- Which problems will most likely be the most useful in addressing the mathematics?
Anticipating is Step 1 of the 5 Practices for Orchestrating Productive Math Discussions. | |

## Assessment Practices Outcome 1Understand CPM and NCTM assessment documents and connect them to instructional practice and assessment decisions | ||

## Assessment Practices Outcome 2Reflect on and make connections between formative assessment and instructional strategies | ||

## Assessment Practices Outcome 3Utilize CPM’s assessment tools and resources | ||

## Assessment Practices Outcome 4Identify appropriate formative and summative assessment topics and strategies for each chapter based on the learning progression | ||

## Assessment Practices Outcome 5Understand the purpose and value of team tests | ||

## Assessment Practices Outcome 6Develop feedback and expectations for all forms of assessments | ||

## Attend to precisionMathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently and express numerical answers with a degree of precision appropriate for the problem context. | |

## B |
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## Board Report
Teacher monitors student progress while students self-assess work and increase mathematical discourse. This is recommended for questions with short solutions, not for all questions from a lesson. If a problem requires choosing a tool and setting up an equation with many steps to solve, it is best if teams only report the end solution, or part of the solution. Teacher creates a space in the classroom to write a row of problem numbers from the lesson. When teams get to the problem listed on the board report, the team writes their answer on a sticky note. A student from the team goes to the board to place the sticky note and compare to other teams. Teacher monitors student work on board and through circulation. Based on work, teachers may ask specific teams to do a Swapmeet, or I Spy. Repeat this process for each problem listed on the board, with a new student placing the sticky note each time.
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## Build Procedural Fluency from Conceptual UnderstandingOne of the eight Mathematics Teaching Practices from Principles to Actions that needs to be a consistent component of every mathematics lesson. Effective teaching of mathematics builds fluency
with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using
procedures flexibly as they solve contextual and mathematical problems. | |

## C |
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## Carousel: Around the World
Teams explore topics or questions displayed on poster paper around the classroom. After a brief discussion—two or three minutes, teams agree on a written statement to add to the poster. Teams rotate several times to discuss additional topics or questions. Teams read the previous written statements before adding to the list. Teacher monitors and determines when to conclude the activity. A Gallery Walk closure provides students time to read all of the written statements. Display topics or questions around the classroom. Provide a different colored marker for each team. Assign one team to each topic or question to start. Teams discuss and agree on a written statement to include about the topic. Teams rotate to the next topic or question and repeat the process every few minutes. - For closure, facilitate a Gallery Walk to view all topics or questions.
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## Carousel: Index Card
Teachers or students write one struggle about learning mathematics including time management, Review and Preview, partner work, teamwork, etc., on separate index cards. The index card rotates to other students that offer suggestions to support the struggles. Students record one struggle/question/comment/concern on an index card. Index card rotates within a team of students or to the next team. Students or teams write suggestions on the index card. Rotate the index card several times. Index card is returned to the original student or can be displayed in class for all to benefit from.
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## Carousel: Station Rotation
Stations include review problems—possibly four to six—placed into a sheet protector. There should be more stations than teams. Teams record written explanations on a prepared sheet—in numerical order—to manage teacher review of work. After teams have completed a written explanation for a station, the paper is submitted to the teacher. Teams rotate to an available station. Stations include several review problems. Set up more stations than teams. Teams record written explanations on a prepared record sheet. Teams check in with the teacher. Teams rotate to an available station. Repeat until time is up or stations are complete.
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## Checkpoint Problem These problems have been identified for determining if students are building skills at the expected level. Checkpoint problems are designed to support students in taking responsibility for the development of their own skills. When students find that they need help with these problems, worked examples and practice problems are available in the Checkpoint Problems section at the back of their book. | ||

## Checkpoint Problems These problems have been identified for determining if students are building skills at the expected level. Checkpoint problems are designed to support students in taking responsibility for the development of their own skills. When students find that they need help with these problems, worked examples and practice problems are available in the Checkpoint Problems section at the back of their book. | |

## Cognitive Delays in ProcessingStudents who have the ability to engage with the mathematics but need more time and supports may struggle to keep up.
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## Collaborative LearningResearch says students learn ideas more deeply when they discuss ideas with classmates. Collaborative learning is evident in a classroom when - Students and teachers are aware of the purpose for and value of working in teams.
- Students and teachers are familiar with team norms and roles.
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## Concept GoalsConcept Goals are those focused on the mathematics students learn. Teachers may have particular success criteria attached to these Learning Goals that explain how and when students may demonstrate their proficiency level. | ||

## Connect-Extend-Challenge protocolThis is a reading strategy used for longer passages where students are asked to make connections to things they already know, extend their thinking by finding new ideas in the material, identify ideas that challenge them, and then share these with their group or the class. | ||

## Connecting- Craft questions to make the math visible.
- Compare and contrast 2 or 3 students' work. What are the mathematical relationships?
- What do parts of a student's work represent in the original problem? The solution? Work done in the past?
Connecting is Step 5 of the 5 Practices for Orchestrating Productive Math Discussions. | |

## Construct viable arguments and critique the reasoning of othersMathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. | ||

## Content GoalsA content goal is linked to a specific content standard or objective. A content goal example might be "creating equations and inequalities in one variable and using them to solve problems." | ||

## Core ProblemsIf time is limited, use these problems to meet the lesson objectives and the Common Core State Standards. A problem that is not listed as part of the core is either an extension, an opportunity for deeper understanding, or further practice. Core Problems for each lesson are listed in the Teacher Notes. | |

## Course NotebookYour course notebook is the place where you record solutions to all of your classwork and Review & Preview problems. Some teachers ensure that their written solutions are complete with the intent of sharing them with their students. Some teachers use this area to take notes about formative and summative assessments, including questioning. You will also want to think about how your students should organize their own Course Notebook. How will you support your students with their notebook organization throughout the school year? | |

## CPM Principles of AssessmentTeachers understand that students learn at different rates and through different experiences. The
CPM materials have been designed to support mastery over time through a student-centered,
problem-based course, and this approach supports students’ different learning styles. But when
changing the materials and changing the methodology, teachers must also change their
assessment practices. Teachers cannot tell students they want them to explain their thinking
during class and then assess them with only a multiple choice test. Students will quickly realize
that “explaining” is not valued enough to be given the time to be assessed. | ||

## CPM WorkshopsCPM Workshops are a partnership created with teachers and site administration to improve instruction through specialized workshops and coaching. | |

## CPM's Position Paper on AssessmentTo ensure all students are afforded the same opportunities for appreciation and success, CPM researches the best practices to support learning. It is on this research that CPM has based its philosophy and methodology for the position paper on assessment. | ||

## CPM's Position Paper on HomeworkCPM's philosophy and methodology surrounding homework. The Review & Preview portion of each lesson is CPM's opportunity for independent practice. | |

## CPM's Three Pillars of ResearchIn the seven years since the original CPM Research Report was posted, the new research has continued to validate the efficacy of the three pillars of CPM pedagogy: - Students learn ideas more deeply when they discuss ideas with classmates.
- Students learn ideas more usefully for other arenas when they learn by attacking problems—ideally from the real world.
- Students learn ideas more permanently when they are required to engage and re-engage with the ideas for months or even years.
These three principles (termed respectively as collaborative learning, problem-based learning and mixed, spaced practice) have driven the development of the CPM textbooks from the beginning, and each year these principles are validated by more research to prove their effectiveness. | ||

## D |
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## DashboardThe dashboard is a virtual location where CPM users can interact with the Learning Management System. It provides at-a-glance views of instructional element outcomes progress. | |

## Deficit MindsetDeficit thinking refers to the notion that students (particularly low income, minority students) fail in school because such students and their families experience deficiencies that obstruct the learning process (e.g. limited intelligence, lack of motivation and inadequate home socialization). Deficit Thinking does not Support Productive Struggle in Learning Mathematics. A deficit mindset often results in Educators rescuing students from difficult tasks, and removing the opportunity for productive struggle. “Teachers sometimes perceive student frustration or lack of immediate success as indicators that they have somehow failed their students. As a result, they jump in to “rescue” students by breaking down the task and guiding students step by step through the difficulties. Although well intentioned, such “rescuing” undermines the efforts of students, lowers the cognitive demand of the task, and deprives students of opportunities to engage fully in making sense of the mathematics” Principles to Action, pg. 48
Mindsets must shift about what it means to be “successful” in mathematics. Productive struggle should be considered as a valuable part of the learning process. “Mathematics classrooms that embrace productive struggle necessitate rethinking on the part of both students and teachers. Students must rethink what it means to be a successful learner of mathematics, and teachers must rethink what it means to be an effective teacher of mathematics. “Principles to Action, pg. 49
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## Depth of Knowledge
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## Descriptive, Effective FeedbackGood feedback improves student learning. It has the following qualities: Specific: It is a tool for future change. Ask yourself, "What worked?" or "What does the student understand?" Then ask, "What needs improvement?" Actionable: Emphasize what could be done differently rather than what is wrong. Actionable feedback is often in the form of a question. "How could you have justified this differently?" Timely: The most effective feedback is immediate and frequent. How can your feedback be timely for both formative and summative work? Respectful: Make an effort to look for the good while still focusing on future changes. How can this work be an asset for future learning not just for this student but other students? | ||

## Discussion PointsSome lessons include questions embedded in the task that study teams should use to guide their discussions, investigations, and problem-solving processes. | |

## Dyad
Student think-alouds are used in a variety of ways. For example, partners take turns talking about feelings of returning to school. Or, students express concern about math topics on upcoming assessments. A Dyad allows students time to talk without interruption. Each student receives equal time. The listener does not talk; a Dyad is not a conversation. Students maintain eye contact and positive body language. Students share—without interruption—with a partner. Each partner shares for an equal amount of time. Listening partner remains quiet and uses positive body language.
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## E |
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## Elevator Talk
Students summarize a concept or topic in a 30-second, relatively short, talk. Basically, if someone gets onto an elevator and asks about a problem, you have until the elevator gets to your floor to explain it. Use Proximity Partners or a Whiparound to share out the summaries. Each individual/partner/team is given a topic or concept. Students summarize the topic or concept into a brief presentation. Teacher facilitates a share out of presentations.
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## Elicit and Use Evidence of Student ThinkingOne of the eight Mathematics Teaching Practices from Principles to Actions that needs to be a consistent component of every mathematics lesson. Effective teaching of mathematics uses evidence of student
thinking to assess progress toward mathematical understanding and to adjust instruction continually in ways that
support and extend learning. | ||

## Error Analysis
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## Establish Math GoalsOne of the eight Mathematics Teaching Practices from Principles to Actions that needs to be a consistent component of every mathematics lesson. Effective teaching of mathematics establishes clear goals for the mathematics that students are learning, situates goals within learning progressions, and uses the goals to guide instructional decisions. | |

## Establish Mathematics Goals to Focus LearningOne of the eight Mathematics Teaching Practices from Principles to Actions that needs to be a consistent component of every mathematics lesson. Effective teaching of mathematics establishes clear goals
for the mathematics that students are learning, situates goals within learning progressions, and uses the goals to
guide instructional decisions. | |

## eToolTechnology tools that provide a way to interact with lessons electronically. The Teacher Notes offer suggestions for incorporating this technology into the classroom experience. Some eTools are intended to be used as part of a whole-class demonstration, while others are meant for individuals or pairs of students to use while exploring. | ||

## Executive FunctionExecutive function is a set of mental skills that include working memory, flexible thinking, and self-control. We use these skills every day to learn, work, and manage daily life. Executive functions have very limited capacity due to working memory. | ||

## Executive Summary of ResearchA summary of the research that supports the Three Pillars of CPM - collaborative learning, problem based learning and mastery over time. | |

## F |
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## Facilitate Meaningful Mathematical DiscourseOne of the eight Mathematics Teaching Practices from Principles to Actions that needs to be a consistent component of every mathematics lesson. Effective teaching of mathematics facilitates discourse
among students to build shared understanding of mathematical ideas by analyzing and comparing student
approaches and arguments. | ||

## FacilitatorThe Typically, a teacher could expect to hear a Facilitator asking:
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## Final Say ProtocolA reading strategy where team members read a selection, highlight important information, and select 3 points/quotes that stood out as most important. In small teams, the first person leads by reading their quote aloud. Others take turns responding individually to the quote shared. First person finally explains why they chose the quote, what it meant to them, and any new connections or new thinking that stemmed from the responses of the other team members. Sharing moves to the other group members in the same manner. | |

## Fishbowl
Teacher facilitates one team modeling an activity for the whole class. Students watch the team collaborate on a task while the teacher highlights positive behaviors and team norms. Questions may be asked about the goals for the lesson. Classmates stand in a circle around one team that models an activity. Teacher highlights positive behaviors and team norms. Questions may be asked. Students return to teams and complete activity.
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## Formative AssessmentAll of those activities undertaken by teachers and/or by their students, which provide information to be used as feedback to modify the teaching and learning activities in which they are engaged (Black and Wiliam 1998). Formative assessment can be viewed as having two parts: checking for understanding and an action taken based on that check. Both parts can be accomplished by both teachers and students. However, the teacher usually orchestrates effective formative assessment. | ||

## Fortune Cookie
Teams receive five or six sentence starters (fortunes) related to topics/concepts/norms in an envelope. Team Member (1) draws a sentence starter from the envelope, reads it aloud, and shares a brief—30 second—explanation. Team Member (2) receives the same sentence starter, makes one comment about Team Member (1)'s explanation, and rotates to continue the process for each member. Team member (2) draws a sentence starter from the envelope, reads it aloud, and shares. Team member (3) receives the same sentence starter, makes one comment about Team Member (2)'s explanation. Continue this rotation for each of the sentence starters in the envelope. Teacher circulates to make instructional decisions about team discussions. For closure, each sentence starter (fortune) is included on a separate poster—not previously mentioned to the students. Teams rotate through each poster location to add one comment/explanation about the sentence starter. Teams receive sentence starters (fortunes) in an envelope. Team Member (1) reads one sentence starter and shares a brief explanation. Team Member (2) receives the same sentence starter and comments on Team Member (1)'s explanation. Rotate to each team member. When complete, Team Member (2) reads one sentence starter and shares. Team Member (3) comments on Team Member (2)'s explanation. Continue the rotation through all sentence starters (fortunes). Posters with each fortune are displayed around the room after the activity is complete. Each team adds one comment to each poster for closure.
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## Four “A’s” ProtocolA reading strategy where the group reads the text silently, highlighting and writing notes in the margin on post-it notes in answer to these four questions: What assumptions does the author of the text hold? What do you agree with in the text? What do you want to argue with in the text? What parts of the text do you want to aspire to? Discussion then occurs within the group to talk about the text in light of each of the “A’s”, taking them one at a time. What do people want to argue with, agree with, and aspire to in the text? | |

## Four Corners Jigsaw
A study team and teaching strategy where teams work collaboratively to understand a large quantity of information. There are various ways to organize the jigsaw activity, but the central concept is that teams of people are assigned or select topics that they teach to others. The teams decide collectively how they are going to share what they know. One way to acquire knowledge from a large amount of material is to break it into smaller pieces. Each student becomes an expert for part of the material and then shares their knowledge with their team. | |

## Fourth CPM Principle of AssessmentFormative assessment is a learning experience for both the student and the teacher. | ||

## Further GuidanceSome lessons include additional support for students immediately following the task statement. This section of the lesson has step-by-step instructions for students to follow. Several large investigations in this course will have this structure. This design allows the teacher either to have teams attack the problem using their own strategies and available tools or to have students follow a more directed approach using Further Guidance. The beginning and end of each Further Guidance section is clearly marked. | |

## G |
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## Gallery Walk
Teams display posters or presentations. Students explain and critique as individuals/teams rotate about the classroom. Rotations are completed quietly—Museum Walk—or through discussion—Gallery Walk. If the teacher decides to allow feedback, students provide positive feedback Two Stars and a Wish or Glow and Grow. Teams display posters or presentations. Students explain and critique displayed work. Students rotate to each location. Feedback is given with Two Stars and a Wish or Glow and Grow.
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## Gaps in Understanding/LearningStudents who understand the bigger picture of the mathematics but have gaps in the skills necessary to complete the task, or students who possess discrete skills but do not understand how to put their skills to use, will struggle to progress.
Does the student need additional learning opportunities to fill in learning gaps? | |

## Give One - Get One
Students explain and critique ideas with members of the class. For example, students write three ideas on separate note cards for creating positive team norms. Students circulate to give one idea to a classmate, while they get one idea. Student names are recorded next to the idea. For closure, a volunteer reads an idea from a classmate, and then the named person continues to share another idea. Allow many to share. Students record three ideas to share about a given topic. Students circulate and share ideas. For each idea the student gives, they get one in return to record on paper - including the name of the student who gives the idea. After many ideas are gathered, the teacher asks a volunteer to read an idea from a classmate and their name. Named classmate then shares the idea of another classmate and the sharing process continues.
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## Glow and Grow
Students use think time to write one topic, team norm, or idea that is a strength and one topic, team norm, or idea where improvement is needed. Students use Glow and Grow to provide feedback team posters, topics, team roles, team norms, assessment, goals. Students share one topic, team norm, or idea that is a strength—Glow. Students share one topic, team norm, or idea where improvement is needed—Grow.
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## Go-Around One ProtocolA reading strategy where persons in the group read the text silently, highlighting or using Post-it notes to identify those parts in the text that raise questions, confirm beliefs, cause “aha” thoughts, conflict with beliefs, cause reconsideration of prior assumptions and/or show constraints of the problem or topic. One person reports one idea that he or she recorded while other group members listen, but do not question. The next person does the same until all group members have reported. The group discusses ideas that were reported. | |

## Golden Line ProtocolA reading strategy where persons in the group read the text silently, highlighting or using Post-it notes to identify those parts in the text that raise questions, confirm beliefs, cause “aha” thoughts, conflict with beliefs, cause reconsideration of prior assumptions and/or show constraints of the problem or topic. Each person then chooses two different “Golden Lines” that they want to share with the group. Taking turns, members direct others to their line, reading it and explaining the significance. Once everyone has shared, the whole group discusses together. | |

## GPS
To help teams navigate through a problem set, the GPS strategy is a visual road map of the tasks. Students know the final destination to achieve the goals for the lesson. One member of the team reports the progress on a one-quadrant grid displayed in the room, where the team travels along one axis and the problem numbers travel the other axis. Teachers adjust instruction throughout the lesson monitoring where each team is located. Teachers may send Ambassadors from teams that are further along in the lesson to deepen understanding, or they may utilize a Swapmeet with teams that are similarly located on the grid. If all teams land at one-point on the grid—unable to continue—the teacher may conduct a Huddle to progress teams past that point. A one-quadrant grid is displayed for the class—on whiteboard, reusable laminated grid, paper or electronic. Teams are informed of the goals of the lesson included within each problem. One team member checks off progress on the grid as each problem is completed. Teacher makes instructional decisions about STTS used to support teams and/or closure.
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## Guiding PrinciplesThese beliefs guide CPM throughout course implementation which are rooted in research about collaborative learning, problem based learning, and mastery over time. - Students' involvement in effective study teams increases their ability to learn mathematics.
- Students have significantly better retention of mathematics when concepts are grounded in context.
- Students deepen their mathematical understanding when they engage with concepts over time.
- Effective study teams are guided, supported, and summarized by a reflective, knowledgeable teacher.
- Assessing what students understand requires more than one method and more than one opportunity.
- When students and stakeholders embrace a growth mindset, they understand that mastery takes time, effort, and support.
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## H |
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## Homework Help | |

## Hosted Gallery WalkA Gallery Walk is a STTS where teams display posters or presentations. Students explain and critique as individuals/teams rotate about the classroom. Rotations are completed quietly—Museum Walk—or through discussion—Gallery Walk. If the teacher decides to allow feedback, students provide positive feedback Two Stars and a Wish or Glow and Grow. - Teams display posters or presentations.
- Students explain and critique displayed work.
- Students rotate to each location.
- Feedback is given with Two Stars and a Wish or Glow and Grow.
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## Hot Potato
Students practice concepts before mastery or review multiple step problems. Teams receive one sheet of paper. Each team member is provided a different colored writing utensil. Team Member (1) records the first step of the strategy, and rotates the paper to Team Member (2). Team Member (2) corrects mistakes, explains the step aloud, and records the second step. The rotation continues until the problem is completed. Teacher decides if each team member signs the paper agreeing to everything that was written down. Team uses one sheet of paper to solve a problem or practice a procedure. Each team member uses a different colored writing utensil. Team member (1) records the first step of the strategy. The paper rotates to the next team member to correct mistakes and explain and write the next step. Continue the rotation until the problem is completed.
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## Hot Seat
Students demonstrate individual accountability by working on a problem individually away from team members. While mastery of a concept is not yet expected when using this strategy, students have worked with similar problems through collaboration in teams. One chair is placed in the front of the classroom for each team. Team Member (1) sits in the chair to work on the same problem as other team members. Team Member (1) remains silent while teams discuss and record strategies. Teacher assesses both Team Member (1)'s work and teamwork. Two points are assigned for Team Member (1) and one point for correct team responses. Team Member (2) rotates to the Hot Seat to continue the strategy. One chair—Hot Seat—is placed in the front of the classroom for each team. Using Numbered Heads, Team Member (1) sits in the Hot Seat. Teacher gives everyone a problem to work on for a specified amount of time. Hot Seat team member (1) remains silent, but teams discuss and record strategies. Teacher assesses individual and team explanations. Hot Seat team member (1) receives two points for correct answers. Teams receive one point for correct answers. - Team Member (2) rotates to the Hot Seat to continue the strategy.
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## Huddle
Teacher monitors the learning progression of teams, disseminates information, or seeks consensus on topics or concepts. Teacher calls one team member to a Huddle in the classroom. Teacher shares a piece of information or checks for understanding. Team member (1) returns from the Huddle to share with other team members. One team member is called to a location in the classroom. Teacher shares a piece of information or checks for understanding. Team member (1) returns to the team and shares information.
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## I |
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## I Have...Who Has...
To review, build vocabulary, connect mathematical representations, or connect mathematical threads, Teacher leads I Have...Who Has... Student receives a card with one problem and one answer. Student (1)—starter card—states, "Who has...[problem]." Student (2)—with the solution—says, "I have...[answer]." This continues throughout the set of cards. This strategy may be modified for independent practice, partners, or teams. Consider time restrictions and multiple rounds. Student receives a card with one problem and one answer to a different problem. Student (1) asks, "Who has..." and states the problem. Student (2)—with the solution—says, "I have..." and states the answer. Process continues until all problems have answers.
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## I Spy
A team becomes stuck while problem solving. The Resource Manager becomes a spy for the team and silently circulates to listen for ideas from other teams. The Resource Manager refrains from any team interactions while spying on other teams. The Resource manager reports back to the team to share information about the problem. Team becomes stuck at one point while problem solving. The Resource Manager circulates around the classroom, silently listening to, or spying on the teams' work. The Resource Manager reports back to the team to share information about the problem.
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## I Used to Think..., Now I Think... ProtocolA reading strategy where group members read text and then reflect using “I used to think...” and “Now I think…” Responses are shared with partners, groups and/or the whole class. | |

## Implement Tasks that Promote Reasoning and Problem SolvingOne of the eight Mathematics Teaching Practices from Principles to Actions that needs to be a consistent component of every mathematics lesson. Effective teaching of mathematics engages
students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow
multiple entry points and varied solution strategies. | ||

## Implementation Support VisitsAlso known as ISVs, these classroom visits are conducted by a trained and experienced CPM specialist using CPM's Implementation Progress Tool in order to provide individual support, non-evaluative feedback, and an opportunity for teacher reflection. Each new teacher is eligible to receive up to two visits per year during the first two years of implementation. | |

## Instructional Strategies Outcome 1Apply knowledge of NCTM’s Mathematics Teaching Practices and connect them to instructional strategies | ||

## Instructional Strategies Outcome 2Implement instructional strategies that support the Standards for Mathematical Practice | ||

## Instructional Strategies Outcome 3Understand how multiple modes of instruction ensure access for all students | ||

## Instructional Strategies Outcome 4Establish and reinforce routines and roles that clearly define expectations for multiple modes of instruction | ||

## Instructional Strategies Outcome 5Understand how intentional circulation and purposeful questioning provide feedback to students and teachers | ||

## Instructional Strategies Outcome 6Experience and reflect on instructional strategies through model lessons and a typical day | ||

## Instructional Strategies Outcome 7Know that lesson closure provides opportunities for students to make connections among key mathematical ideas and provides opportunities to reflect on the math goal | ||

## Instructional Strategies Outcome 8Understand the importance of using a variety of instructional strategies and activities to engage students in chapter closure | ||

## Intentional Planning Outcome 1Create, implement and reflect on purposefully planned CPM lessons | ||

## Intentional Planning Outcome 10Integrate Desmos and other eTools into purposefully planned lessons that engage students with content | ||

## Intentional Planning Outcome 2Create, implement, reflect on and revise an Implementation Action Plan that will guide classroom procedures and expectations | ||

## Intentional Planning Outcome 3Incorporate suggested Universal Access strategies to support all students | ||

## Intentional Planning Outcome 4Plan for intentional use of instructional strategies that support formative assessment | ||

## Intentional Planning Outcome 5Use the course preparation resources to inform individual school decisions | ||

## Intentional Planning Outcome 6Plan each chapter using the opening teacher notes | ||

## Intentional Planning Outcome 7Plan for intentional use of instructional strategies that support status and equity | ||

## Intentional Planning Outcome 8Incorporate multiple modes of instruction to support all learners | ||

## Intentional Planning Outcome 9Utilize the Implementation Progress Tool to reflect on student learning and instructional strategies | ||

## IREThe most common type of teacher to student discourse is IRE. The teacher | ||

## J |
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## Jigsaw
Each team member becomes an expert for one part of a topic or concept. Large reading assignments are broken into four parts, and each member of the team receives one part of the reading. Use Numbered Heads to assign a number to each team member. Team Member (1) learns about one part and prepares to share that learning. Team Members (2), (3), and (4) repeat this same process.
Each team member becomes an expert for one part of the material presented. Each team member reports to the corner assigned, reads the assigned part, discusses with other students in that corner, decides on an explanation, and returns to the original team. Team members then take turns sharing with their team. Each team member is assigned a different part of a topic or concept. Team member (1) learns about the topic or concept. Team member (1) presents the information to the team. Team Members (2), (3), and (4) repeat this same process.
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## L |
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## Lack of Mathematical ConfidenceA student who lacks the ability to produce the desired result or perceives they lack the ability to do so, will be less likely to try when failure is certain.
Does the student have a fixed mindset or in rare cases a significant learning disability? Are there a variety of opportunities and methods for students to demonstrate their mathematical understanding? | |

## Lack of MotivationStudents may appear to be unmotivated when they have several root causes of unproductive struggle. Additionally, a student’s priorities may lead to lack of motivation in class.
What matters to this student? Is there a way to relate the problem to something he or she cares about, or allow them to use their talent/interest in a way that benefits the team? | |

## Learning GoalsLearning Goals specify the learning that is intended for a lesson. Learning goals are usually restricted to a single lesson and may refer to understanding (i.e. a portion of the Lesson Objective), knowledge, skills, or applications. They may also reference a process for doing math such as the Standards for Math Practice, or behaviors such as modeling quality collaboration. These goals may use words such as know, develop, become fluent, apply, understand, use, or extend. They are often accompanied by success criteria. They can also be identified by their function: concept goals, process goals, or product goals. | ||

## Learning Log
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## Learning Management SystemAlso referred to as the LMS, is a software application system that monitors professional development for CPM users; keeps track of progress; allows interaction with others and CPM specialists. | ||

## Listening Post
In teams, two team members are mathematicians and two team members are observers. Team Member (1) and Team Member (2) problem solve, sharing explanations aloud. Team Member (3) listens to Team Members (1) and (2) and asks clarifying questions, as needed. Team Member (4) records observations about explanations and attitudes of participants, but Team Member (4) remains silent throughout the activity. After the assigned time—15 minutes—Team Member (4) shares notes and observations. Team Members (1), (2), and (3) may share their perspectives, as well. Variations of this activity include multiple rounds with the roles rotated to other members. Team Member (1) and Team Member (2) work on a math problem, explaining aloud. Team Member (3) listens and may ask clarifying questions. Team Member (4) only records what is discussed and verbalized—observes body language, team norms, or another focus area. After 15 minutes, Team Member (4) shares notes and observations.
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## Literacy TabA section in the Teacher tab of the Navigation Bar in the eBook that gives teachers strategies to support students struggling with the English language, specifically reading and writing; includes Introduction, Literacy Guide, Student Strategies, Team Strategies, and Reading Strategies. | |

## Look for and express regularity in repeated reasoningMathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation ( | |

## Look for and make use of structureMathematically proficient students look closely to discern a pattern or structure. Students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression | |

## M |
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## Make sense of problems and persevere in solving themMathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. | ||

## Mastery Over TimeCPM's Mixed, Spaced Practice provides students an opportunity to achieve conceptual understanding over time. Students must have the opportunity to engage meaningfully with and make sense of concepts before they are expected to have mastery. | |

## Math Chat
Students participate in the silent activity to reflect, summarize ideas, generate ideas, assess learning, or solve problems. Display posters with one topic or concept on each poster. Students use a writing utensil and circulate to each poster. Student (1) adds one brief note or explanation to the poster. Time for activity varies depending on the topic. You may want to consider using a timer to help pace the time at each poster. Display posters with one topic or concept per poster. Student has one writing utensil. Silently, Student (1) circulates to each poster, writing a brief note or explanation on each one. After rotation is complete, students return to seats.
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## Math NotesAppearing routinely throughout the text, Math Notes consolidate core content ideas, provide definitions, explanations, examples, instructions about notation, formalizations of topics, and occasionally interesting extensions or applications of mathematical concepts. These boxes enable students to reference ideas that they missed or have forgotten. | |

## Mathematical Content Outcome 1Experience team-worthy math problems | ||

## Mathematical Content Outcome 2Work through lessons to understand how the learning progressions support the coherence of the program | ||

## Mathematical Content Outcome 3Identify and provide opportunities for students to make sense of the math goal throughout the lesson | ||

## Mathematical Content Outcome 4Experience how a conceptual understanding of math leads to procedural fluency | ||

## Mathematical Content Outcome 5Understand the use of mathematical strategies, structures and tools to develop conceptual understanding | ||

## Mathematical Content Outcome 6Engage with the opening and closure activities through the chapter snapshot | ||

## Mathematical Content Outcome 7Experience Desmos activities and eTools that support conceptual understanding | ||

## Mathematical Content Outcome 8Understand the value and purpose of chapter 1 | ||

## Mathematical Content Outcome 9Deepen their own content knowledge | ||

## Mathematics AgencyMath Agency is Math Identity in action and the presentation of one’s identity to the world. | ||

## Mathematics IdentityMathematics identity includes beliefs about one’s self as a mathematics learner, one’s perceptions of how others perceive them as a mathematics learner, the nature of mathematics, engagement in mathematics, and perception of self as a potential participant in mathematics. | ||

## Mathematics Teaching PracticesResearch indicates that these eight practices need to be consistent components of every mathematics lesson - Establish Mathematics Goals to Focus Learning
- Implement Tasks that Promote Reasoning and Problem Solving
- Use and Connect Mathematical Representations
- Facilitate Meaningful Mathematical Discourse
- Pose Purposeful Questions
- Build Procedural Fluency from Conceptual Understanding
- Support Productive Struggle in Learning Mathematics
- Elicit and Use Evidence of Student Thinking
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## Metacognition IconAlthough a metacognitive approach is embedded throughout CPM's curriculum, many lessons in the Pre Calculus text contain specific prompts that encourage students to stop and reflect on becoming a better learner. | |

## Micro Lab ProtocolA reading strategy where group members read a text and then share individually for two minutes while other members listen attentively without comment or interruption. Pause for 30 seconds of silence to take in what was said. This is repeated and then a group discussion occurs referencing the comments that have been made and making connections between the responses. | |

## Mixed, Spaced PracticeResearch says students learn ideas more permanently when they are required to engage and re-engage with those ideas for months or even years. Mixed, Spaced Practice is evident in a classroom when - both individual lessons and chapters are followed, using suggested pacing.
- Review & Preview problems are assigned and valued as an essential part of learning.
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## Model with mathematicsMathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. | |

## Monitoring- Listen, observe, identify key strategies
- Keep track of approaches
- Ask questions of students to get them back on track or to think more deeply
Monitoring is Step 2 of the 5 Practices for Orchestrating Productive Math Discussions | |

## More Knowledgeable OtherDescribes the role of the teacher as proactively supporting students' learning through co-participation. Stresses the importance of designing learning environments that support problematizing mathematical ideas, giving students mathematical authority, holding students accountable to others and to shared disciplinary norms, and providing students with relevant resources (Engle & Conant, 2002). | |

## Multiple Modes of InstructionTeachers use a variety of instructional strategies to engage students in teamwork, partner work, individual work, teacher-led discussions, presentations, and more. Multiple modes of instruction provides differentiated learning opportunities for student engagement in a collaborative classroom. | |

## MULTIPLE WAYS TO ACCESS INFORMATION AND KNOWLEDGEThis UDL principle connects to the Second Math Teaching Practice: Practice 2: Implement tasks that promote reasoning and problem solving. Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies. | |

## N |
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## Notice & Wonder
Students view a picture, math problem, peer work, favorite mistake. Students critique a team poster, observe a team for their teamwork, etc. Students are prompted with the questions—What do you notice? and What do you wonder? Student (1) receives a topic, picture, piece of work, math problem, sample student work, reading, etc. Complete the prompt: I notice... Complete the prompt: I wonder…
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## Numbered Heads
Numbered Heads is an effective strategy for keeping all students involved and accountable during teamwork. Students number off in teams. Then they are given a problem to solve, a question to answer, or any task to complete. The team members work together making sure that each student in the team understands what they are doing, knows the answers, and can explain the team's work. Students number themselves 1 - 4 in their team. Each team member is assigned a team role, task, or problem number that they are responsible for. Teacher circulates, stops at a group, and asks a random number to answer a checking for understanding question. The Teacher can use a random number to have students share out at the end of class.
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## P |
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## Pairs Check
Use when practicing a new skill or procedure, or to pre-assess a topic that will soon be taught. Within each team of four, students work in pairs to solve problems and then check solutions with the other pair. Each set of partners has one sheet of paper and one pencil. While one student writes, the other student explains. If the student writing disagrees with the explanation, then a discussion happens before the step is recorded on the paper. When finished the role of writer is rotated to the other student, and the process continues. After problems are complete, the partners check the explanations of the other team members. If both pairs agree, a checkmark is added to the paper. If pairs disagree, teams conduct error analysis. Team Member (1) writes while Team Member (2) explains the first problem. Team Member (1) asks clarifying questions to Team Member (2). The partners check with the other partners from the team—if they agree, put a ✅, if they disagree, find mistakes. Team Member (1) rotates the paper to Team Member (2), and roles are reversed for the next problem.
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## Parent Guide with Extra PracticeThe | |

## Participation Quiz
During a team worthy task, the teacher picks a class norm on which to focus and provide students with feedback. The teacher explains to students how the feedback will be presented (rubric, laminated feedback cards, overhead, posters, chalkboard, electronic, etc.). The teacher may want to pause to debrief in the middle and end of class, or just debrief at the end of the class period. Teacher displays the team norm that will be the focus of the lesson. Teacher explains to teams how they will monitor teamwork. Teacher records comments while students are working. Debrief together as a class.
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## Peer Edit
This strategy allows peers to give each other feedback either orally or in writing. It can be used when writing Toolkit or Learning Log entries, or any problem that asks for an explanation or justification. Peers should be positive with their comments and specific with their feedback. They should highlight the things that they like about what was written. They should share ideas to improve their partner’s writing. Students complete a rough draft of their writing entry. Students trade papers with a partner and read their partner’s work. Students use another color to make edits, provide comments and suggestions, ask clarifying questions, or provide praise. Student (1) shares out to Student (2), what they like about the writing, and any additional notes or feedback. Student (2) shares out to Student (1), what they like about the writing, and any additional notes or feedback. Students make changes or additions to their rough draft thinking.
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## Pick Three
The purpose of this activity is to provide a quick reminder at the beginning of class about the importance of having every team member contribute to the team's work, and of what is involved in good mathematics work. Examples of what the list of strengths may include are: Looking for patterns, Asking questions, Understanding vocabulary, Making a drawing or model, Acting out the problem, Helping others, Explaining my thinking and justifying answers, Noticing details, Organizing, Predicting, Writing equations from patterns, Looking at things in different ways, Reading aloud, Keeping people on task, Following directions, Learning from our mistakes, Remembering a similar problem, Encouraging your team members to persevere. Teacher posts a list of strengths. Each student selects and writes down three strengths they can contribute to their team. Students take turns sharing their strengths with their team. Students use strengths as they work on the lesson.
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## PlanningThinking about and organizing the activities, sequence, manner of presentation, study team teaching strategies and materials needed to implement a lesson. The article on Purposefully Planning a Lesson can be found here. | |

## Players-Coach
Team Member (1) acts as a coach to teach the other members about a topic or concept. Team member (1) shares self-generated strategies to build team comprehension. The team members acting in the role of player may ask clarifying questions. Team Member (1) assumes the role of coach. Team Members (2), (3), and (4) assume the role of player. Team Member explains a topic or concept. Team Members (2), (3), and (4) ask clarifying questions.
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## Pose Purposeful QuestionsOne of the eight Mathematics Teaching Practices from Principles to Actions that needs to be a consistent component of every mathematics lesson. Effective teaching of mathematics uses purposeful questions to assess and
advance students’ reasoning and sense making about important mathematical ideas and relationships. | ||

## Position PapersArticles written that present a position about a topic to an audience that the opinion presented is valid and worth listening to. | |

## Principles of AssessmentGuidance for teachers and all stakeholders regarding assessment practices and suggest teachers create their own tests, work through all assessments, only assess material students have had ample time to engage with, formatively assess as a learning experience for both students and themselves, and be flexible in grading to allow differences in reaching mastery. | |

## Principles to ActionsAlso known as PtA, strategies for teachers to engage students in mathematical thinking, reasoning, and sense making to significantly strengthen teaching and learning’ put forth by the National Council Teachers of Mathematics to offer guidance to teachers, mathematics coaches, administrators, parents, and policymakers. | |

## Principles to Actions Executive SummaryAn introduction which sets out the Mathematics Teaching Practices - consistent components research states are needed for every mathematics lesson; productive and unproductive beliefs for facing obstacles as well as suggestions for combating the issues; and a call to action to recognize the critical need in education to develop understanding in math education and confidence for all students. | |

## Problem-Based LearningResearch says students learn ideas more usefully for other arenas when they learn by attacking problems. Problems-Based Learning is evident in a classroom when - Students and teachers share math authority as they value and engage in productive struggle.
- Teachers guide without taking over the thinking.
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## Process GoalsProcess goals are designed to build habits, stick to consistent routines, and define success as growth in one's skills and abilities. For example, process goals might refer to how students complete their work using the Standards for Mathematical Practice. Teachers may have particular success criteria attached to these Learning Goals that explain how and when students may demonstrate their proficiency level. | ||

## Product GoalsProduct goals are project-oriented, stick to firm deadlines, and define success by the completion of great work. For example, product goals might refer to which math problems that students should complete by the end of the day or class period. Teachers may have particular success criteria attached to these Learning Goals that explain how and when students may demonstrate their proficiency level. | ||

## Productive StruggleOne of the eight Mathematics Teaching Practices from Principles to Actions that needs to be a consistent component of every mathematics lesson. Effective teaching of mathematics consistently provides students, individually and collectively, with opportunities and supports to engage in productive struggle as they grapple with mathematical ideas and relationships. “…productive struggle comprises the work that
students do to make sense of a situation and
determine a course of action when a solution
strategy is not stated, implied, or immediately
obvious. From an equity perspective, this implies that
each and every student must have the opportunity
to struggle with challenging mathematics and to
receive support that encourages their persistence
without removing the challenge.” Boston, Melissa D., Fredrick Dillon, Margaret S. Smith, and Stephen Miller. Taking
Action: Implementing Effective Mathematics Teaching Practices in Grades 9-12.
Reston, VA: National Council of Teachers of Mathematics, 2017. [p.208] | |

## Professional NoticingProfessional Noticing requires that the teacher be able to: identify relevant aspects of the teaching situation; use knowledge to interpret the events, and establish connections between specific aspects of teaching and learning situations and more general principles and ideas about teaching and learning. Professional noticing is a crucial component of CPM math teacher competency and requires not only knowledge and expertise with mathematics, but also knowledge of the pedagogy associated with using the curriculum as intended. | |

## Professional Outcome 2Establish professional relationships and learning communities that support lifelong professional growth and a commitment to mathematics education | ||

## Professional Outcome 3Reflect on the efficacy of their instructional practices and share instructional challenges and successes with colleagues | ||

## Professional Outcome 4Recognize the importance of being transparent with their instructional practices with students, colleagues, and administration | ||

## Professional Outcome 5Hold themselves accountable to be prepared to teach their course and commit to purposefully planning each chapter and lesson | ||

## Professional Outcome 6Understand that professional growth develops over time and requires an ongoing commitment to engage in professional learning | ||

## Professionalism Outcome 1Recognize that effective teaching requires implementation of research-based instructional strategies that advance student learning | ||

## Proximity Partner
To find proximity partners, students stand up, push in chairs, touch 2 tables/desks, three walls, and a chair. The two students closest together are proximity partners. Partners discuss a topic or concept for a period of time—one or two minutes. Partners thank each other and return to the teams.
To find a partner, students stand up, push in chairs, put a hand up, make eye contact, and put hand down. After all hands are down, partners move to meet and discuss a topic or concept. Students stand up and push in chairs. Students touch 3 walls, 2 tables/desks, and 1 chair. The two students closest to one another are proximity partners. Proximity partners briefly discuss a topic or concept. Proximity Partners return to teams.
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## R |
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## Reading protocolsStrategies useful for reading and processing longer passages in a team. | |

## Reason abstractly and quantitativelyMathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to | |

## Reciprocal Teaching
This is an activity that can be done at any time during the class period to check for understanding. At the beginning of the period, students could be asked to explain a concept or definition from the previous day or week. It would serve to remind them of what they have recently learned. Or this might be a way to introduce the topic of the day by remembering some of the parts from previous units which lead up to the new concept. If reciprocal teaching is used during the middle of class it can be used to check the comprehension of what is being learned in that day's lesson. The teacher listens to as many pairs as possible to formatively assess what they have learned and to identify questions and points of confusion. Students work in pairs. Partner (1) pretends that Partner (2) was absent and explains a concept. Switch roles and Partner (2) pretends that Partner (1) was absent and explains a second concept. Teacher circulates to listen to as many pairs as possible.
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## Recorder/ReporterThe Typically, a teacher could expect to hear a Recorder/Reporter asking:
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## Red Light - Green Light
Teacher ensures that teams verify the correct solution to each problem before proceeding to the next problem. Each team indicates their status—red indicates finished and green indicates in progress. The teacher circulates to red light teams to pose purposeful questions to check for understanding. Students continue to the next problem as instructed by the teacher. Each team has a red cup and green cup, or laminated double-sided red/green card to indicate status. While the team works together on a problem or problems, status is set to a green light. When the team finishes a problem or gets stuck, status is set to red light. Teacher goes over to a red light and poses questions to check for understanding. After speaking with the teacher, teams go to the next problem or set of problems.
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## Research Connections Outcome 1Gain knowledge of CPM’s Three Pillars of Research and how they are incorporated into the design of CPM curriculum | ||

## Research Connections Outcome 2Understand and apply strategies and systems to establish CPM’s Three Pillars of Research in their classroom using the Implementation Progress Tool | ||

## Research Connections Outcome 3Build an understanding of NCTM’s Mathematics Teaching Practices, connecting them to the design of CPM curriculum and to their own instructional practice | ||

## Research Connections Outcome 4Incorporate knowledge of CPM’s Three Pillars of Research to intentionally build formative and summative assessments | ||

## Research Connections Outcome 5Assimilate content from the CPM Newsletters, Position Papers, Principles, and the Teacher Research Corp’s (TRC) action research to reflect on and guide instructional practices | ||

## Research Connections Outcome 6Use and connect research to instructional practices to formatively assess understanding | ||

## Research Connections Outcome 7Examine beliefs about teaching and learning mathematics and its impact on all students based on research | ||

## Research Connections Outcome 8Recognize the importance of the Triangle of Teacher Support and how all three components support effective implementation | ||

## Resource ManagerThe Typically, a teacher could expect to hear a Resource Manager asking:
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## Review & PreviewThe section after a lesson consisting of six to ten problems on a variety of topics and skills; a mixed spaced practice approach that leads to higher learning and better long–term retention. | |

## Review & Preview Feedback StrategiesThese are strategies for processing Review & Preview problems done independently. | |

## Rick RolledRickrolling, alternatively | |

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## Says-Means-Matters ProtocolA reading strategy used with a longer reading passage to aid in understanding where a reader reads a passage/article, describes what it says, interprets what they think it means and writes why they think it matters. | |

## Second CPM Principle of AssessmentTeachers need to read and work through all assessment items carefully before giving
them to students, making sure it is clear what kind of response is expected and that there
are no errors. | ||

## Selecting- CRUCIAL STEP - What did you want to highlight?
- Purposefully select those that will advance mathematical ideas
Selecting is Step 3 of the 5 Practices for Orchestrating Productive Math Discussions | |

## Sentence-Phrase-Word ProtocolA reading strategy where group members read the text silently selecting a meaningful sentence that captures a core idea; a moving, engaging or provoking phrase; and a powerful word or one that captures attention. Discuss and record group choices. Looking at the groups’ choices of words, phrases, and sentences, reflect on the conversation by identifying emerging themes, implications and/or aspects of the text not yet captured. | |

## Sequencing- In what order do you want to present the student work samples?
- Do you want the most common? Present misconceptions first?
- How will students share their work? Draw on board? Put under doc cam?
Sequencing is Step 4 of the 5 Practices for Orchestrating Productive Math Discussions | |

## Share Math AuthorityThe idea that authority should be “shared” between the teacher and the students—that authority should be openly co-constructed between all the individuals involved in the classroom. This is an important step in getting students to take ownership in team collaboration and in their own learning. | ||

## Shareable Content Object Reference ModelAlso known as a SCORM, is a collection of standards and specifications for web-based electronic educational technology. It defines communications between client side content and a host system, which is commonly supported by a learning management system. | |

## Shared AuthorityThe idea that
authority should be “shared” between the teacher and the students—that authority should
be openly co-constructed between all the individuals involved in the classroom. This is an important step in getting students to take ownership in team collaboration and in their own learning. | ||

## Silent Debate
To improve writing and communications skills, students are prompted to write clear and concise statements about topics. The process is similar to oral debates, except that it is silent. Partners are assigned a topic and one partner writes pro statements while the other responds with con statements. One paper and pencil is shared by the partners. The pro partner begins and writes a statement in favor of the prompt. The con partner reads the statement and writes a statement against it or against the original prompt. The process continues. Students work in pairs. Partner (1) is assigned the pro (for) position, Partner (2) takes the con (against) position. Partners share a pencil and one sheet of paper. A prompt or topic is given by the teacher. Partner (1) makes a pro, or supportive statement in writing. Partner (2) reads the statement, and writes a comment against. Process continues—three or four times.
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## Six Word Synthesis ProtocolA reading strategy where text is read and marked to gain an understanding of the ideas and applications. Ideas about the reading are synthesized into only six words which could be a sentence, phrase, connection, personal learning, or an Aha. Each member then shares his/her words with the group along with an explanation. The group could then create a six word synthesis with all of the words. | |

## Standards for Mathematical PracticeAlso known as SMPs, are enumerated in the Common Core State Standards and describe varieties of expertise that math teachers should strive to develop in their students. CPM lessons are aligned to the SMPs, which can be found in the Mathematical Practices section of the Teacher Notes. - Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with mathematics.
- Use appropriate tools strategically.
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning.
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## StatusStatus is the perception of students’ academic capability and social desirability. Status will play a role in all classrooms and in all teams. To support Collaborative Learning, teachers must continually monitor status and take action to raise a student's status using strategies such as Team Roles and STTS. | |

## Stoplight Problems
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## Strength in NumbersA book on collaborative learning in secondary mathematics by Ilana Seidel Horn. | |

## Study Team and Teaching StrategiesSometimes referred to as STTS, these strategies help structure effective collaboration among students. They are set up with particular ways for students to interact. Some are useful for brainstorming, for creating individual think time before team discussion, or for ensuring that all students have an opportunity to be vocal in a discussion. | ||

## Success CriteriaSuccess Criteria explain how students can demonstrate a Learning Goal. Success criteria often use words such as explain, describe, model, show, write, justify, or create. In instances where hinge questions are used, success criteria may designate a particular part of the lesson. Students will be able to explain the rule and growth after question 4-14b. | ||

## Suggested Assessment PlanThe Suggested Assessment Plan is in the Teacher Notes of each Chapter Opening. It provides suggestions for Team Assessments, Participation Quizzes, and Individual Assessments. The problems listed in this plan can be shared with students via a Learning Management System in order for teachers to be transparent about the connections between Review & Preview and Summative Assessments. | |

## Support Productive Struggle in Learning MathematicsOne of the eight Mathematics Teaching Practices from Principles to Actions that needs to be a consistent component of every mathematics lesson. Effective teaching of mathematics consistently
provides students, individually and collectively, with opportunities and supports to engage in productive
struggle as they grapple with mathematical ideas and relationships. | |

## Swapmeet
Teams work on problems where ideas are generated, strategies are formed, and solutions are developed. At times, teams reach a point of impasse. When this happens, this strategy can be used to allow team members to swap ideas with other teams. Two team members rotate to the next team to exchange strategies and ideas with that team. After sharing, students return to their original teams to share new understanding. When a team task is partially finished, one pair from each team rotates to the next team. Pairs from the two teams exchange ideas, solutions, thinking, etc. Pairs return to their original teams and each pair shares what they learned from other teams. Teams continue to work on problems.
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## T |
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## Talk MovesAlthough IRE is the most common type of Teacher-Student discourse, talk moves are a better option. The secret to talk moves is
Turn and Talk "Share your thinking with your partner" Revoicing "Are you saying that.....?" Say More "Could you give us an example?" or "Would someone else say that in their own words?" or "I'm not sure I understand what you are saying, could you say more?"
Repeating "Who can repeat what was said?" Sharing Out "What did your partner think?" Surveying Access "Can everyone hear what is being said?" Focusing Attention on Student Thinking "As we listen to this response, think about how it is the same or different than what we talked about yesterday?"
Press for Reasoning "Why do you think that?" Having Reasoning Repeated in Multiple Ways "Who can put that in their own words?" Find a Student Who is Unconvinced "Erin is not convinced, Who can explain why that is true?" Turn and Talk, Prompting students to make sense of reasoning "Talk about Marla's idea"
Press for reactions "Do you agree or disagree? Why?" or "What does Marla's statement make you think of?" Compare or Contrast "Is what Erin said the same or different than what Marla said? How?" Invite Challenges "Who sees it differently?" or "Can someone make a counter-argument?" Turn and Talk "Talk to your partner about whether or not you agree with Erin?"
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## Task ManagerThe Typically, a teacher could expect to hear a Task Manager saying:
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## Tasks that Promote ReasoningOne of the eight Mathematics Teaching Practices from Principles to Actions that needs to be a consistent component of every mathematics lesson. Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies. | |

## Teacher ToolkitThe Teacher Toolkit - Collaboration, Pacing, and Routines is a module within the Professional Learning Portal that provides teacher testimonials from experienced teachers regarding the routines and procedures they use to support student learning in their own classrooms. | ||

## Teacher TransparencyWhen implementing new instruction strategies, it is not only important to make obvious the intellectual practices involved in completing and evaluating learning tasks, but also to explain the intent of your practice to your students. Each strategy and classroom expectation should be accompanied by an explanation of how that intentional act will positively impact the students' learning. Read the article | ||

## Team RolesCPM resources are designed around four Team Roles: Resource Manager, Facilitator, Recorder/Reporter, and Task Manager. Click on the individual roles to see their descriptions. | |

## Teammates Consult
Teammates Consult is an effective strategy to use for problem solving and concept development situations. It allows the students an opportunity to think and discuss the problem before actually writing anything down. All pencils and calculators are set aside (no writing). Students read the problem or question individually. Students get approximately 1 minute of individual think time. Students take turns sharing and discussing the problem for clarity. Students share possible strategies or next steps. Teacher gives okay for pencils to be picked up and written work to begin.
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## They Say..., I Say..., So What...? ProtocolA reading strategy where text is read silently and individuals are asked to describe what they say (the authors say) about the topic; interpret what the reader thinks about the topic (I say); and then the reader writes what the topic means to them (so what). This is shared with a partner, group and/or whole class. | |

## THINK INK PAIR SHARE-ASYNCHRONOUSThis could be done on your Learning Management System or on a document that all students have access to. You could assign each student a Team Role ahead of time and then pair up team roles (i.e. Facilitators are paired with Resource Manager) In your LMS, create a Forum with your prompt. Students write an entry and then read and comment on another a partners entry. Partners are determined by their team role. On a Google Doc, put the prompt at the top. Have each student write an entry and then read their partners and comment or question on what they read. | |

## Think Ink Pair Share-Socially DistancedThis could be done with whiteboards so that the writing is large enough for distanced students to see each others work. It could also be done with a Google Doc. The teacher posts a prompt in a document or sheet. Have each student write an entry and then read their partners and comment or question on what they read. Teachers would monitor the time and give students verbal feedback about the amount of time left. | |

## Think Ink Pair Share-SynchronousThis could be done on a document or a spreadsheet. The advantage of a spreadsheet is that you could lock some of the columns or rows so that students could not change the data. The teacher posts a prompt in a document or sheet. Have each student write an entry and then read their partners and comment or question on what they read. Teachers would monitor the time and give students verbal feedback about the amount of time left. You could also do this with Private Chats or Breakout Rooms but be mindful that you may not always be able to monitor these chats. | |

## Think-Ink-Pair-Share
To emphasize the importance of think-time, the teacher poses a question/problem for students to silently think about. After a short period of time, students write an explanation to share. When the teacher indicates, partners share explanations. Partners may share within the team or the whole class.
Students receive a question—possibly about concepts covered in a unit, Diamond Problems, or mental math—and silently think for a short period of time. Without writing, partners discuss explanations of the question. Partners may then share out with the rest of the team or class. Teacher poses a question/problem. Students think for a period of time—one or two minutes. Students silently prepare an explanation in writing to share. Partners take turns sharing written explanations. Partners may then share out with the rest of the team or class.
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## Third CPM Principle of AssessmentStudents should be assessed only on content with which they have been meaningfully
engaged, and with which they have had ample time to make sense of. | ||

## ThreadRelated problems and/or lessons intentionally sequenced within and between courses to help students both deepen conceptual knowledge and build procedural fluency. | |

## Toolkit
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## Tools and Technology Outcome 1Learn how to access CPM’s Synthesis of Research and course preparation resources | ||

## Tools and Technology Outcome 2Experience how to engage students with content using Desmos and other eTools | ||

## Tools and Technology Outcome 3Understand the structure and organization of the teacher and student ebooks | ||

## Tools and Technology Outcome 4Locate Closure and Assessment resources | ||

## Tools and Technology Outcome 5Know how to utilize Chapter Opening and Lesson Teacher Notes | ||

## Tools and Technology Outcome 6Understand how to access Team Support and Strategies to establish and maintain classroom expectations | ||

## Tools and Technology Outcome 7Examine Universal Access and Literacy resources to support the learning of all students | ||

## Tools and Technology Outcome 8Learn how to access and navigate CPM’s Learning Management System | ||

## Tools and Technology Outcome 9Locate resources to support parent and public relations | ||

## Traveling Salesperson
Teams receive a topic or problem—same/different from other teams—and a presentation is prepared. Teams plan and practice within teams. Then one team member—Traveling Salesperson—rotates to another team to pitch the presentation. The team asks clarifying questions. Then the Traveling Salesperson returns to the team so that roles may rotate. The use of Numbered Heads allows all team members to rotate. The teacher circulates to make informed instructional decisions about which team member is assigned the role of Traveling Salesperson. Teams receive a topic or problem from the teacher. Teams complete the problem by planning a presentation. Team Member (1) shares the presentation with another team. The process continues with another topic or problem, and roles may rotate.
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## Tuning Protocol
Team Member (1) presents an explanation to a problem to other team members in a short period of time—one or two minutes. The presentation may be a portfolio project, a report, a math problem, Learning Log, etc. When the time is up, team members ask Team Member (1) any questions or clarifications—about one minute. Then Team Member (1) turns away from the team while team members discuss the presentation and deepen understanding of the problem—five minutes—, while Team Member (1) listens and writes notes. Team Member (1) turns back and shares a reflection of the discussion. The role may rotate to Team Member (2) and the process continues. The teacher uses discretion for length of activity. Team Member (1) presents an explanation to a problem in teams. Team Members ask Team Member (1) questions or clarifications. Team Member (1) turns away from the team while listening and writing notes. Team Member (1) shares a reflection of the discussion. The role is rotated to Team Member (2) and the process continues.
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## Turn and Talk
To discuss a procedure or concept without writing, one team member explains while the other team member listens. If there is a disagreement, students continue to discuss the solution and agree on a single explanation. When partners have an explanation, they share with the rest of the team. Students work in pairs. Team Member (1) explains while Team Member (2) listens. Team Member (2) asks clarifying questions to Team Member (1). Partners agree on one explanation to share with other members of the team. Roles are reversed for the next problem.
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## Two Stars and a Wish
During any presentation, students record two things they really liked—Two Stars—and one suggestion that might improve the presentation—a Wish. At times, a team member may be present to explain or answer questions about the presentation. Students record two things they like about a presentation. Students record one thing that would improve the presentation.
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## U |
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## Universal AccessSupport for teachers with students of varied abilities and backgrounds across a broad range of abilities. This section in the teacher’s edition ebook under the teacher tab includes an Introduction, Success for Students, Student Struggle, More Help, Special Needs, ELL, Advanced Learners, Unprepared Students and a Conclusion. | |

## Use and Connect Mathematical RepresentationsOne of the eight Mathematics Teaching Practices from Principles to Actions that needs to be a consistent component of every mathematics lesson. Effective teaching of mathematics engages students in
making connections among mathematical representations to deepen understanding of mathematics concepts
and procedures and as tools for problem solving. | |

## Use appropriate tools strategicallyMathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations | |

## W |
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## Walk and Talk
To promote movement during learning, partners receive a topic or concept to discuss while walking—about the classroom, in the hallway, or outside. Partners summarize learning, clarify ideas, or ask questions. Partners report any remaining questions to the teacher after returning to desks. Partners receive topics or concepts to discuss. Partners summarize, clarify ideas, or ask questions while walking. Partners share any remaining questions with the teacher after returning to their desks.
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## Whiparound
Students participate in a teacher-led discussion to share a final comment on a topic, concept, or lesson. Students turn in the direction of the speaker—possibly forming a circle around the classroom. Teacher states the topic or problem, and students take turns sharing brief comments. Students have the option to pass on their turn to share.
Teams participate in a teacher-led discussion to share a final comment. Teams share out only one agreed upon response per team. Teacher provides a prompt. Students take turns sharing brief comments. Students listen while others share.
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## Word WallA word wall is a collection of words which are displayed in large visible letters on a wall, bulletin board, or other display surface in a classroom. | |