A typical lesson has four phases:

1. Warm-up
2. Instructional activities
3. Lesson Synthesis
4. Cool-down

Warm-up

The first activity in every lesson is a Warm-up. The Warm-up routines provide opportunities for students to bring their personal experiences as well as their mathematical knowledge to problems and discussions. These routines place value on the voices of students as they communicate their developing ideas, ask questions, justify their responses, and critique the reasoning of others. 

The Warm-up serves one or both of two purposes:

• Helps students get ready for the day’s lesson.
• Gives students an opportunity to strengthen their number sense or procedural fluency.

A Warm-up that helps students get ready for the day’s lesson might serve to remind them of a context they have seen before, get them thinking about where the previous lesson left off, or preview a calculation that will happen in the lesson so that the calculation doesn't get in the way of learning new mathematics.

A Warm-up designed to strengthen number sense or procedural fluency asks students to do mental arithmetic or reason numerically or algebraically. It gives them a chance to make deeper connections or become more flexible in their thinking.

Three instructional routines frequently used in a Warm-up are Math Talk, Notice and Wonder, and Which Three Go Together? In addition to the mathematical purposes, these routines serve the additional purpose of strengthening students’ skills in listening and speaking about mathematics.

Once it becomes familiar, the Warm-up should take 5–10 minutes. If these routines frequently take much longer than that, work on concrete moves to more efficiently accomplish the goal of the Warm-up.

Instructional Activities

After the Warm-up, lessons consist of a sequence of 1–3 instructional activities. The activities are the heart of the mathematical experience and make up the majority of the time spent in class.

An activity can serve one or more purposes:

• Provide experience with a new context.
• Introduce a new concept and associated language.
• Introduce a new representation.
• Formalize the definition of a term for an idea previously encountered informally.
• Identify and resolve common mistakes and misconceptions that people make.
• Practice using mathematical language.
• Work toward mastery of a concept or a procedure.
• Provide an opportunity to apply mathematics to an open-ended problem, such as modeling.

The purpose of each activity is described in its Activity Narrative.

Each instructional activity has three phases:

• Launch
  During the Launch, make sure that students understand the context of the given problem (if there is a context) and what the problem is asking them to do. This is not the same as making sure students know how to do the problem—part of the work that students should do for themselves is to figure out how to solve the problem. The Launch invites students into the lesson and helps them connect to contexts with which they are unfamiliar.

• Student Work Time
  The Launch of an activity frequently includes suggestions for grouping students. At different times, students are given opportunities to work individually, with a partner, and in small groups.

• Activity Synthesis
  During the Activity Synthesis, allow time for students to incorporate and make connections to what they have learned. This time ensures that all students have an opportunity to understand the mathematical punch line of the activity and to situate the new learning within their previous understanding.

Some activities have two versions: print and digital. An activity with a digital version is identified in the print version, at the start of the Activity Narrative. Choosing which version to use with students depends on device availability, students’ needs, and mathematical goals. For example, digital versions of activities may reduce barriers for students who either need support with fine-motor skills or benefit from extra processing time, or they may help students see relationships in dynamic ways.

A note about optional activities: A relatively small number of activities throughout the course have been marked “optional.” Common reasons an activity is optional include:

• The activity addresses a concept or a skill that is below grade level, but students commonly need a chance to refocus on it before encountering grade-level material. If the pre-unit diagnostic assessment (Check Your Readiness) indicates that students don’t need this review, skip an activity such as this.
• The activity addresses a concept or a skill that goes beyond the requirements of a standard. The activity is nice to do if there is time, but students won’t miss anything important if the activity is skipped.
• The activity provides an opportunity for additional practice on a concept or a skill that many students (but not necessarily all students) need. Use your judgment about whether class time is needed for such an activity.

Lesson Synthesis

After the activities for the day are completed, students should take time to synthesize what they have learned. This portion of class should take 5–10 minutes before students start working on the Cool-down. Each lesson includes a Lesson Synthesis that assists with ways to help students incorporate new insights gained during the activities into their big-picture understanding. Use this time in any number of ways, including posing questions verbally and calling on volunteers to respond, asking students to respond to prompts in a written journal, asking students to add on to a graphic organizer or concept map, or adding a new component to a persistent display such as a word wall.

Cool-Down

Give students a Cool-down task at the end of the lesson. Students are meant to work independently on the Cool-down for about 5 minutes and then turn it in. The Cool-down serves as a brief formative assessment to determine whether students understand the lesson. Use students’ responses to the Cool-down to make adjustments to further instruction.

Student Lesson Summary

After a lesson is complete, the Student Lesson Summary serves as a follow-up to the lesson. This summary includes important math concepts from the day along with example problems and any new glossary terms, which appear in boldface type and are defined for reference.

The story of each grade is told across the units in the narratives. Each unit has a narrative that describes the mathematical work that will unfold in that unit. Each section within a unit has a narrative that describes the mathematical work in the section. Each lesson and each activity in a unit also have narratives. 

The Lesson Narrative explains:

• The mathematical content of the lesson and its place in the learning sequence.
• The meaning of any new terms introduced in the lesson.
• How the mathematical practices come into play, as appropriate.

The Activity Narrative explains:

• The mathematical purpose of the activity and its place in the learning sequence.
• What students are doing during the activity.
• What to look for, while students are working on an activity, to orchestrate an effective Activity Synthesis.
• Connections to the mathematical practices, when appropriate.

Instructional Routines (IRs) are designs for interaction that invite all students to engage in the mathematics of each lesson. They provide opportunities for students to bring their personal experiences as well as their mathematical knowledge to problems and discussions. Instructional routines place value on the voices of students as they communicate their developing ideas, ask questions, justify their responses, and critique the reasoning of others.

Instructional routines have a predictable structure and flow. They provide structure for both teachers and students. A finite set of routines supports the pacing of lessons. As the routines become familiar and save time in classroom choreography, students can spend less time learning how to execute lesson directions and more time learning mathematics. Some of the instructional routines, known as Mathematical Language Routines (MLRs), were developed by the Stanford University UL/SCALE team. MLRs are written into each lesson, either as an embedded structure of a lesson activity in which all students engage, or as a suggested optional support specifically for English learners.

The first occurrence of each routine in a course includes detailed guidance for how to successfully conduct the routine. Subsequent instances include an abbreviated run-through, so as not to unnecessarily inflate the word count of the Teacher Guide.

Digital Routines (DRs) indicate required or suggested applications of technology, appearing repeatedly throughout the curriculum. Activities using the routines are flagged, which is helpful for lesson planning and for focusing the work of professional development.

• 5 Practices
  

  What: The 5 Practices for Orchestrating Productive Mathematical Discussions (Smith & Stein, 2011) are: anticipate, monitor, select, sequence, and make connections between students’ responses. In this curriculum, much of the work of anticipating, sequencing, and connecting is handled by the materials in the Activity Narrative, the Launch, and the Activity Synthesis. Prepare for and conduct whole-class discussions. 

  Why: In a problem-based curriculum, many activities fit the description of “do math and talk about it,” but the 5 Practices lend structure to these activities so that they more reliably result in students making connections and learning new mathematics. Lessons that include this routine are designed to allow students to solve problems in ways that make sense to them. During the activity, monitor to uncover and nurture conceptual understandings as students engage in problems in meaningful ways. During the Activity Synthesis, students collectively reveal multiple approaches to a problem and make connections between these approaches (MP3).

• Card Sort
  

  What: A Card Sort uses cards or slips of paper that are manipulated and moved around (or the same functionality enacted with a computer interface). Students sort cards on their own or in groups of 2–4. They organize objects into categories or groups, based on shared characteristics or connections. Combine this routine with the Take Turns routine so that each time a student sorts a card into a category or makes a match, they are expected to explain their rationale while the group listens for understanding. The first few times students engage in these activities, demonstrate them. Once students are familiar with these structures, less set-up will be necessary. While they are working, ask students to restate their question more clearly or paraphrase what their partner said.

  Why: A Card Sort provides opportunities to attend to mathematical connections, using ready-made representations that save time and effort. A card-sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7). As students work, monitor for the different ways groups choose their categories, and encourage increasingly precise mathematical language (MP6).

• Math Talk
  

  What: In these Warm-up routines, one problem is displayed at a time. Give students a few moments to quietly think and to signal when they have an answer and a strategy. Select students to share different strategies for each problem. Ask: “Who thought about it a different way?” Record students’ explanations for all to see. Press students to provide more details about why they decided to approach a problem a certain way. Gather two or three distinctive strategies per problem if it is not possible to share every strategy in the given time. Problems are purposefully chosen to elicit different approaches, often in a way that builds from one problem to the next.

  Why: The Math Talk builds fluency by encouraging students to think about the numbers, the shapes, or the algebraic expressions, and to rely on what they know about structure, patterns, and properties of operations to mentally solve a problem. While participating in these activities, students are precise in their word choice and use of language (MP6). Additionally a Math Talk often provides opportunities to notice and make use of structure (MP7).

• MLR1: Stronger and Clearer Each Time
  

  Adapted from Zwiers (2014)

  What: The routine provides a structured and interactive opportunity for students to revise and refine their ideas and their verbal and written output. The main idea of the Stronger and Clearer Each Time routine is for students to think about or write a response individually, to use a structured pairing strategy for multiple opportunities to refine and clarify their response through conversations with different partners, and then finally to revise their original written response. Throughout this process, press students for details, and encourage them to press each other for details. 

  Why: This routine provides a purpose for students’ conversation—through the use of a discussion-worthy and iteration-worthy prompt—as well as fortifies output. 

  How It Happens

  Response – First Draft: Students think and write individually about a thought-provoking question or prompt. (2–3 minutes)

  Structured Partner Meetings: A structured pairing strategy facilitates students meeting with 2 or 3 different partners. During each meeting, students take turns at playing the roles of the speaker, who shares their ideas (without looking at their first draft, when possible), and the listener, who asks clarifying questions and gives feedback. (1–2 minutes per meeting)

  Response – Second Draft: Students write a final draft that is stronger (showing evidence of incorporating or addressing new ideas, examples, and reasoning about mathematical content) and clearer (showing evidence of greater precision, organization, and refinement of language). When finished, students compare their first and second drafts. (2–3 minutes)
• MLR2: Collect and Display
  

  What: The purpose of Collect and Display is to capture students’ oral words and phrases into a stable, collective reference. Listen for and record students’ output, using written words, diagrams and pictures. Organize, revoice, or explicitly connect to other terms in a display for all students to use. Throughout the course of a unit (and beyond), reference the displayed language as a model, update and revise the display as students’ language changes, and make bridges between students’ prior language and new disciplinary language (Zwiers et al., 2017). 

  Why: The intent of this routine is to stabilize the fleeting language that students use, during partner, small-group, or whole-class activities, in order to use students’ own output as a reference in developing their mathematical language. This routine provides feedback for students in a way that increases accessibility while simultaneously supporting meta-awareness of language. The display provides an opportunity to showcase connections between students’ ideas and new vocabulary. It also provides an opportunity to highlight examples of students using disciplinary language functions beyond vocabulary words.

  How It Happens

  Collect: Circulate and listen to students talk during partner, small-group, or whole-class discussion. Jot down the words, phrases, drawings, or writing that students use. Capture a variety of uses of language that connect to the lesson content goals, as well as to the relevant disciplinary language functions. 

  Display: Organize the collected output in a display for the whole class to use as a reference during further discussions throughout the unit. Encourage students to suggest revisions, updates, and connections to add to the display as they develop new mathematical ideas and new ways of communicating ideas, over time.
• MLR3: Critique, Correct, Clarify
  

  What: In Critique, Correct, Clarify,  students are given a piece of mathematical writing that is not their own to analyze, reflect on, and develop. Demonstrate how to effectively and respectfully critique the work of others, with meta-think-alouds, and press for details when necessary.

  Why: The intent is to prompt students’ reflection, with an incorrect, incomplete, or ambiguous written argument or explanation, and for students to improve the written work by correcting errors and clarifying meaning. This routine fortifies output and engages students in meta-awareness. More than just error analysis, this routine purposefully engages students in considering the author’s mathematical thinking as well as the features of their communication.

  How It Happens

  Original Statement: Students critique a written mathematical statement that intentionally includes conceptual (or common) errors in mathematical thinking as well as ambiguities in language. (1–2 minutes)

  Partner Discussion: Students revise the original statement, working as partners to create an “improved statement,” resolving any mathematical errors or misconceptions, and correcting ambiguous language. (2–4 minutes)

  Improved Statement: Invite one or two partner groups to share their revised statements. Encourage the whole class to contribute additional language or edits to make the final drafts more clear and convincing. (3–5 minutes)

• MLR4: Information Gap Cards
  

  Adapted from Zwiers (2014)

  What: The Information Gap (also Info Gap) routine cultivates conversation by creating an authentic need for students to communicate and conduct a dialog in a specific way (Gibbons, 2002). In an Info Gap, students share ideas and information in order to bridge a gap and accomplish a task that they could not have done alone. Working as partners or in groups, they are given different pieces of necessary information that must be used together to solve a problem. One partner gets a Problem Card, with a math question that doesn’t have enough given information, and the other partner gets a Data Card, with information relevant to the problem. Students ask each other questions, such as: “What information do you need?” and are expected to explain what they will do with the information. The first few times students engage in these activities, demonstrate, with a partner, how to ask for and share information, how to justify a request for information, and how to clarify and elaborate on information. Once students are familiar with these structures, less set-up will be necessary.

  Why: This activity structure is designed to strengthen the opportunities and supports for high-quality mathematical conversations. Mathematical language is learned by using mathematical language for real and engaging purposes. These activities were designed such that students need to communicate in order to bridge information gaps. During effective discussions, support students  to take the following actions: pose and answer questions, clarify what is asked and what is happening in a problem, build common understandings, and share experiences relevant to the topic. 

  How It Happens

  Problem/Data Cards: Students work together as partners. Partner A is given a card with a problem to solve, and Partner B has information needed to solve the problem on a Data Card. Neither student should read or show their card to their partner. 

  Bridging the Gap: Partner A determines what information is needed, and asks Partner B for that specific information. Partner B should not share the information unless Partner A specifically asks, and justifies the need for, the information. Because partners don’t have the same information, Partner A must work to produce clear and specific requests, and Partner B must work to understand more about the problem through Partner A’s requests and justifications.

  Solving the Problem: Partner A shares the Problem Card, and both students solve the problem independently. Partner B shares the Data Card, and students compare strategies and solutions.

  

• MLR5: Co-Craft Questions
  

  What: The Co-Craft Questions routine  allows students to explore a context before feeling pressure to produce answers, and creates space for students to produce the language of mathematical questions themselves. Students analyze how different mathematical forms and symbols represent different situations. Push for clarity and revoice oral responses as necessary. 

  Why: Through this routine, students use conversation skills to generate, choose (argue for the best one), and improve questions and situations, as well as develop meta-awareness of the language in mathematical questions and problems. 

  How It Happens

  Hook: Present the context. Display a problem stem, a graph, a video, an image, or a list of interesting facts. Optional: Students keep books and devices closed.

  Students Write Questions: Students write mathematical questions about the situation. These questions should be answerable by doing math and could be about the situation, information that is unknown, or assumptions that students think are important. (1–2 minutes)

  Students Compare Questions: Students compare, with a partner, the questions they generated (1–2 minutes), before sharing questions with the whole class. Invite students to make comparisons between the questions, and amplify questions that focus on the lesson learning goals. (2–3 minutes)

  Actual Question(s) Revealed/Identified: Reveal the questions that students are expected to work on. Students compare these to the prompts they generated.

• MLR6: Three Reads
  

  What: The Three Reads ensure that students know what they are asked to do, and create an opportunity for students to reflect on the ways mathematical questions are presented, and equip students with the tools to actively make sense of mathematical situations and information (Kelemanik, Lucenta, & Creighton, 2016). Students take time to understand mathematical situations and story problems, and plan their strategies before finding solutions. 

  Why: This routine supports reading comprehension, sense-making of problems and meta-awareness of mathematical language. It also supports negotiating information in a text, with a partner, in mathematical conversation.

  How It Happens

  In this routine, students are supported in reading, three times, a mathematical text, a situation, or a word problem, each time with a particular focus. The intended question or main prompt is intentionally withheld until the third read so that students can concentrate on making sense of what is happening in the text before rushing to a solution or method.

  First Read: Shared Reading After a shared reading, students describe the situation or context. This is the time to identify and resolve any challenges with non-mathematical vocabulary. (1 minute)

  Second Read: Individual, Partner, or Shared Reading After the second read, students list all quantities in the situation that are countable or measurable. Examples: “number of people in a room” rather than “people,” “number of markers remaining” instead of “markers”). Record the quantities as a reference to use when solving the problem after the third read. (3–5 minutes)

  Third Read: Individual, Partner, or Shared Reading During the third read, the final question or prompt is revealed. Students discuss possible solution strategies. They may find it helpful to create diagrams to represent the relationships among quantities identified in the second read, or to represent the situation with a picture (Asturias, 2012). (1–2 minutes)
• MLR7: Compare and Connect
  

  What: The purpose of a Compare and Connect routine is to foster students’ meta-awareness—leveraging the powerful mix of disciplinary representations available in mathematics as a resource for language development—as they identify, compare, and contrast different mathematical approaches, representations, and language. In this routine, students make sense of mathematical strategies other than their own by relating and connecting other approaches to their own. Demonstrate thinking out loud (for example, exploring why a mathematical problem is approached, represented, or said in a certain way, questioning an idea, wondering how an idea compares or connects to other ideas or language). Prompt students to reflect on, and verbally respond to, these comparisons (for example, by exploring why or when to approach, represent, or say a mathematical problem a certain way, by identifying and explaining correspondences between different mathematical representations or methods, or by wondering how a certain concept compares or connects to other concepts). 

  Why: This routine supports meta-cognitive and meta-linguistic awareness, and also supports mathematical conversation.

  How It Happens

  Students Display Their Work: Give students a problem they can approach and solve, using multiple strategies, or a situation they can model, using multiple representations. Students prepare a display of how they made sense of the problem and why their solution makes sense.

  Compare: Students investigate each others’ work, pointing out important mathematical features, and making comparisons. These comparisons should focus on the typical structures, purposes, and affordances of the different approaches or representations: what worked well in this or that approach, or what is especially clear in this or that representation.

  Connect: Students find correspondences in how specific mathematical relationships, operations, quantities, or values appear in each approach or representation. During the discussion, amplify language students use to communicate about mathematical features that are important for solving the problem or modeling the situation. Call attention to the similarities and the differences between the ways those features appear.
• MLR8: Discussion Supports
  

  What: The Discussion Supports include a variety of teacher moves to support rich and inclusive discussions about mathematical ideas, representations, contexts, and strategies (Chapin, O’Connor, & Anderson, 2009). Combine and use the moves and strategies in this collection with any of the other routines to support discussion during almost any activity. Continue to demonstrate, and students should begin using these strategies themselves to prompt each other to engage more deeply in discussions. 

  Why: Multi-modal strategies help students comprehend complex language and ideas. These strategies make classroom communication accessible, invite and incentivize students’ participation, conversation, and meta-awareness of language, and demonstrate how students can enhance their own communication and construction of ideas.

  How It Happens

  Unlike the other routines, this MLR includes a collection of strategies and moves to combine and use to support discussion during almost any activity.

  Examples of Possible Strategies:

  • Revoice students’ ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students.

  • Press for details in their explanations by requesting that students challenge or elaborate on an idea, or give an example.

  • Show central concepts multi-modally by using different types of sensory inputs: act out scenarios or invite students to do so, show videos or images, use gestures, and talk about the context of what is happening.

  • Practice phrases or words through choral response.

  • Think aloud by talking through thinking about a mathematical concept while solving a related problem or working on a task.

  • Demonstrate uses of disciplinary language functions, such as detailing steps, describing and justifying reasoning, and questioning strategies.

  • Give students time to make sure that everyone in the group can explain or justify each step or part of the problem. Then make sure to vary who is called on to represent the work of the group, so students get accustomed to preparing each other to fill that role.

  • Prompt students to think about different possible audiences for the statement, and about the level of specificity or formality needed, for a classmate versus a mathematician, for example. This thinking works well with the “convince yourself, convince a friend, convince a skeptic” strategy (Mason, Burton, & Stacey, 2010).

• Notice and Wonder
  

  What: This routine can appear as a Warm-up or in the Launch or the Activity Synthesis of an instructional activity. Students are shown some media or a mathematical representation. The Notice and Wonder routine invites all students into a mathematical task, with two low-stakes prompts: “What do you notice?” and “What do you wonder?” Give students a few minutes to think of, and share with a partner, the things they notice and the things they wonder. Then ask several students to share the things they noticed and the things they wondered, and record these for all to see. At opportune times, steer the conversation to wondering about the mathematics on which the class is about to focus.

  Why: The Notice and Wonder routine makes a mathematical task accessible to all students, using two approachable questions. By thinking about them and responding, students gain entry into the context and may have their curiosity piqued. By taking steps to become familiar with the context and the mathematics involved, students learn to make sense of problems (MP1). Note: Notice and Wonder and I Notice/I Wonder, trademarks of the National Council of Teachers of Mathematics (NCTM) and the Math Forum, are used in these materials with permission.

• Poll the Class
  

  What: Poll the Class is used to register an initial response or an estimate, most often in the Launch of an activity or to kick off a discussion. It also is used when collecting data from each student in class, for example, "What is the length of your ear in centimeters?" Every student in class reports a response to the prompt. Develop a mechanism by which poll results are collected and displayed so that this frequent form of classroom interaction is seamless. For smaller classes, conduct a roll call by voice. For larger classes, give students mini whiteboards or a set of colored index cards to hold up. Free and paid commercial tools are also readily available.

  Why: Collecting data from the class to use in an activity makes the outcome of the activity more interesting. In other cases, going on record with an estimate makes students want to know if they are right, and increases investment in the outcome. If coming up with an estimate is daunting for students, ask them for a guess that they are sure is too low or too high. Putting some boundaries on possible outcomes of a problem is an important skill for mathematical modeling (MP4).

• Take Turns
  

  What: In the Take Turns routine, students work with a partner or in a small group. They take turns in the work of the activity, including spotting matches, explaining, justifying, agreeing or disagreeing, or asking clarifying questions. If students disagree, they are expected to support their case and listen to their partner’s arguments. The first few times students engage in these activities, demonstrate the discussion, with a partner. Once students are familiar with these structures, less set-up will be necessary. While students are working, ask them to restate their question more clearly or paraphrase what their partner said.

  Why: Building in the expectation, through the routine, that students explain the rationale for their choices and listen to another's rationale deepens the understanding achieved through these activities. Specifying that students take turns deciding, explaining, and listening inherently limits the phenomenon of one student taking over and the other not participating. Taking turns also can give students more opportunities to construct logical arguments and critique others’ reasoning (MP3).

• Which Three Go Together?
  

  What: Students are presented with four figures, diagrams, graphs, or expressions with the prompt “Which three go together?” Typically, each option for a group of three “go together” for a different reason, and the similarities and the differences are mathematically significant. Students are prompted to explain their rationale for deciding why their group of three goes together and given opportunities to make their rationale more precise. Encourage students first to identify why all four representations go together and then to look closely at how the representations are alike and how they are different while being as specific as they can in their responses. 

  Why: Which Three Go Together? fosters a need for students to define terms carefully and to use words precisely (MP6) in order to compare and contrast a group of geometric figures or other mathematical representations.

Each lesson includes an associated set of practice problems. Assign practice problems for homework or for extra practice in class. Collect and score the problems, or provide students with answers ahead of time for self-assessment. Decide which problems to assign (including assigning none at all).

The set of practice problems associated with each lesson includes a few questions about the contents of that lesson, plus additional problems that review material from earlier in the unit and previous units. Distributed practice (revisiting the same content over time) is more effective than massed practice (a large amount of practice on one topic all at once).