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A typical lesson has four phases:
The first event in every lesson is a Warm-up. The Warm-up is an instructional routine that invites all students to engage in the mathematics of the lesson. The Warm-up routines offer 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.
A Warm-up serves one or both of two purposes:
A Warm-up that helps students get ready for today’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 context or idea that will come up in the lesson so that it 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.
At the beginning of the year, consider establishing a small, discreet hand signal that students can display to indicate they have an answer they can support with reasoning. Examples of signals include a thumbs up and showing the number of fingers to indicate the number of responses for the problem. This is a quick way to see if students have had enough time to think about the problem, and to keep them from getting distracted or rushed by classmates’ raised hands.
Act It Out is an IM Kindergarten and IM Grade 1 routine that invites students to represent story problems (MP4). Students listen to a story problem and act it out, connecting language to mathematical representations. This routine provides an opportunity for students to connect with the storytelling tradition, typically found in ethnically diverse cultures.
While a Choral Count offers students the opportunity to practice verbal counting, the recorded count is the primary focus of the routine. As students reflect on the recorded count, they make observations, predict upcoming numbers in the count, and justify their reasoning (MP7 and MP3).
An Estimation Exploration encourages students to use what they know and what they can see to problem-solve for a rough evaluation of a quantity rather than giving a “wild guess.” The estimates are in the context of measurement, computation, or numerosity—estimating about a large group of objects (MP2).
The How Many Do You See? routine helps early math learners develop an understanding of counting and of quantity, through subitizing and combining parts of sets to find the total in a whole collection. In later grades, this routine encourages students to use operations and groupings to find the total number of dots in less time. Through these recorded strategies, students look for relationships between the operations and their properties (MP7).
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?” By thinking about and responding to these questions, students gain entry into the context and may have their curiosity piqued. Students learn to make sense of problems (MP1) by taking steps to become familiar with the context and the mathematics involved. 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.
The sequence of problems in a Number Talk encourages students to look for structure and use repeated reasoning to evaluate expressions and develop computational fluency (MP7 and MP8). As students share their strategies, they make connections and build on one another’s ideas, developing conceptual understanding.
Questions about Us is an IM Kindergarten routine that allows students to consider number concepts in a familiar context. Students analyze data collected about their class and answer such questions as ”What do you notice?” and “What do you wonder?” Using data that represents students helps them to see math in the world around them (MP2).
The structure of the True or False? routine encourages students to reason about numerical expressions and equations, using base-ten structure, the meaning and the properties of operations, and the meaning of the equal sign. Often, students can determine whether an equation or an inequality is true or false, without doing any direct computation (MP7).
The What Do You Know about _____? routine elicits students’ ideas of numbers, place value, operations, and groupings through images that show a quantity, expressions, and other representations. It is an invitational prompt that could include objectives such as understanding where students see numbers embedded in various contexts and how students compare and order numbers.
The Which Three Go Together? routine fosters a need for students to identify defining attributes and use language precisely in order to compare and contrast a carefully chosen group of geometric figures, images, or other mathematical representations (MP6).
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 serves one or more purposes:
The purpose of each activity is described in its Activity Narrative. Read more about how activities serve these different purposes in the section on design principles.
Each instructional activity has three phases.
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.
The Cool-down task is given to students 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 understood the lesson. Use students’ responses to the Cool-down to make adjustments to further instruction.
The Response to Student Thinking supplies guidance on how to make adjustments, based on specific student responses to a Cool-down. A Next-Day Support, such as providing students access to specific manipulatives or having students discuss their reasoning with a partner, is recommended for addressing Cool-down responses while continuing on to the next lesson. Teachers are directed to appropriate prior grade-level support for Cool-down responses that may need more attention.
The story of each grade is told in eight or nine units. Each unit has a narrative that describes the mathematical work that will unfold in that unit. Each lesson and each activity in the unit also has a narrative.
The Lesson Narrative explains:
The Activity Narrative explains:
Instructional routines (IRs) offer opportunities for all students to engage in and contribute to mathematical conversations. IRs are invitational, promote discourse, and are predictable in nature. They are “enacted in classrooms to structure the relationship between the teacher and the students around content in ways that consistently maintain high expectations of student learning while adapting to the contingencies of particular instructional interactions.” (Kazemi, Franke, & Lampert, 2009)
In the materials, a small set of intentionally chosen, instructional routines ensures that they are used frequently enough to become truly routine. The focused number of routines benefits teachers as well as students. Consistently using a small set of carefully chosen routines is just one way to lower the cognitive load for teachers. Refocus the energy—previously used on structuring an activity—on other objectives, such as understanding students’ thinking and how mathematical ideas are playing out.
Throughout the curriculum, routines are introduced in a purposeful way to build a collective understanding of their structure. They are selected for activities, based on their alignment with the unit, lesson, or activity learning goals. While each routine serves a different specific purpose, they all have the general purpose of supporting students in accessing the mathematics, and all require students to think and communicate mathematically. The Instructional Routines section of this Course Guide gives more details on the specific routines used in the curriculum.
To help identify when a particular routine appears in the curriculum, each activity is tagged with the name of the routine so teachers can search for upcoming opportunities to try out or focus on a particular instructional routine. Professional learning for the curriculum materials includes video of the routines in classrooms, so teachers understand what the routines look like when they are enacted. Teachers also have opportunities, in curriculum workshops and professional-learning communities, to practice and reflect on their own enactment of the routines.
Students analyze, reflect on, and develop a piece of mathematical writing that is not their own. MLR3 is embedded in IM Grades 3–5.
Notice and Wonder invites all students into a mathematical task with two low-stakes prompts: “What do you notice? What do you wonder?” By thinking about things they notice and wonder, students gain entry into the context and might have their curiosity piqued. Students learn to make sense of problems (MP1) by taking steps to become familiar with a context and the mathematics that might be involved. Note: Notice and Wonder and I Notice/I Wonder are trademarks of NCTM and the Math Forum and are used in these materials with permission.
The sequence of problems in a Number Talk encourages students to look for structure and use repeated reasoning to evaluate expressions and develop computational fluency (MP7 and MP8). As students share their strategies, they make connections and build on one another’s ideas, developing conceptual understanding.
Which Three Go Together fosters a need for students to identify defining attributes and use language precisely in order to compare and contrast a carefully chosen group of geometric figures, images, or other mathematical representations (MP6).
Each section in a unit (starting in Kindergarten, Unit 2) includes an associated set of practice problems. There are three types of practice problems: pre-unit, lesson, and exploration. Assign practice problems for homework or for extra practice in class. Collect and score them, or supply students with answers ahead of time for self-assessment. Decide which problems to assign (including assigning none at all).
The first section in each unit has several pre-unit practice problems that relate to content from a prior unit or grade level. Use these problems to review prerequisite material or as a pre-unit assessment, if desired.
Each lesson has one or more practice problems associated with the content of the lesson. Use these problems for in-class practice or homework, or to assess learning.
Each section has two or more exploration practice problems that offer differentiation for students ready for a greater challenge. There are two types of exploration problems. One type is a hands-on activity directly related to the material of the unit that students complete either in class if they have free time, or at home. The second type of exploration problem is more open ended and challenging. These problems go deeper into grade-level mathematics. They are not routine or procedural, and they are not “the same thing again but with harder numbers.”
Students use exploration problems on an opt-in basis if they finish a main instructional activity early or want to do more mathematics on their own. It is not expected that an entire class engages in exploration problems, and it is not expected that any individual student works on all of them. Exploration problems also are good fodder for a Problem of the Week or similar structure.
In addition to Practice Problems, each unit has a Checkpoint of problems or a checklist after each section, depending on the grade level, and an End-of-Unit Assessment. Units also have aligned center activities to support the unit content and ongoing procedural fluency. Each of these components is described in other sections of this Course Guide.