A note about optional activities: A relatively small number of activities throughout the course have been marked "optional." Some common reasons an activity might be optional include:

• The activity addresses a concept or skill that is below grade level, but we know that it is common for students to need a chance to focus on it before encountering grade-level material. If the pre-unit diagnostic assessment (Check Your Readiness) indicates that students don't need this review, an activity like this can be safely skipped.
• The activity addresses a concept or 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 skill that we know many students (but not necessarily all students) need. Teachers should use their judgment about whether class time is needed for such an activity.

A typical lesson has four phases:

1. a warm-up
2. one or more instructional activities
3. the lesson synthesis
4. a cool-down

The Warm-Up

The first event in every lesson is a warm-up. A warm-up either:

• helps students get ready for the day’s lesson, or
• gives students an opportunity to strengthen their number sense or procedural fluency

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 calculation that will happen in the lesson so that the calculation doesn't get in the way of learning new mathematics.

A warm-up that is meant 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.

Instructional routines frequently used in warm-ups are Number Talks, Notice and Wonder, and Which One Doesn't Belong. In addition to the mathematical purposes, these routines serve the additional purpose of strengthening students’ skills in listening and speaking about mathematics.

Once students and teachers become used to the routine, warm-ups should take 5–10 minutes. If warm-ups frequently take much longer than that, the teacher should work on concrete moves to more efficiently accomplish the goal of the warm-up.

At the beginning of the year, consider establishing a small, discreet hand signal students can display to indicate they have an answer they can support with reasoning. This signal could be a thumbs up, or students could show the number of fingers that indicate the number of responses they have for the problem. This is a quick way to see if students have had enough time to think about the problem and keep them from being distracted or rushed by classmates’ raised hands.

Classroom Activities

After the warm-up, lessons consist of a sequence of one or more classroom 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 of many purposes.

• Provide experience with a new context.
• Introduce a new concept and associated language.
• Introduce a new representation.
• Formalize a 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 procedure.
• Provide an opportunity to apply mathematics to a modeling or other application problem.

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.

Lesson Synthesis

After the activities for the day are done, 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 section that assists the teacher with ways to help students incorporate new insights gained during the activities into their big-picture understanding. Teachers can 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 like a word wall.

Cool-Down

Each lesson includes a cool-down task to be given to students at the end of the lesson. Students are meant to work on the cool-down for about 5 minutes independently and turn it in. The cool-down serves as a brief formative assessment to determine whether students understood the lesson. Students' responses to the cool-down can be used to make adjustments to further instruction.