## How to Use the Materials

### Each Lesson and Unit Tells a Story

The story of each course is told in 7 or 8 units. Each unit has a narrative that describes the mathematical work that will unfold in that unit. Each lesson in the unit also has a narrative. Lesson Narratives explain:

- A description of 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.

Activities within lessons also have a narrative, which explain:

- The mathematical purpose of the activity and its place in the learning sequence.
- What students are doing during the activity.
- What teacher needs to look for while students are working on an activity to orchestrate an effective synthesis.
- Connections to the mathematical practices when appropriate.

### Launch - Work - Synthesize

Each classroom activity has three phases.

### The Launch

During the launch, the teacher makes sure that students understand the context (if there is one) and *what the problem is asking them to do*. This is not the same as making sure the students know *how to do the problem*—part of the work that students should be doing for themselves is figuring out how to solve the problem.

### Student Work Time

The launch for an activity frequently includes suggestions for grouping students. This gives students the opportunity to work individually, with a partner, or in small groups.

### Activity Synthesis

During the activity synthesis, the teacher orchestrates some time for students to synthesize what they have learned. This time is used to ensure that all students have an opportunity to understand the mathematical goal of the activity and situate the new learning within students' previous understanding.

### Practice Problems

Each lesson includes an associated set of practice problems. Teachers may decide to assign practice problems for homework or for extra practice in class. They may decide to collect and score it or to provide students with answers ahead of time for self-assessment. It is up to teachers to decide which problems to assign (including assigning none at all).

The practice problem set 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, but all at once).

### Are You Ready For More?

Select classroom activities include an opportunity for differentiation for students ready for more of a challenge. We think of them as the "mathematical dessert" to follow the "mathematical entrée" of a classroom activity.

Every extension problem is made available to all students with the heading "Are You Ready for More?" These problems go deeper into grade-level mathematics and often make connections between the topic at hand and other concepts. Some of these problems extend the work of the associated activity, but some of them involve work from prior grades, prior units in the course, or reflect work that is related to the K–12 curriculum but a type of problem not required by the standards. They are not routine or procedural, and they are not just "the same thing again but with harder numbers."

They are intended to be used on an opt-in basis by students if they finish the main class activity early or want to do more mathematics on their own. It is not expected that an entire class engages in *Are You Ready for More? *problems, and it is not expected that any student works on all of them. *Are You Ready for More?* problems may also be good fodder for a Problem of the Week or similar structure.

### Instructional Routines

The kind of instruction appropriate in any particular lesson depends on the learning goals of that lesson. Some lessons may be devoted to developing a concept, others to mastering a procedural skill, yet others to applying mathematics to a real-world problem. These aspects of mathematical proficiency are interwoven. These materials include a small set of activity structures and reference a small, high-leverage set of teacher moves that become more and more familiar to teachers and students as the year progresses.

The first instance of each routine in a course includes more detailed guidance for how to successfully conduct the routine. Subsequent instances include more abbreviated guidance, so as not to unnecessarily inflate the word count of the teacher guide.

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

Some of the instructional routines, known as Mathematical Language Routines (MLR), were developed by the Stanford University UL/SCALE team. The purpose of each MLR is described here, but you can read more about supports for students with emerging English language proficiency in the Supporting English Language Learners section.

- Analyze It
- Anticipate, Monitor, Select, Sequence, Connect
- Aspects of Mathematical Modeling
- Card Sort
- Construct It
- Draw It
- Estimation
- Extend It
- Fit It
- Graph It
- Math Talk
- MLR1: Stronger and Clearer Each Time
- MLR2: Collect and Display
- MLR3: Clarify, Critique, Correct
- MLR4: Information Gap Cards
- MLR5: Co-Craft Questions
- MLR6: Three Reads
- MLR7: Compare and Connect
- MLR8: Discussion Supports
- Notice and Wonder
- Poll the Class
- Take Turns
- Think Pair Share
- Which One Doesn’t Belong?