How to Use the Materials

Each Lesson and Unit Tells a Story

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 in the unit also has a narrative. Lesson narratives explain:
  • 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 narratives, which explain:
  • the mathematical purpose of the activity and its place in the learning sequence
  • what students are doing during the activity
  • what the 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.

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. The launch invites students into the lesson and helps them connect to contexts that may be unfamiliar. 

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 punch line of the activity and situate the new learning within students’ previous understanding.

Practice Problems

Each section in a unit includes an associated set of practice problems. There are 3 types of practice problems: pre-unit, lesson, and exploration. Teachers may decide to assign practice problems for homework or for extra practice in class. They may decide to collect and score them 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).

Pre-unit Problems
The practice problem set associated with the first section of each unit includes several prior grade-level questions. These questions can be used to review prerequisite material from the previous grade or as a pre-unit assessment, if desired.
Lesson Practice Problems
The practice problem set associated with each section typically includes one question for each lesson in the section.  
Exploration Problems
Each practice problem set also includes exploration questions that provide an opportunity for differentiation for students ready for more of a challenge. There are two types of exploration questions. One type is a hands-on activity directly related to the material of the unit that students can do either in class if they have free time, or at home. The second type of exploration is more open-ended and challenging. These problems go deeper into grade-level mathematics. They are not routine or procedural, and they are not just “the same thing again but with harder numbers.”
Exploration questions are intended to be used on an opt-in basis by students if they finish a main class 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 student works on all of them. Exploration problems may also be good fodder for a Problem of the Week or similar structure.

Instructional Routines

Instructional Routines 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. They place value on students’ voices as they communicate their developing ideas, ask questions, justify their responses, and critique the reasoning of others.

As mentioned in the Design Principles, instructional routines have a predictable structure and flow. They provide structure for both the teacher and the students. A finite set of routines support the pacing of lessons as they become familiar and save time in classroom choreography, so students can spend less time learning how to execute lesson directions, and more time on learning mathematics. Some of the instructional routines, known as Mathematical Language Routines (MLRs), were developed by the Stanford University UL/SCALE team. 

There are two types of Instructional Routines used in the materials: Warm-up Routines and Lesson Activity Routines. A list of the routines within each type is outlined in this table.

Warm-up Routines Lesson Activity Routines
Act It Out Math Language Routines (MLRs)
Choral Count MLR1: Stronger and Clearer Each Time
Estimation Exploration MLR2: Collect and Display
How Many Do You See? MLR3: Clarify, Critique, Correct
Notice and Wonder MLR4: Information Gap
Number Talk MLR5: Co-craft Questions
Questions About Us MLR6: Three Reads
True or False? MLR7: Compare and Connect
What Do You Know About _____? MLR8: Discussion Supports
Which One Doesn’t Belong? Other Lesson Activity Routines
5 Practices
Card Sort

Each lesson begins with a Warm-up Routine intentionally designed to elicit student discussions around the mathematical goal of the lesson. The Lesson Activity Routines embed structures within the tasks of the lessons that allow students to engage in the content, and collaborate in ways that support the development of student thinking and precision with language. 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.

Below is a list of each routine with a brief description of its purpose.

  • 5 Practices
  • Act It Out
  • Card Sort
  • Choral Count
  • Estimation Exploration
  • How Many Do You See?
  • MLR1 Stronger and Clearer Each Time
  • MLR2 Collect and Display
  • MLR3 Clarify, Critique, Correct
  • MLR4 Information Gap
  • MLR5 Co-craft Questions
  • MLR6 Three Reads
  • MLR7 Compare and Connect
  • MLR8 Discussion Supports
  • Notice and Wonder
  • Number Talk
  • Questions About Us
  • True or False
  • What Do You Know About _____?
  • Which One Doesn’t Belong?