Given the nature of math classrooms, students come with differing math identities. Some students see themselves as doers of mathematics, and others do not. Furthermore, apparent inequities in math instruction suggest that some students have opportunities to bring their voice into the classroom, and others do not. In order to extend the invitation to do mathematics to all students, explicit development of the math learning community is required.

Classroom environments that foster a sense of community that allows students to express their mathematical ideas—together with norms for students to communicate their mathematical thinking, both orally and in writing, to their peers and their teacher, using the language of mathematics—positively affect participation and engagement among all students (NCTM, 2014).

To support development of the math learning community, the first six lessons of each course embed structures to collectively identify what it looks and sounds like to do math together, and to create and reflect on classroom norms that support those actions.

Beyond the first six days, revisit these norms at least once a week to sustain the math learning community. Consistently returning to these ideas shows students that the math learning community is valued as much as the math content. Provide students with opportunities to reflect on the norms by asking them to state which norms are the most challenging for them and why. Teacher-reflection questions offer periodic reminders at points in a unit when it may be helpful to reflect on these norms.

Additional teaching moves support the development of math learning communities throughout the school year. The table highlights teaching moves, put forth by Phil Daro and the SERP Institute, that support students’ engagement in the mathematical practices. A solid math learning community exists when all students display the vital student actions described in the table.

Teaching Moves to Support Math Community

student vital actions	teacher moves
All students participate.	Assign rotating roles, and provide routines for collaboration so that every student is actively engaged in each task and has experience in all roles over time. Ask confused students to show where they got lost, or ask a question that can help them move forward (so that they respond with more than “I don’t get it” or “How do you do it?”).  Check to see if there are recognizable patterns between participation and prior achievement or social groups (for example, English language proficiency, race/ethnicity, or gender).
Students say a second sentence.	Ask and encourage students to ask:  “Can you tell me more about that?” “Why do you think that?” “What changed and what stayed the same?” “Is that an answer that makes sense for this problem? How do you know?” “How did you get that answer? Why did you (reference students' work)?” “Is it always true? Sometimes true?”
Students talk about each other’s thinking.	Show and discuss students’ work with mathematics concepts. Questions that may be used to prompt students: “Did anyone approach the problem a different way?” “How is your thinking different from theirs?” “What does their way of thinking help you understand?” “Do you think their method would work with this kind of problem? Why or why not?” Try responding to questions from groups only when no member of the group can answer the question and every member of the group can ask it.
Students revise their thinking.	Play the role of not understanding when a student presents an explanation. Ask:, “Could you help me make sense of your thinking? Could you revise your explanation?” Invite a student to quote a classmate’s statement that inspired them to revise.  Organize students to confer in small groups after whole-class presentations, to revise and refine their way of thinking.
Students engage and persevere.	Ask a student, who has given a wrong answer, additional questions to explore their thinking. Demonstrate curiosity about that thinking. Invite students to share their thinking and attempts, even when they have not found a viable solution.  Encourage groups that are “finished” earlier than others to analyze their work and to revise their explanation so more students will understand it, or to look for an alternative approach.
Students use general and discipline-specific academic language.	Give students sentence frames and probing questions that feature important terms before beginning small-group work. Accept students’ everyday way of talking as a starting point for joining the math conversation.  Refer to students’ statements, using students’ language while strategically incorporating more precise academic language, with the addition of a key word or phrase.
English learners produce language.	Provide multiple contexts for everyday words that have precise mathematical meaning, and invite students to explain what the word refers to in each context. Ask them to use the word to make connections between the different representations. Encourage students to use language to construct meaning from representations, with prompts such as:  “Explain where you see (length, ten, oranges) in the (figure, equation, table). How do you know it represents the same thing?”  Ensure that every student speaks, listens, reads, and writes.

For more details and a full list of teaching moves, visit the SERP Institute site: https://www.serpinstitute.org/5x8-card/vital-student-actions 

Opportunities for communication—in particular classroom discourse—are foundational to the problem-based structure of the IM K–5 Math curriculum. The National Council of Teachers of Mathematics’s Principles and Standards for School Mathematics (NCTM, 2000) states, “Students, who have opportunities, encouragement, and support for speaking, writing, reading, and listening in mathematics classes, reap dual benefits: they communicate to learn mathematics, and they learn to communicate mathematically.” Opportunities for each action are intentionally embedded directly into the curriculum materials, through the student-task structures, and supported by the accompanying teacher directions.

One highly visible form of discourse is students’ discussion during the lesson. Another, less visible form of discourse is writing. While this often is  seen only as the written responses in a student book, journal writing offers an additional opportunity to support each student in their learning of mathematics.

Writing is a useful catalyst in learning mathematics because not only does it supply students with an opportunity to describe their feelings, thinking, and ideas clearly, but also it serves as a means of communicating with other people (Baxter, Woodward, Olson, & Robyns, 2002; Liedtke & Sales, 2001; NCTM, 2000). The NCTM (1989) suggests that writing about mathematics helps students clarify their ideas and develop a deeper understanding of the mathematics at hand.
 

To encourage journal writing in math class, use the included list of journal prompts at any point in time during a unit and across the year. These prompts are divided into two overarching categories: Reflecting on Content and Practices and Reflecting on Learning and Feelings about Math.

Prompts for the first category focus on students’ learning, or on specific learning objectives in each lesson. Students reflect on the mathematical content because the act of writing generally entails careful analysis, encouraging the explicit connection between what is known and new knowledge, which becomes incorporated into a consciously constructed network of meaning (Vygotsky, 1987). For example, when students write about ways in which the math they learned in class that day is connected to something they knew from an earlier unit or grade, they explicitly connect their prior and new understandings.

Prompts for the second category are more metacognitive and focus on students’ feelings, mindset, and thinking around using mathematics. Writing about these subjects promotes metacognitive frameworks that extend students’ reflection and analysis (Pugalee, 2001, 2004). For example, as students describe an aspect of a lesson they found challenging, they have the chance to reflect on the factors that made it a challenge.

John Dewey (1933) asserted that students make sense of the world through metacognition, making connections between their lived experiences and their knowledge base, and argued that education should offer students opportunities to make connections between school and their lived experiences in the world. Ladson-Billings encourages the idea that teachers must help students effectively connect their culturally- and community-based knowledge to the learning experiences taking place in the classroom. These beliefs support the need for students to reflect continually not only on the mathematics, but on their own beliefs and experiences as well.

The prompts live in the Course Guide. Use them at any point during the year, regardless of category. In the early grades, use them as discussion prompts between partners, or ask students to respond to a prompt in the form of a drawing or an example from their work of the day. In later grades, students establish a math journal at the beginning of the year and record their reflections at the beginning, in the middle, or at the end of lesson, depending on the prompt. For schools or districts that require homework, the prompts serve as a nice way for students to reflect on the learning of the day or to ask questions they may not have asked during the class period.

Journal writing not only encourages explicit connections between current and new knowledge and promotes metacognitive frameworks to extend ideas, but also offers opportunities for teachers to learn more about each student’s identity and math experiences. Writing in mathematics offers a means for teachers to forge connections with students who typically drift—or run rapidly—away from mathematics, and offers students the opportunity to continually relate mathematical ideas to their own lives (Baxter, Woodward, and Olson, 2005). Writing prompts and journaling work well because students, who may not advocate for themselves when they are struggling, get their voices heard in a different way, and thus their needs met (Miller, 1991).

Use these questions and prompts with the intention that students communicate to learn mathematics and learn to communicate mathematically.
 

Reflecting on Content and Practices

• What math did you learn and do today that connected to something you knew from an earlier unit or grade? 
• Describe a time when you used math outside of school.
• How did your thinking change about something in math today?
• How did any predictions you made in class today work out? Why do you think that happened?
• What questions do you still have about the math today? What new questions do you have?
• Describe the way you solved a problem in class today that makes you proud. 
• Where do you see the math you did in class today outside of school? (MP4) 
• What math tool did you find most helpful today? Why? (MP5) 
• What patterns did you notice in the mathematics today? Why did that pattern happen? (MP7, MP8)
• Starter prompts:
  • The most important thing I learned today is . . . 
  • Today, I struggled and worked through a problem when . . . (MP1) 
  • I could use the math that I learned today outside of school when I . . . 
  • At the end of this unit, I want to know how to . . . 
  • I knew my answer was right today when . . . 
  • Another strategy I could have used to solve a problem today is . . . 
  • The most important point to remember about the strategy we used today is . . .

Reflecting on Learning and Feelings about Math

• Describe something you understand well after today’s lesson, or describe something that was confusing or challenging. 
• In math class, it’s important to explain your thinking. Describe a time when you explained your ideas to other people in your class. (MP3) 
• In math class, it’s important to listen to other people’s ideas. Describe a time when you learned something by listening carefully to someone in your class. (MP3) 
• What does it mean to be “good at math”? 
• Describe a time when you asked a question about the math that you were doing. How did asking a question help you? 
• If you could change anything about math class, what would you change? Why?
• Starter prompts:
  • I learned from a mistake today in math when . . .
  • When it comes to math, I find it hard to . . .
  • I love math because . . .
  • I felt heard during class today when . . .
  • I felt my ideas were valued during class today when . . .
  • The most helpful thing that happened today was . . .

Centers are intended to give students time to practice skills and concepts that are developed across the year. There are two types of centers. Addressing centers address the work of a lesson or a section of a unit. Supporting centers review prior unit or prior grade-level understandings and fluencies.

Each center builds across multiple stages that may span several grades. For example, Get Your Numbers in Order, a center in which students use their understanding of relative magnitude to order numbers, has five stages that span IM Grades 1–5. These centers build toward the content in a lesson or a section, develop fluency across a grade level, or preview content for an upcoming unit. In IM Kindergarten and IM Grade 1, centers are an integral part of the lessons, so additional suggested centers are not included in each lesson. Note: Early center stages in IM Kindergarten may build toward the aligned kindergarten grade-level standards.

Structure of Center Time

In IM Kindergarten and IM Grade 1, center time is built into the lessons so that students have a chance to spend more time on topics that require time to develop understanding. New centers are introduced, and students are given a choice to work on previously introduced centers.

In IM Grades 1 and 2, there is a center day at the end of each section of a unit. In IM Grade 2, these lessons are optional. New centers are introduced, and students also choose between previously introduced centers that reinforce content from the unit or build grade-level fluencies.

In IM Grades 3–5, center time is in addition to regular class time, as desired by the teacher. Occasionally, optional center-day lessons are included in a unit to introduce a center to students, but in general, centers are provided as an extra resource for teachers.

Use centers in a variety of additional ways. Students work on centers if a lesson is completed and there is class time remaining. Entire class sessions dedicated to centers allow students to practice, or solidify the mathematical ideas of a unit. Students work on center activities during morning work time, or any other free periods throughout the day. Use centers to support students when practice with prior grade-level standards is needed.
 

Teaching mathematics requires continual learning. Teachers must be adept at moment-to-moment decision making in order to engage students in rich discussions of mathematical content (O’Connor & Snow, 2018). This learning, embedded within the IM teacher’s daily work, is a collective experience within professional-learning communities (PLCs). To support teachers and coaches in this collective work, each unit section has an activity identified as a “PLC activity.” This activity either highlights an important mathematical idea in the unit, or has complex facilitation that benefits from teachers planning and rehearsing the activity together. Also included is a structure for teachers to use as they work together in professional-learning communities.

The suggested structure is categorized as pre-, during-, and post-lesson, offering teachers opportunities to experiment with instruction, during both planning and the classroom enactment, by collectively discussing instructional decisions in the moment (Gibbons, Kazemi, Hintz, & Hartmann, 2017). These suggestions are meant to provide guidance to a professional-learning community of teachers and coaches, who meet to plan for upcoming lessons. While using all suggestions in the given structure is ideal, they are flexible enough to adapt to fit any schedule or context.

Suggested before a professional-learning community meeting

• Read the upcoming lesson that is the focus of the meeting.
• Review the Cool-down of previous lessons.
• Discuss:
  • Students’ current understandings.
  • Ways in which these understandings build toward the PLC activity.

Suggested during a professional-learning community meeting

• Do the math of the PLC activity individually.
• Read the CCSS and the learning goal addressed by the activity.
• Discuss how the standard and the learning goal are reflected in what is asked of students in the activity. Think about:
  • Are students conceptually explaining a new, or developing, understanding?
  • Are students making connections between a conceptual understanding and a procedure or process?
• Based on the Cool-down in previous lessons, discuss one or two ways students may complete the activity.
• Discuss:
  • How students’ responses reflect the CCSS and the lesson learning goal.
  • Any unfinished learning that students have.
• Based on these discussions, make a plan to:
  • Monitor for look-fors during students’ work time.
  • Ask questions that assess and advance students’ thinking.
  • Invite students to share their work and discussion, during the Activity Synthesis.

Suggested after a professional-learning community meeting

• Record observations as students work.
• Review the Cool-down in relation to the learning goal of the lesson.

Productive Discussions

Promoting productive and meaningful conversations between students and teachers is essential to success in a problem-based classroom. To facilitate these conversations, the IM curriculum incorporates the framework presented in 5 Practices for Orchestrating Productive Mathematical Discussions (Smith & Stein, 2011). The 5 Practices are: anticipating, monitoring, selecting, sequencing, and making connections between students’ responses. All IM lessons support the practices of anticipating, monitoring, and selecting students’ work to share during whole-group discussions. In lessons in which students make connections between representations, strategies, concepts, and procedures, the Lesson Narrative and the Activity Narrative support the practices of sequencing and connecting as well, and the lesson is tagged so that these opportunities are easily identifiable. For additional opportunities to connect students’ work, look for activities tagged with MLR7 Compare and Connect. Similar to the 5 Practices routine, MLR7 supports the practices of monitoring, selecting, and making connections. In curriculum workshops and PLCs, rehearse and reflect on enacting the 5 Practices.

Teacher Learning Through Curriculum Materials

For all students to have access to mathematical learning opportunities, it is important for teachers to believe that every student can learn mathematics. And it is the responsibility of teachers to provide equitable instruction and to position all students in a way that supports learning. Getting better at teaching requires planning at the course level, the unit level, and the lesson level, and reflecting on and improving each day’s instruction. Equitable instruction requires the development of teachers’ knowledge of mathematics and the socio-cultural contexts of their students in order to deepen the learning for all. Key components of the materials support teachers in understanding the mathematics they are teaching, the students they are teaching, or in some cases, both.

Unit, Lesson, and Activity Narratives

The narratives included in the materials afford teachers with a deeper understanding of the mathematics and its progression within the materials.

Authentic Use of Contexts and Suggested Launch Adaptations

The use of authentic contexts and adaptations offer students opportunities to bring their own experiences to the lesson activities and to see themselves in the materials and the mathematics. When academic knowledge and skills are taught within students’ lived experiences and frames of reference, “they are more personally meaningful, have higher interest appeal, and are learned more easily and thoroughly” (Gay, 2010). By design, lessons include contexts in which students see themselves in the activities or learn more about others’ cultures and experiences. In places where there are opportunities to adapt a context to be more relevant to students, suggested prompts elicit these ideas.

Two sections within each lesson plan support teachers in learning more about what each student knows and provide guidance on ways to respond to students’ understandings and ideas.

Advancing Student Thinking
This section offers look-fors and questions to support students as they engage in an activity. Effective teaching requires supporting students as they work on challenging tasks, without taking over the process of thinking for them (Stein, Smith, Henningsen, & Silver, 2000). As teachers monitor during the course of an activity, they gain insight into what students know and are able to do. Based on these insights, the Advancing Student Thinking section provides questions that advance students’ understanding of mathematical concepts, strategies, or connections between representations.

Response to Student Thinking
Most lessons end with a Cool-down to formatively assess students’ thinking in relation to the learning goal of the day. The materials offer guidance to support students in meeting the learning goals. This guidance falls into one of two categories, Next-Day Support or Prior-Unit Support, based on anticipated student responses. This guidance offers ways to continue teaching grade-level content, with appropriate and aligned practice and support for students. These suggestions range from providing students with more concrete representations in the next day’s lesson to recommending a section from a prior unit, with activities that directly connect to the concepts in the lesson.

Teacher-Reflection Questions

To encourage reflection on the classroom teaching and learning, each lesson includes a teacher-directed reflection question on the mathematical work or pedagogical practices of the lesson. The questions are drawn from four categories: mathematical content, pedagogy, student thinking, and beliefs and positioning. The questions are designed to be used by individuals, grade-level teams, coaches, and anyone who supports teachers.

To ensure that all students have access to an equitable mathematics program, educators need to identify, acknowledge, and discuss the mindsets and beliefs they have about students’ abilities (NCTM, 2014). The beliefs-and-positioning questions support the identification and acknowledgment of teachers’ mindsets and beliefs. These questions prompt reflection, and challenge the assumptions teachers make—about mathematics, learners of mathematics, and the communication of mathematics in their classrooms.

Coherent Progression

To support students in making connections to prior understandings and upcoming grade-level work, it is important that teachers understand the progressions in the materials. Grade-level, unit, lesson, and activity narratives describe decisions about the organization of mathematical ideas, connections to prior and upcoming grade-level work, and the purpose of each lesson and activity. When appropriate, the narratives explain whether a decision about the scope and sequence is required by the standards or a choice made by the authors.

The basic architecture of the materials supports all learners through a coherent progression of the mathematics, based both on the standards and on research-based learning trajectories. Activities and lessons are parts of a mathematical story that spans units and grade levels. This coherence allows students to view mathematics as a connected set of ideas that makes sense.

Every unit, lesson, and activity has the same overarching design structure: The learning begins with an invitation to the mathematics, is followed by a deep study of concepts and procedures, and concludes with an opportunity to consolidate understanding of mathematical ideas. The invitation to the mathematics is particularly important because it offers students access to the mathematics. It builds on prior knowledge and encourages students to use their own language to make sense of ideas before formal language is introduced, both of which are consistent with the principles of Universal Design for Learning.

The overarching design structure at each level is as follows:

• Each unit starts with an invitation to the mathematics. The first few lessons provide an accessible entry point for all students and offer the opportunity to observe students’ prior understandings.
• Each lesson starts with a Warm-up to activate students’ prior knowledge and to set up the work of the day. This is followed by instructional activities in which students are introduced to new concepts, procedures, contexts, or representations, or make connections between them. The Lesson Synthesis at the end consolidates understanding and makes the learning goals of the lesson explicit. In the Cool-down that follows, students apply what they learned.
• Each activity starts with a Launch that gives all students access to the task. Independent work time follows, allowing them to grapple with problems individually before working in small groups. In the Activity Synthesis at the end, students consolidate their learning by making connections between their work and the mathematical goals. Each activity includes carefully chosen contexts and numbers that support the coherent sequence of learning goals in the lesson.

Purposeful Representations

“The power of a representation can . . . be described as its capacity, in the hands of a learner, to connect matters that, on the surface, seem quite separate. This is especially crucial in mathematics” (Bruner, 1966).

Mathematical representations in the materials serve two main purposes: to help students develop an understanding of mathematical concepts and procedures, and to help them solve problems. For example, in IM Grade 3, equal-groups drawings introduce students to the concept of multiplication. Later on, students make equal-groups drawings to find the total number of objects in situations involving equal groups.

Curriculum representations, and the grade levels at which they are used, are determined by their usefulness for particular mathematical learning goals. More concrete representations are introduced before those that are more abstract. For example, in IM Kindergarten, students begin by counting and moving objects—before they represent these objects in 5- and 10-frames—to lay the foundation for understanding the base-ten system. In later grades, these familiar representations are extended so that as students encounter greater numbers, they use place-value diagrams and more symbolic methods, such as equations, to represent their understanding. When appropriate, the reasoning behind the selection of certain representations in the materials is made explicit.

Across lessons and units, students are systematically introduced to representations and encouraged to use those that make sense to them. As their learning progresses, students make connections between different representations and the concepts and procedures they show. Over time, they see and understand more efficient methods of representing and solving problems, which supports the development of procedural fluency.

A couple of key progressions of representations occur across grade bands in different domains. These progressions, described here, support students who have unfinished learning and would benefit from more concrete representations to make sense of mathematical concepts.

 

Two-Color Counters (K–1)

Counters of one color are used frequently to represent quantities in the early grades. Students use the two-color counters to support their work in comparing, counting, combining, and decomposing quantities. In later grades, the counters visually represent properties of operations.

5-frame and 10-frame (K–2)

5- and 10-frames provide students with a way of seeing not only the numbers 5 and 10 as units, but also the combinations that make these units. Because they use the base-ten number system, it is critical for students to have robust mental representations of the numbers 5 and 10. They learn that a “ten” is when the frame is full of 10 individual counters, and when they cannot fill another full ten, the “extra” counters are ones, supporting a foundational understanding of the base-ten number system. The use of multiple 10-frames supports students in extending the base-ten number system to greater numbers.

Connecting Cubes (K–5)

Like counters, cubes are used in the early grades for comparing, counting, combining, and decomposing numbers. They are used in later grades to represent multiplication and division, and in IM Grade 5, to study volume. Use cubes that connect on multiple sides to develop students’ understanding of volume in grade 5.

Connecting Cubes in Towers of 10 (1–2)

Cubes in towers of 10 support students in using place-value structure for adding, subtracting, and comparing numbers. Connecting cubes have the advantage of students physically composing and decomposing numbers, unlike place-value blocks or Cuisenaire rods. The cubes are a helpful physical representation as students begin to unitize. For example, students come to understand that 10 of the single cubes are the same as 1 ten and 10 of the tens are the same as 1 hundred.

Base-Ten Blocks (2–5)

Base-ten blocks are used after students have had the physical experience of composing and decomposing towers of 10 cubes. The blocks offer students a way to physically represent concepts of place value and operations of whole numbers and decimals. Because students cannot break apart the blocks, as they can the connecting-cube towers, they must focus on the unit. As students regroup the blocks, they develop a visual representation of the algorithms. The size relationships among the place-value blocks and the continuous nature of the larger blocks allow students to investigate number concepts more deeply. The blocks are used to represent whole numbers and, in IM Grades 4 and 5, decimals, by defining different-size blocks as the whole.

Base-Ten Diagram (2–5)

Base-ten diagrams represent base-ten blocks after students no longer need concrete representations. Although individual units may be shown, the advantage of a place-value diagram is that it can serve as a “quick sketch” of representing numbers and operations.

 

Array (2–3)

An array is an arrangement of objects or images, in rows and columns, used to represent multiplication and division. Each column contains the same number of objects as the other columns, and each row has the same number of objects as the other rows.

Inch Tiles (2–4)

Inch tiles offer students a way to create physical representations of flat figures that have a certain area, and a way to cover a flat figure, with square units, to determine its area. Students organize inch tiles into rows and columns to connect the area of a rectangle to multiplication and division.

Tape Diagram (2–5)

Tape diagrams, resembling a segment of tape, primarily are used to represent the operations of adding, subtracting, multiplying, and dividing. Students use them, first, with whole numbers and, later, with fractions and decimal numbers, to emphasize the idea that the meaning and the properties of operations are true as the number system expands. Tape diagrams help students represent problems, visualize relationships between quantities, and solve mathematical problems.

Number-Line Diagram (2–5)

Number-line diagrams are used to represent and compare numbers, and also to represent operations. Understanding of number-line diagrams is built on students’ experience with rulers in IM Grade 2. Students begin by working with number lines, with tick marks, to represent the whole numbers. Then they work with number lines, with tick marks corresponding to multiples of 10, 100, or 1,000, to develop an understanding of place value and relative magnitude. In later grades, students understand that there are numbers between the whole numbers. They extend their work with whole-number operations on the number line to include fractions and decimals.

Fraction Strips (3–4)

Fraction strips are rectangular pieces of paper or cardboard, used to represent different parts of the same whole. They help students concretely visualize and explore fraction relationships. As students partition the same whole into different-size parts, they develop a sense for the relative sizes of fractions and for equivalence. Experience with fraction strips facilitates students’ understanding of fractions on the number line.

Area Diagram (3–5)

An area diagram is a rectangular diagram used to represent multiplication and division of whole numbers, fractions, and decimals. The area diagram, overlaid with a grid, shows individual units. As students move from working with an area diagram overlaid with a grid to one without, they move from a more concrete to a more abstract understanding of area. In an area diagram without a grid, the unit squares are not explicitly represented, which makes this diagram useful when working with greater numbers or fractions and making connections to the distributive property and algorithms.

As the factors of products become greater, area diagrams are difficult to read if the ones, tens, and eventually hundreds are shown accurately. This diagram shows a way to visualize the product of \(53 \times 31\).

It shows how to decompose the product into four parts, represented in the diagram as smaller rectangles. The size of each smaller rectangle in the diagram does not represent its actual size. The segment labeled 30 is not 30 times as long as the segment labeled 1. Even though the small rectangles do not have the correct relative sizes, the diagram still can be used to correctly decompose the product \(53 \times 31\),

\(\displaystyle 53 \times 31 = (50 \times 30) + (50 \times 1) +(3 \times 30) + (3 \times 1)\)

The diagram helps visualize geometrically why the equation is true.

Developing Conceptual Understanding and Procedural Fluency

Three aspects of rigor are essential to mathematics: conceptual understanding, procedural fluency, and the ability to apply these concepts and skills to mathematical problems with and without real-world contexts. These aspects, developed together, are interconnected in the materials in ways that support students’ understanding.

Opportunities to connect new representations and language to prior learning support students in building conceptual understanding. Access to new mathematics and problems prompts students to apply their conceptual understanding and procedural fluency to novel situations. Warm-up routines, practice problems, centers, and other built-in activities help students develop procedural fluency, which develops over time.

Task Complexity

Mathematical tasks are complex in different ways, with the source of complexity varying, based on students’ prior understandings, backgrounds, and experiences. In the curriculum, careful attention is given to the complexity of contexts, numbers, and required computation, as well as to students’ potential familiarity with given contexts and representations. To help students navigate possible complexities, without losing the intended mathematics, look to the Warm-up routine and the activity Launch for built-in preparation, and to teacher-facing narratives for further guidance.

In addition to tasks that provide all students access to the mathematics, the materials include guidance on how to ensure that all students engage in the mathematical practices during the tasks. Teacher-reflection questions and other fields in the lesson plans help assure that all students have not only access to the mathematics, but also the opportunity to truly engage in the mathematics.