The following information comes directly from Open Up Resources Course Guide on the Open Up Resources Website. Some edits have been made for clarity. This information is intended to give you a better idea of how the Math 6 course is designed. Math 6 Plus is designed in a similar way, but combines content from 6th and 7th grade standards.
Grade 6 begins with a unit on reasoning about area and understanding and applying concepts of surface area. It is common to begin the year by reviewing the arithmetic learned in previous grades, but starting instead with a mathematical idea that students haven’t seen before sets up opportunities for students to surprise the teacher and themselves with the connections they make. Instead of front-loading with review and practice from prior grades, these materials incorporate opportunities to practice elementary arithmetic concepts and skills through warm-ups, in the context of instructional tasks, and in practice problems as they are reinforcing the concepts they are learning in the unit.
One of the design principles of these materials is that students should encounter plenty of examples of a mathematical or statistical idea in various contexts before that idea is named and studied as an object in its own right. For example, in the first unit, students will generalize arithmetic by writing simple expressions before they study algebraic expressions as a class of objects in the sixth unit. Sometimes this principle is put into play several units before a concept is developed more fully, and sometimes in the first several lessons of a unit, where students have a chance to explore ideas informally and concretely, building toward a more formal and abstract understanding later in the unit.
These materials were authored by Illustrative Mathematics in collaboration with Open Up Resources. They were piloted and revised in the 2016–2017 school year.
The course contains nine units; each of the first eight are anchored by a few big ideas in grade 6 mathematics. Units contain between 15 and 20 lesson plans. Each unit has a diagnostic assessment for the beginning of the unit and an end-of-unit assessment. Longer units also have a mid-unit assessment. Unit 9 in the course is structured differently, and contains optional lessons that help students apply and tie together big ideas from the year.
Each lesson plan is designed to fit within a 45–50 minute period. Some lessons contain optional activities that provide additional practice for teachers to use at their discretion.
There are two ways students can interact with these materials. Students can work solely with printed workbooks or pdfs. Alternatively, if all students have access to an appropriate device, students can look at the task statements on that device and write their responses in a notebook or the print companion for the digital materials. It is recommended that if students are to access the materials this way, they keep the notebook carefully organized so that they can go back to their work later.
Many activities are written in a card sort, matching, or info gap format that requires teachers to provide students with a set of cards or slips of paper that have been photocopied and cut up ahead of time. Teachers might stock up on two sizes of resealable plastic bags: sandwich size and gallon size.
In a problem-based curriculum, students work on carefully crafted and sequenced mathematics problems during most of the instructional time. Teachers help students understand the problems and guide discussions to be sure that the mathematical take-aways are clear to all. Not all mathematical knowledge can be discovered, so direct instruction is sometimes appropriate. On the other hand, some concepts and procedures follow from definitions and prior knowledge and students can, with appropriately constructed problems, see this for themselves. In the process, they explain their ideas and reasoning and learn to communicate mathematical ideas. The goal is to give students just enough background and tools to solve initial problems successfully, and then set them to increasingly sophisticated problems as their expertise increases.
A problem-based approach may require a significant realignment of the way math class is understood by all stakeholders in a student’s education. Families, students, teachers, and administrators may need support making this shift. These materials are designed to support professional learning that is undertaken by teachers either in professional learning communities at their schools or in more formal settings. The value of a problem-based approach is that students spend most of their time in math class doing mathematics: making sense of problems, estimating, trying different approaches, selecting and using appropriate tools, evaluating the reasonableness of their answers, interpreting the significance of their answers, noticing patterns and making generalizations, explaining their reasoning verbally and in writing, listening to the reasoning of others, and building their understanding. Mathematics is not a spectator sport.
A typical lesson has four phases:
The first event in every lesson is a warm-up. A warm-up either:
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.
Four instructional routines frequently used in warm-ups are Number Talks, Notice and Wonder, Which One Doesn’t Belong, and True or False. In addition to the mathematical purposes, these routines serve the additional purpose of strengthening students’ skills in listening and speaking about mathematics.
After the warm-up, lessons consist of a sequence of one to three 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.
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.
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.
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.
Each unit begins with a pre-assessment that helps teachers gauge what students know about both prerequisite and upcoming concepts and skills, so that teachers can gauge where students are and make adjustments accordingly. The initial lesson in a unit is designed to activate prior knowledge and provide an easy entry to point to new concepts, so that students at different levels of both mathematical and English language proficiency can engage in the work. As the unit progresses, students are systematically introduced to representations, contexts, concepts, language and notation. As their learning progresses, they make connections between different representations and strategies, consolidating their conceptual understanding, and see and understand more efficient methods of solving problems, supporting the shift towards procedural fluency. The distributed practice problems give students ongoing practice, which also supports developing procedural proficiency.
Students have opportunities to make connections to real-world contexts throughout the materials. Frequently, carefully-chosen anchor contexts are used to motivate new mathematical concepts, and students have many opportunities to make connections between contexts and the concepts they are learning. Additionally, most units include a real-world application lesson at the end. In some cases, students spend more time developing mathematical concepts before tackling more complex application problems, and the focus is on mathematical contexts. The first unit on geometry is an example of this.
Selected activities are structured using Five Practices for Orchestrating Productive Mathematical Discussions (Smith & Stein, 2011), also described in Principles to Actions: Ensuring Mathematical Success for All (NCTM, 2014), and Intentional Talk: How to Structure and Lead Productive Mathematical Discussions (Kazemi & Hintz, 2014). These activities include a presentation of a task or problem where student approaches are anticipated ahead of time. Students first engage in independent think-time followed by partner or small-group work on the problem. The teacher circulates as students are working and notes groups using different approaches. Groups or individuals are selected in a specific, recommended sequence to share their approach with the class, and finally the teacher leads a whole-class discussion to make connections and highlight important ideas.
Activities that give all students experience with a new context ensure that students are ready to make sense of the concrete before encountering the abstract. For example, as their first encounter with a constant speed context in grade 6, students move a pre-measured distance at a constant speed and time each other (or watch a demonstration of this).
Activities that introduce a new concept and associated language build on what students already know and ask them to notice or put words to something new. For example, in grade 8, students use a picture of a figure and the same figure transformed to describe the move as, for example, a “quarter turn.” Over the course of later activities, they formalize the idea of a rotation about a point by a specific angle.
Activities that introduce a new representation often present the new representation of a familiar idea first and ask students to interpret it. Where appropriate, new representations are connected to familiar representations (as in tables of equivalent ratios and double number line diagrams) or extended from familiar representations (as in extending the number line to the left to show negative numbers in grade 6). Students are then given clear instructions on how to create such a representation as a tool for understanding or for solving problems. For subsequent activities and lessons, students are given opportunities to practice using these representations and to choose which representation to use for a particular problem.
Activities that formalize a definition take a concept that students have already encountered through examples, and give it a more general definition. For example, the term π is not defined until the end of the third lesson about measuring circles in grade 7.
Activities that give students a chance to identify and resolve common mistakes and misconceptions usually present some incorrect work and ask students to identify it as such and explain what is incorrect about it. Students deepen their understanding of key mathematical concepts as they analyze and critique the reasoning of others.
Activities that provide an opportunity to practice using mathematical language are focused on that as the primary goal rather than having a primarily mathematical learning goal. They are intended to give students a reason to use mathematical language to communicate. These frequently use the Info Gap instructional routine.
Activities where students work toward mastery are included for topics where experience shows students often need some additional time to work with the ideas. These activities are marked as optional because no new mathematics is covered, so if a teacher were to skip them, no new topics would be missed.
Activities that provide an opportunity to apply mathematics to a modeling or other application problem are most often found toward the end of a unit. Their purpose is to give students experience using mathematics to reason about a problem or situation that one might encounter naturally outside of a mathematics classroom.
This story of grade 6 mathematics is told in 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:
Activities within lessons also have a narrative, which explain:
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 for an activity frequently includes suggestions for grouping students. This gives students the opportunity to work individually, with a partner, or in small groups.
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.
Select classroom activities include an opportunity for differentiation for students ready for more of a challenge. 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 at grade level or that are outside of the standard K-12 curriculum. 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.
Each lesson includes an associated set of practice problems. The set includes a few problems from that day’s lesson along with a mix of topics from previous lessons. 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). Teachers may decide to assign the 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 also up to teachers whether to assign the entire set or to choose a subset to assign (including assigning none at all).
The materials contain many opportunities and tools for both formative and summative assessment. Some things are purely formative, but the tools that can be used for summative assessment can also be used formatively.