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Grade 6 How to Use the Materials

Guidance for Accelerating Students in Mathematics

We believe that understanding the concepts addressed in K–8 mathematics is extremely important, and students need time to learn this content properly. Schools should provide students who are ready for more of a challenge throughout K–8 with extension and enrichment opportunities, aiming for deeper understanding (which is great for future learning), before acceleration is considered (which can undermine future learning). (Note that classroom teachers need resources, training, funding, and structures that support this.)

There is recent evidence that too many students are being accelerated in middle school mathematics. When more students were placed into a grade 7 or 8 Algebra 1 course, the pass rates of Algebra 1 declined and the students were significantly less likely to pass Geometry and Algebra 2 (Clotfelter, 2012). Students who were de-tracked in middle school performed better in high school, even the higher achievers (Boaler, 2000). 44% of 8th graders who took Algebra 1 had to repeat it with mixed results in improvement among groups (Fong, et al., 2014). 30% of a sample of accelerated students had to retake Algebra 1 between grades 7 and 12, with very little improvement the second time (Finkelstein, et al., 2012).

Are Your Current Acceleration Practices Good for Students?

Considering historical practices and data can help show whether acceleration has been good for students. Here are some questions schools and districts should investigate before making choices about acceleration policies:

  • What are the explicit and implicit reasons for accelerating students in math?

  • What are the criteria for selecting students for an accelerated pathway, and what are the possible sources of bias in the selection process?

  • What opportunities do students have to enter an accelerated mathematics pathway later in their schooling, if they are not selected for early opportunities to join an accelerated cohort?

  • What do your current data say? Of the students accelerated to high school work in grade 8 or earlier:

    • What proportion repeats one or more high school level courses?

    • What proportion enrolls in and completes a calculus course (or other challenging, advanced math course) in high school?

    • How does their racial and ethnic composition compare to the student population of the whole district?

    • For all of these questions, what proportions are tolerable?

Use Multiple Metrics

When identifying students who may be good candidates for an accelerated pathway, it is important to use multiple metrics, including student and parent self-assessment, teacher recommendation, and assessment scores to make placement decisions for accelerated coursework. If that decision process results in disparities in the racial and ethnic composition of the group of students placed in accelerated courses, then consider both how to address opportunity gaps in K–5, as well as how to support students with the interest and aptitude but unfinished learning to enter an accelerated pathway. Providing accelerated pathways that start in high school, or offering acceleration through extra math courses or summer options, will increase the likelihood that mathematically interested students with unfinished learning have access to the same opportunities going forward as their peers who have had more opportunities from the beginning.

Additionally, consider building in a checkpoint with families after the first year of acceleration, so that students who might be better served by building a stronger foundation have a well-defined opportunity to not continue on an accelerated path.

Each Lesson and Unit Tells a Story

The story of each grade 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:

  • 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 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 punch line 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 used 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 lesson plans 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.

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 Supports for English Language Learners section.

  • Algebra Talk

  • Anticipate, Monitor, Select, Sequence, Connect

  • Group Presentations

  • 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

  • Number Talk

  • Poll the Class

  • Take Turns

  • Think Pair Share

  • True or False

  • Which One Doesn’t Belong?

Algebra Talk

What: One expression is displayed at a time. Students are given a few minutes to quietly think and give a signal when they have an answer and a strategy. The teacher selects students to share different strategies for each one, “Who thought about it a different way?” Their explanations are recorded for all to see. Students might be pressed to provide more details about why they decided to approach a problem a certain way. It may not be possible to share every possible strategy for the given limited time; the teacher may only gather two or three distinctive strategies per problem. Problems are purposefully chosen to elicit different approaches.

Where: Warm Ups

Why: Algebra Talks build algebraic thinking by encouraging students to think about the numbers and variables in an expression and rely on what they know about structure, patterns, and properties of operations to mentally solve a problem. Algebra Talks promote seeing structure in expressions and thinking about how changing one number affects others in an equation. While participating in these activities, students need to be precise in their word choice and use of language (MP6).

Anticipate, Monitor, Select, Sequence, Connect

What: These are the 5 Practices for Orchestrating Productive Mathematical Discussions (Smith and Stein, 2011). In this curriculum, much of the work of anticipating, sequencing, and connecting is handled by the materials in the activity narrative, launch, and synthesis sections. Teachers need to prepare for and conduct whole-class discussions. 

Where: Many classroom activities lend themselves to this structure.

Why: In a problem-based curriculum, many activities can be described as “do math and talk about it,” but the 5 Practices lend more structure to these activities so that they more reliably result in students making connections and learning new mathematics.

Group Presentations

Some activities instruct students to work in small groups to solve a problem with mathematical modeling, invent a new problem, design something, or organize and display data, and then create a visual display of their work. Teachers need to help groups organize their work so that others can follow it, and then facilitate different groups’ presentation of work to the class. Teachers can develop specific questioning skills to help more students make connections and walk away from these experiences with desired mathematical learning. For example, instead of asking if anyone has any questions for the group, it is often more productive to ask a member of the class to restate their understanding of the group’s findings in their own words.

MLR1: Stronger and Clearer Each Time

To provide a structured and interactive opportunity for students to revise and refine both their ideas and their verbal and written output. This routine provides a purpose for student conversation as well as fortifies output. The main idea is to have students think or write individually about a response, use a structured pairing strategy to have multiple opportunities to refine and clarify the response through conversation, and then finally revise their original written response. Throughout this process, students should be pressed for details, and encouraged to press each other for details.

MLR2: Collect and Display

To capture students’ oral words and phrases into a stable, collective reference. The intent of this routine is to stabilize the fleeting language that students use during partner, small-group, or whole-class activities in order for student’s own output to be used as a reference in developing their mathematical language. The teacher listens for, and scribes, the student output using written words, diagrams and pictures; this collected output can be organized, revoiced, or explicitly connected to other language in a display for all students to use. This routine provides feedback for students in a way that increases accessibility while simultaneously supporting meta-awareness of language.

MLR3: Clarify, Critique, Correct

To give students a piece of mathematical writing that is not their own to analyze, reflect on, and develop. The intent is to prompt student reflection with an incorrect, incomplete, or ambiguous written argument or explanation, and for students to improve upon the written work by correcting errors and clarifying meaning. This routine fortifies output and engages students in meta-awareness. Teachers can demonstrate with meta-think-alouds and press for details when necessary.

MLR4: Information Gap Cards

What: Students conduct a dialog in a specific way. In an Info Gap, one partner gets a problem card with a math question that doesn’t have enough given information, and the other partner gets a data card with information relevant to the problem card. Students ask each other questions like “What information do you need?” and are expected to explain what they will do with the information. The first few times students engage in these activities, the teacher should demonstrate, with a partner, how the discussion is expected to go. Once students are familiar with these structures, less set-up will be necessary.

Why: This activity structure is designed to strengthen the opportunities and supports for high-quality mathematical conversations. Mathematical language is learned by using mathematical language for real and engaging purposes. These activities were designed such that students need to communicate in order to bridge information gaps. During effective discussions, students should be supported to do the following: pose and answer questions, clarify what is asked and happening in a problem, build common understandings, and share experiences relevant to the topic.

MLR5: Co-Craft Questions

To allow students to get inside of a context before feeling pressure to produce answers, and to create space for students to produce the language of mathematical questions themselves. Through this routine, students are able to use conversation skills as well as develop meta-awareness of the language used in mathematical questions and problems. Teachers should push for clarity and revoice oral responses as necessary.

MLR6: Three Reads

To ensure that students know what they are being asked to do, and to create an opportunity for students to reflect on the ways mathematical questions are presented. This routine supports reading comprehension of problems and meta-awareness of mathematical language. It also supports negotiating information in a text with a partner in mathematical conversation.

MLR7: Compare and Connect

To foster students’ meta-awareness as they identify, compare, and contrast different mathematical approaches, representations, and language. Teachers should demonstrate thinking out loud (e.g., exploring why we one might do or say it this way, questioning an idea, wondering how an idea compares or connects to other ideas or language), and students should be prompted to reflect and respond. This routine supports meta-cognitive and meta-linguistic awareness, and also supports mathematical conversation.

To support rich discussions about mathematical ideas, representations, contexts, and strategies. The examples provided can be combined and used together with any of the other routines. They include multi-modal strategies for helping students comprehend complex language and ideas, and can be used to make classroom communication accessible, to foster meta-awareness of language, and to demonstrate strategies students can use to enhance their own communication and construction of ideas.

Notice and Wonder

What: This routine can appear as a Warm Up or in the launch or synthesis of a classroom activity. Students are shown some media or a mathematical representation. The prompt to students is “What do you notice? What do you wonder?” Students are given a few minutes to think of things they notice and things they wonder, and share them with a partner. Then, the teacher asks several students to share things they noticed and things they wondered; these are recorded by the teacher for all to see. Sometimes, the teacher steers the conversation to wondering about something mathematical that the class is about to focus on.

Where: Appears frequently in Warm Ups but also appears in launches to classroom activities.

Why: The purpose is to make a mathematical task accessible to all students with these two approachable questions. By thinking about them and responding, students gain entry into the context and might get their curiosity piqued. Taking steps to become familiar with a context and the mathematics that might be involved is making sense of problems (MP1). Note: Notice and Wonder and I Notice/I Wonder are trademarks of NCTM and the Math Forum and used in these materials with permission.

Number Talk

What: One problem is displayed at a time. Students are given a few minutes to quietly think and give a signal when they have an answer and a strategy. The teacher selects students to share different strategies for each problem, “Who thought about it a different way?” Their explanations are recorded for all to see. Students might be pressed to provide more details about why they decided to approach a problem a certain way. It may not be possible to share every possible strategy for the given limited time; the teacher may only gather two or three distinctive strategies per problem. Problems are purposefully chosen to elicit different approaches, often in a way that builds from one problem to the next.

Where: Warm Ups

Why: Number talks build computational fluency by encouraging students to think about the numbers in a computation problem and rely on what they know about structure, patterns, and properties of operations to mentally solve a problem. Dot images are similar to number talks, except the image used is an arrangement of dots that students might count in different ways. While participating in these activities, students need to be precise in their word choice and use of language (MP6).

Poll the Class

What: This routine is used to register an initial response or an estimate, most often in activity launches or to kick off a discussion. It can also be used when data needs to be collected from each student in class, for example, “What is the length of your ear in centimeters?” Every student in class reports a response to the prompt. Teachers need to develop a mechanism by which poll results are collected and displayed so that this frequent form of classroom interaction is seamless. Smaller classes might be able to conduct a roll call by voice. For larger classes, students might be given mini-whiteboards or a set of colored index cards to hold up. Free and paid commercial tools are also readily available.

Why: Collecting data from the class to use in an activity makes the outcome of the activity more interesting. In other cases, going on record with an estimate makes people want to know if they were right and increases investment in the outcome. If coming up with an estimate is too daunting, ask students for a guess that they are sure is too low or too high. Putting some boundaries on possible outcomes of a problem is an important skill for mathematical modeling (MP4).

Take Turns

What: Students work with a partner or small group. They take turns in the work of the activity, whether it be spotting matches, explaining, justifying, agreeing or disagreeing, or asking clarifying questions. If they disagree, they are expected to support their case and listen to their partner’s arguments. The first few times students engage in these activities, the teacher should demonstrate, with a partner, how the discussion is expected to go. Once students are familiar with these structures, less set-up will be necessary. While students are working, the teacher can ask students to restate their question more clearly or paraphrase what their partner said.

Why: Building in an expectation, through the routine, that students explain the rationale for their choices and listen to another’s rationale deepens the understanding that can be achieved through these activities. Specifying that students take turns deciding, explaining, and listening limits the phenomenon where one student takes over and the other does not participate. Taking turns can also give students more opportunities to construct logical arguments and critique others’ reasoning (MP3).

Think Pair Share

What: Students have quiet time to think about a problem and work on it individually, and then time to share their response or their progress with a partner. Once these partner conversations have taken place, some students are selected to share their thoughts with the class.

Why: This is a teaching routine useful in many contexts whose purpose is to give all students enough time to think about a prompt and form a response before they are expected to try to verbalize their thinking. First they have an opportunity to share their thinking in a low-stakes way with one partner, so that when they share with the class they can feel calm and confident, as well as say something meaningful that might advance everyone’s understanding. Additionally, the teacher has an opportunity to eavesdrop on the partner conversations so that they can purposefully select students to share with the class.

True or False

What: One statement is displayed at a time. Students are given a few minutes to quietly think and give a signal when they have an answer. The teacher selects students to share different ways of reasoning for each statement. “Who thought about it a different way?” While students may evaluate each side of the equation to determine if it is true or false, encourage students to think about ways to reason that do not require complete computations. It may not be possible to share every possible reasoning approach for the given limited time; the teacher may only gather two or three distinctive strategies per problem. Statements are purposefully chosen to elicit different approaches, often in a way that builds from one statement to the next.

Where: Warm Ups

Why: Depending on the purpose of the task, the true or false structure encourages students to reason about numeric and algebraic expressions using base-ten structure, the meaning of fractions, meaning and properties of operations, and the meaning of comparison symbols. While the structure of a true or false is similar to that of a number talk, number talks are often focused on computational strategies, while true or false tasks are more likely to focus on more structural aspects of the expressions. Often students can determine whether an equation, an inequality, or a statement is true or false without doing any direct computation. While participating in these activities, students need to be precise in their word choice and use of language (MP6).

Which One Doesn’t Belong?

What: Students are presented with four figures, diagrams, graphs, or expressions with the prompt “Which one doesn’t belong?” Typically, each of the four options “doesn’t belong” for a different reason, and the similarities and differences are mathematically significant. Students are prompted to explain their rationale for deciding that one option doesn’t belong and given opportunities to make their rationale more precise.

Where: Warm Ups

Why: Which One Doesn’t Belong fosters a need to define terms carefully and use words precisely (MP6) in order to compare and contrast a group of geometric figures or other mathematical representations.