Math 8

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 8 course is designed.

Course Information

Students begin grade 8 with transformational geometry. They study rigid transformations and congruence, then dilations and similarity (this provides background for understanding the slope of a line in the coordinate plane). Next, they build on their understanding of proportional relationships from grade 7 to study linear relationships. They express linear relationships using equations, tables, and graphs, and make connections across these representations. They expand their ability to work with linear equations in one and two variables. Building on their understanding of a solution to an equation in one or two variables, they understand what is meant by a solution to a system of equations in two variables. They learn that linear relationships are an example of a special kind of relationship called a function. They apply their understanding of linear relationships and functions to contexts involving data with variability. They extend the definition of exponents to include all integers, and in the process codify the properties of exponents. They learn about orders of magnitude and scientific notation in order to represent and compute with very large and very small quantities. They encounter irrational numbers for the first time and informally extend the rational number system to the real number system, motivated by their work with the Pythagorean Theorem.

What is a Problem-Based Curriculum?

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

A typical lesson has four phases:

  1. A warm-up
  2. One or more instructional activities
  3. The lesson synthesis
  4. A cool-down

The Warm-Up

The first event in every lesson is a warm-up. A warm-up either:

  • helps students get ready for the day’s lesson, or
  • gives students an opportunity to strengthen their number sense or procedural fluency.

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.

Classroom Activities

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.

  • Provide experience with a new context.
  • Introduce a new concept and associated language.
  • Introduce a new representation.
  • Formalize a definition of a term for an idea previously encountered informally.
  • Identify and resolve common mistakes and misconceptions that people make.
  • Practice using mathematical language.
  • Work toward mastery of a concept or procedure.
  • Provide an opportunity to apply mathematics to a modeling or other application problem.

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.

Lesson Synthesis

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.

Cool-Down

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.

Design Principles

Developing Conceptual Understanding and Fluency

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.

Applying Mathematics

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.

The Five Practices

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.

Task Purposes

  • Provide Experience with a New Context

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).

  • Introduce a New Concept and Associated Language

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.

  • Introduce a New Representation

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.

  • Formalize a Definition of a Term for an Idea Previously Encountered Informally

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.

  • Identify and Resolve Common Mistakes and Misconceptions that People Make

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.

  • Practice Using Mathematical Language

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.

  • Work Toward Mastery of a Concept or Procedure

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.

  • Provide an Opportunity to Apply Mathematics to a Modeling or Other Application Problem

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.

How to Use These Materials

Each Lesson and Unit Tells a Story

This story of grade 7 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:

  • 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.

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.

Are You Ready for More?

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.

Practice Problems

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).

Assessments

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.

  • Each unit begins with a diagnostic assessment of concepts and skills that are prerequisite to the unit as well as a few items that assess what students already know of the key contexts and concepts that will be addressed by the unit.
  • Each instructional task is accompanied by commentary about expected student responses and potential misconceptions so that teachers can adjust their instruction depending on what students are doing in response to the task. Often there are suggested questions to help teachers better understand students’ thinking.
  • Each lesson includes a cool-down (analogous to an exit slip or exit ticket) to assess whether students understood the work of that day’s lesson. Teachers may use this as a formative assessment to provide feedback or to plan further instruction.
  • A set of cumulative practice problems is provided for each lesson that can be used for homework or in-class practice. The teacher can choose to collect and grade these or simply provide feedback to students.
  • Each unit includes an end-of-unit written assessment that is intended for students to complete individually to assess what they have learned at the conclusion of the unit. Longer units also include a mid-unit assessment. The mid-unit assessment states which lesson in the middle of the unit it is designed to follow.