Grade 8 About These Materials
These materials were created by Illustrative Mathematics in collaboration with Open Up Resources. They were piloted and revised in the 2016–2017 school year.
Further revisions were made in 2019. In addition to many small edits made for clarity, consistency, ease of use, and correctness, these revisions include:
Many more lesson-level supports for English Language Learners.
Enhancements to the student glossary, including language modifications for accessibility and many more illustrations and diagrams.
Overview of the progression of academic language provided for each instructional unit.
Updated lesson learning goals that highlight the language demands of the lesson.
Redesigned lesson-level Supports for Students with Disabilities. These include an updated framework based on principles of Universal Design for Learning, along with activity-specific instructional moves that provide teachers with actionable ways to support access to the curriculum.
Alt text for all images to enhance accessibility for users with visual impairments.
Item-by-item guidance for how to use the results of pre-unit diagnostic assessments. Each item on these assessments now includes an indication of which lesson students need the skill or concept for, and guidance on what to do if students struggle or do well on the item. Based on feedback from our community, we also changed the name of these assessments from Pre-unit Diagnostic Assessments to Check Your Readiness.
New culminating lessons for units that didn’t have one:
Grade 6, Unit 6, Lesson 19 Tables, Equations, and Graphs, Oh My!
Grade 6, Unit 7, Lesson 19 Drawing on the Coordinate Plane
Grade 7, Unit 6, Lesson 23 Applications of Expressions
Grade 8, Unit 8, Lesson 16 When Is the Same Size Not the Same Size?
New assessments. Based on feedback from our community, we learned that these units would benefit from the addition of a mid-unit assessment:
Grade 6, Unit 6, Mid-Unit Assessment
Grade 7, Unit 8, Mid-Unit Assessment
New info gap activities strategically placed to give students more frequent opportunities to practice this math language routine without waiting too long in between:
Grade 6, Unit 4, Lesson 12, Activity 2 Info Gap: How Many Would It Take?
Grade 6, Unit 7, Lesson 7, Activity 3 Info Gap: Points on the Number Line
Grade 8, Unit 4, Lesson 15, Activity 3 Info Gap: Racing and Play Tickets
Grade 8, Unit 7, Lesson 14, Activity 3 Info Gap: Distances in the Solar System
Also based on feedback from our community, we made improvements to:
image quality
consistency of Blackline Master formatting
pagination and work space for students
Each course contains nine units. Each of the first eight are anchored by a few big ideas in grade-level mathematics. Units contain between 11 and 23 lesson plans. Each unit has a diagnostic assessment for the beginning of the unit (Check Your Readiness) and an end-of-unit assessment. Longer units also have a mid-unit assessment. The last unit in each course is structured differently, and contains optional lessons that help students apply and tie together big ideas from the year.
The time estimates in these materials refer to instructional time. Each lesson plan is designed to fit within a class period that is at least 45 minutes long. Some lessons contain optional activities that provide additional scaffolding or 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.
Teachers can access the teacher materials either in print or in a browser. A classroom with a digital projector is recommended.
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. For a given activity, one set of cards can go in each small bag, and then the small bags for one class can be placed in a large bag. If these are labeled and stored in an organized manner, it can facilitate preparing ahead of time and re-using card sets between classes. Additionally, if possible, it is often helpful to print the slips for different parts of an activity on different color paper. This helps facilitate quickly sorting the cards between classes.
Design Principles
Developing Conceptual Understanding and Procedural 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 productively 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 (may be print or other media) 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.
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.
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 or extended from familiar representations. 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.
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. Often 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.
A note about standards alignments: There are three kinds of alignments to standards in these materials: building on, addressing, and building towards. Oftentimes a particular standard requires weeks, months, or years to achieve, in many cases building on work in prior grade-levels. When an activity reflects the work of prior grades but is being used to bridge to a grade-level standard, alignments are indicated as “building on.” When an activity is laying the foundation for a grade-level standard but has not yet reached the level of the standard, the alignment is indicated as “building towards.” When a task is focused on the grade-level work, the alignment is indicated as “addressing.”
A note about mathematical diagrams: Everything in a mathematical diagram has a mathematical meaning. Students are sense makers looking for connections. The mathematical diagrams provided in activities were designed to include only components with mathematical meaning. For example, while it is not uncommon to see arrows on the ends of a graph of a function, the arrows add no mathematical meaning to the graph. Arrows are typically used to imply a sense of direction, but a graph of a function is representation of all the points that make the function true, so there is no direction to imply. It is also possible for students to infer meaning that isn’t there, such as assuming the arrows mean the function continues forever in a specific direction. While this idea works for linear functions, it does not work with functions whose graphs curve or are periodic.
What Is a “Problem-Based” Curriculum?
What Students Should Know and Be Able to Do
Our ultimate purpose is to impact student learning and achievement. First, we define the attitudes and beliefs about mathematics and mathematics learning we want to cultivate in students, and what mathematics students should know and be able to do.
Attitudes and Beliefs We Want to Cultivate
Many people think that mathematical knowledge and skills exclusively belong to “math people.” Yet research shows that students who believe that hard work is more important than innate talent learn more mathematics.¹ We want students to believe anyone can do mathematics and that persevering at mathematics will result in understanding and success. In the words of the NRC report Adding It Up, we want students to develop a “productive disposition—[the] habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.”²
Knowledge
Conceptual understanding: Students need to understand the why behind the how in mathematics. Concepts build on experience with concrete contexts. Students should access these concepts from a number of perspectives in order to see math as more than a set of disconnected procedures.
Procedural fluency: We view procedural fluency as solving problems expected by the standards with speed, accuracy, and flexibility.
Application: Application means applying mathematical or statistical concepts and skills to a novel mathematical or real-world context.
These three aspects of mathematical proficiency are interconnected: procedural fluency is supported by understanding, and deep understanding often requires procedural fluency. In order to be successful in applying mathematics, students must both understand and be able to do the mathematics.
Mathematical Practices
In a mathematics class, students should not just learn about mathematics, they should do mathematics. This can be defined as engaging in the mathematical practices: making sense of problems, reasoning abstractly and quantitatively, making arguments and critiquing the reasoning of others, modeling with mathematics, making appropriate use of tools, attending to precision in their use of language, looking for and making use of structure, and expressing regularity in repeated reasoning.
What Teaching and Learning Should Look Like
How teachers should teach depends on what we want students to learn. To understand what teachers need to know and be able to do, we need to understand how students develop the different (but intertwined) strands of mathematical proficiency, and what kind of instructional moves support that development.
Principles for Mathematics Teaching and Learning
Active Learning is Best
Students learn best and retain what they learn better by solving problems. Often, mathematics instruction is shaped by the belief that if teachers tell students how to solve problems and then students practice, students will learn how to do mathematics.
Decades of research tells us that the traditional model of instruction is flawed. Traditional instructional methods may get short-term results with procedural skills, but students tend to forget the procedural skills and do not develop problem solving skills, deep conceptual understanding, or a mental framework for how ideas fit together. They also don’t develop strategies for tackling non-routine problems, including a propensity for engaging in productive struggle to make sense of problems and persevere in solving them.
In order to learn mathematics, students should spend time in math class doing mathematics.
“Students learn mathematics as a result of solving problems. Mathematical ideas are the outcomes of the problem-solving experience rather than the elements that must be taught before problem solving.”³
Students should take an active role, both individually and in groups, to see what they can figure out before having things explained to them or being told what to do. Teachers play a critical role in mediating student learning, but that role looks different than simply showing, telling, and correcting. The teacher’s role is
to ensure students understand the context and what is being asked,
ask questions to advance students’ thinking in productive ways,
help students share their work and understand others’ work through orchestrating productive discussions, and
synthesize the learning with students at the end of activities and lessons.
Teachers Should Build on What Students Know
New mathematical ideas are built on what students already know about mathematics and the world, and as they learn new ideas, students need to make connections between them.⁴ In order to do this, teachers need to understand what knowledge students bring to the classroom and monitor what they do and do not understand as they are learning. Teachers must themselves know how the mathematical ideas connect in order to mediate students’ learning.
Good Instruction Starts with Explicit Learning Goals
Learning goals must be clear not only to teachers, but also to students, and they must influence the activities in which students participate. Without a clear understanding of what students should be learning, activities in the classroom, implemented haphazardly, have little impact on advancing students’ understanding. Strategic negotiation of whole-class discussion on the part of the teacher during an activity synthesis is crucial to making the intended learning goals explicit. Teachers need to have a clear idea of the destination for the day, week, month, and year, and select and sequence instructional activities (or use well-sequenced materials) that will get the class to their destinations. If you are going to a party, you need to know the address and also plan a route to get there; driving around aimlessly will not get you where you need to go.
Different Learning Goals Require Different Instructional Moves
The kind of instruction that is appropriate at any given time depends on the learning goals of a particular lesson. Lessons and activities can:
Introduce students to a new topic of study and invite them to the mathematics
Study new concepts and procedures deeply
Integrate and connect representations, concepts, and procedures
Work towards mastery
Apply mathematics
Lessons should be designed based on what the intended learning outcomes are. This means that teachers should have a toolbox of instructional moves that they can use as appropriate.
Each and Every Student Should Have Access to the Mathematical Work
With proper structures, accommodations, and supports, all students can learn mathematics. Teachers’ instructional toolboxes should include knowledge of and skill in implementing supports for different learners.
Critical Practices
Intentional Planning
Because different learning goals require different instructional moves, teachers need to be able to plan their instruction appropriately. While a high-quality curriculum does reduce the burden for teachers to create or curate lessons and tasks, it does not reduce the need to spend deliberate time planning lessons and tasks. Instead, teachers’ planning time can shift to high-leverage practices (practices that teachers without a high-quality curriculum often report wishing they had more time for): reading and understanding the high-quality curriculum materials; identifying connections to prior and upcoming work; diagnosing students’ readiness to do the work; leveraging instructional routines to address different student needs and differentiate instruction; anticipating student responses that will be important to move the learning forward; planning questions and prompts that will help students attend to, make sense of, and learn from each other’s work; planning supports and extensions to give as many students as possible access to the main mathematical goals; figuring out timing, pacing, and opportunities for practice; preparing necessary supplies; and the never-ending task of giving feedback on student work.
Establishing Norms
Norms around doing math together and sharing understandings play an important role in the success of a problem-based curriculum. For example, students must feel safe taking risks, listen to each other, disagree respectfully, and honor equal air time when working together in groups. Establishing norms helps teachers cultivate a community of learners where making thinking visible is both expected and valued.
Building a Shared Understanding of a Small Set of Instructional Routines
Instructional routines allow the students and teacher to become familiar with the classroom choreography and what they are expected to do. This means that they can pay less attention to what they are supposed to do and more attention to the mathematics to be learned. Routines can provide a structure that helps strengthen students’ skills in communicating their mathematical ideas.
Using High Quality Curriculum
A growing body of evidence suggests that using a high-quality, coherent curriculum can have a significant impact on student learning. Creating a coherent, effective instructional sequence from the ground up takes significant time, effort, and expertise. Teaching is already a full-time job, and adding curriculum development on top of that means teachers are overloaded or shortchanging their students.
Ongoing Formative Assessment
Teachers should know what mathematics their students come into the classroom already understanding, and use that information to plan their lessons. As students work on problems, teachers should ask questions to better understand students’ thinking, and use expected student responses and potential misconceptions to build on students’ mathematical understanding during the lesson. Teachers should monitor what their students have learned at the end of the lesson and use this information to provide feedback and plan further instruction.
¹ Uttal, D.H. (1997). Beliefs about genetic influences on mathematics achievement: a cross-cultural comparison. Genetica, 99(2-3), 165-172.
² National Research Council. (2001). Adding it up: Helping children learn mathematics. J.Kilpatrick, J. Swafford, and B.Findell (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.
³ Hiebert, J., et. al. (1996). Problem solving as a basis for reform in curriculum and instruction: the case of mathematics. Educational Researcher 25(4), 12-21.
⁴ National Research Council. (2001). Adding it up: Helping children learn mathematics. J.Kilpatrick, J. Swafford, and B.Findell (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.
⁵ Steiner, D. (2017). Curriculum research: What we know and where we need to go. Standards Work. Retrieved from
A Typical Lesson
A note about optional activities: A relatively small number of activities throughout the course have been marked “optional.” Some common reasons an activity might be optional include:
The activity addresses a concept or skill that is below grade level, but we know that it is common for students to need a chance to focus on it before encountering grade-level material. If the pre-unit diagnostic assessment (”Check Your Readiness”) indicates that students don’t need this review, an activity like this can be safely skipped.
The activity addresses a concept or skill that goes beyond the requirements of a standard. The activity is nice to do if there is time, but students won’t miss anything important if the activity is skipped.
The activity provides an opportunity for additional practice on a concept or skill that we know many students (but not necessarily all students) need. Teachers should use their judgment about whether class time is needed for such an activity.
A typical lesson has four phases:
A Warm Up
One or more instructional activities
The lesson synthesis
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.
Once students and teachers become used to the routine, Warm Ups should take 5–10 minutes. If Warm Ups frequently take much longer than that, the teacher should work on concrete moves to more efficiently accomplish the goal of the Warm Up.
At the beginning of the year, consider establishing a small, discreet hand signal students can display to indicate they have an answer they can support with reasoning. This signal could be a thumbs up, or students could show the number of fingers that indicate the number of responses they have for the problem. This is a quick way to see if students have had enough time to think about the problem and keeps them from being distracted or rushed by classmates’ raised hands.
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.
Additionally, each Cool Down provides a responding to student thinking section that includes one of three types of suggestions for follow-up:
More Chances: Provides reminders regarding future opportunities for students to understand the mathematical ideas in the Cool Down. These reminders should be used to ensure that pacing is not slowed down or additional work added to next lessons for ideas that will be reinforced in later lessons.
Points to Emphasize: Emphasizes components of tasks in the current lesson or future lessons to support students who struggle with specific aspects of the Cool Down.
Press Pause: Provides guidance regarding revisiting content from prior lessons or to pause and reflect upon misunderstandings in the Cool Down by using an alternative method.