8th Grade - Gateway 3

The materials reviewed for Math Nation Grade 8 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students to guide their mathematical development. 

Examples of where and how the materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials include:

• Course Overview: A Course Overview (Unit 0) is found at the beginning of each course.  Within each Course Overview there is a Course Narrative, which contains a summary of the mathematical content contained in each course, and a Course Guide. The Course Guide contains the following sections: Introduction, About These Materials, How to Use These Materials, Assessment Overview, Scope and Sequence, Required Resources, Corrections, and Cool-Down Guidance. Each of these sections contains specific guidance for teachers on implementing lesson instruction. For example, in the About These Materials section, teachers can find an outline of and detailed information about the components of a typical lesson, including Warm-Up, Classroom Activities, Lesson Synthesis, and Cool-Down. The How to Use These Materials section contains guidance about the three phases of classroom activities (Launch-Work-Synthesize) and utilizing instructional routines. In the Scope and Sequence section, teachers will find a Pacing Guide which contains time estimates for coverage of each of the units.

• Teacher Edition: There is a Teacher Edition section for each unit that contains a unit introduction, unit assessments, and unit-level downloads. The Unit Introduction contains a summary of the mathematical content to be found in the unit. The Assessment component contains downloads for multiple types of assessments (Check Your Readiness, Mid-Unit, and End-of-Unit Assessment). Unit Level Downloads include: Student Task Statements Cool-downs, Practice Problems, Blackline Masters, and My Reflections all of which provide support for teacher planning. Each lesson has a Teacher Edition component that contains guidance for Lesson Preparation, Cool-down Guidance, and a Lesson Narrative. The Lesson Preparation component includes a Teacher Prep Video, Learning Goal(s), Required Material(s), and Full Lesson Plan downloads. Cool-down Guidance provides teachers with guidance on what to look for or emphasize over the next several lessons to support students in advancing their current understanding. The Lesson Narrative provides specific guidance about how students can work with the lesson activities.

• Full Lesson Plan: Within each Teacher Edition lesson component, teachers can find a Full Lesson Plan that contains lesson learning goals and targets, a lesson narrative, and specific guidance for implementing each of the lesson activities. The Lesson Narrative contains the purpose of the lesson, standards and mathematical practices alignments, specific instructional routines, and required materials related to the lesson. Teachers are given guidance for implementing these routines as a way of introducing students to the learning targets. There is also teacher guidance for launching lesson activities, such as suggestions for grouping students, working with a partner, or whole group discussion. The planning section identifies possible student errors and misconceptions that could occur. There is also guidance on how to support English Language Learners and Students with Disabilities.

Materials include sufficient and useful annotations and suggestions that are presented within the context of specific learning objectives. Preparation and lesson narratives within the Course Guide, Lesson Plans, Lesson Narratives, Overviews, and Warm-up provide useful annotations. Examples include:

• Course Guide, Assessments Overview, “Pre-Unit Diagnostic Assessments At the start of each unit is a pre-unit diagnostic assessment. These assessments vary in length. Most of the problems in the pre-unit diagnostic assessment address prerequisite concepts and skills for the unit. Teachers can use these problems to identify students with particular below-grade needs, or topics to carefully address during the unit. Teachers are encouraged to address below-grade skills while continuing to work through the on-grade tasks and concepts of each unit, instead of abandoning the current work in favor of material that only addresses below-grade skills…What if a large number of students can’t do the same pre-unit assessment problem? Look for opportunities within the upcoming unit where the target skill could be addressed in context…What if all students do really well on the pre-unit diagnostic assessment? Great! That means that they are ready for the work ahead, and special attention likely doesn’t need to be paid to below-grade skills.”

• Unit 1, Lesson 4,  Full Lesson Plan, 1.4.3 Exploration Activity, “Anticipated Misconceptions Students may struggle drawing the image under transformation from the quick flashes of the image because they are trying to count the number of spaces each vertex moves. Encourage these students to use the line in the image to help them reflect the image.”

• Unit 2, Lesson 5, Full Lesson Plan, “Lesson NarrativeIn previous lessons, students learned what a dilation is and practice dilating points and figures on a circular grid, on a square grid, on a coordinate grid, and with no grid. In this lesson, they work on a coordinate grid and use the coordinates to communicate precisely the information needed to perform a dilation. Students use the info gap structure. The student with the problem card needs to dilate a polygon on the coordinate grid. In order to do so, they need to request the coordinates of the polygon's vertices and the center of dilation as well as the scale factor. After obtaining all of this information from the partner with the data card, the student performs the dilation. The focus here is on deciding what information is needed and communicating clearly to request the information and explain why it is needed. One important use of coordinates in geometry is to facilitate precise and concise communication about the location of points (MP6). This allows students to indicate where the center of the dilation is and also to communicate the vertices of the polygon that is dilated.”

The materials reviewed for Math Nation Grade 8 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.

The Course Guide, About These Materials sections, states the following note about standards alignment, “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.’” All lessons in the materials have this correlation information. An example:

• Unit 7, Lesson 4, Full Lesson Plan, Lesson Standards Alignment, Building on 5.NF.5b; Addressing 8.EE.1; Building Towards 8.EE.1.

Explanations of the role of the specific grade-level mathematics in the context of the series can be found throughout the materials including but not limited to the Course Guide, Scope and Sequence section, the Course Overview, Unit Introduction, Lesson Narrative and Full Lesson Plan. Examples include:

• Course Guide, Scope and Sequence, Unit 1: Rigid Transformations and Congruence, “Work with transformations of plane figures in grade 8 draws on earlier work with geometry and geometric measurement. Students began to learn about two- and three-dimensional shapes in kindergarten, and continued this work in grades 1 and 2, composing, decomposing, and identifying shapes. Students' work with geometric measurement began with length and continued with area. Students learned to "structure two-dimensional space," that is, to see a rectangle with whole-number side lengths as composed of an array of unit squares or composed of iterated rows or iterated columns of unit squares. In grade 3, students distinguished between perimeter and area…In grade 6, students combined their knowledge of geometry and geometric measurement to produce formulas for the areas of parallelograms and triangles, using these formulas to find surface areas of polyhedra. In grade 7, students worked with scaled copies and scale drawings, learning that angle measures are preserved in scaled copies, but areas increase or decrease proportionally to the square of the scale factor…”

• Course Guide, Scope and Sequence, Unit 3: Linear Relationships, “Work with linear relationships in grade 8 builds on earlier work with rates and proportional relationships in grade 7, and grade 8 work with geometry. At the end of the previous unit on dilations, students learned the terms ‘slope’ and ‘slope triangle,’ used the similarity of slope triangles on the same line to understand that any two distinct points on a line determine the same slope, and found an equation for a line with a positive slope and vertical intercept…A proportional relationship is a collection of equivalent ratios. In high school-after their study of ratios, rates, and proportional relationships-students discard the term ‘unit rate,’referring to a to b, a:b, and ab as ‘ratios.’...”

The materials reviewed for Math Nation Grade 8 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. Instructional approaches of the program and identification of the research-based strategies can be found throughout the materials, but particularly in the Course Guide, About These Materials, and How to Use These Materials sections. 

The About These Materials section states the following about the instructional approach of the program, “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 ensure the mathematical takeaways are clear to all. Some concepts and procedures follow from definitions and prior knowledge so 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. However, not all mathematical knowledge can be discovered, so direct instruction is sometimes appropriate. 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. The materials are designed with these supports in mind. Family materials are included for each unit and assist with the big mathematical ideas within the unit. Lesson and activity narratives, Anticipated Misconceptions, and instructional supports provide professional learning opportunities for teachers and leaders. 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.”

Examples of materials including and referencing research-based strategies include:

• “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…”

• “Supporting English Language Learners This curriculum builds on foundational principles for supporting language development for all students. This section aims to provide guidance to help teachers recognize and support students' language development in the context of mathematical sense-making. Embedded within the curriculum are instructional supports and practices to help teachers address the specialized academic language demands in math when planning and delivering lessons, including the demands of reading, writing, speaking, listening, conversing, and representing in math (Aguirre & Bunch, 2012).”

• “Instructional Routines … Some of the instructional routines, known as Mathematical Language Routines (MLR), were developed by the Stanford University UL/SCALE team…”

Within the Course Guide, How to Use These Materials, a Reference section is included. 

The materials reviewed for Math Nation Grade 8 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. Comprehensive lists of supplies needed to support the instructional activities can be found in Course Guides (Required Materials), Teacher Editions, for each lesson, under Lesson Preparation (Required Material(s)), and in Teacher Guides for specific lessons. Examples include:

• Unit 1, Lesson 1, Lesson Preparation, Required Materials: “Blackline master for Activity 1.2, Cool-down, copies of blackline master, geometry toolkits (tracing paper, graph paper, colored pencils, scissors, and an index card)”

• Unit 5, Lesson 13, Lesson Preparation, Required Materials: “Cool-down, colored pencils”

• Unit 8, Lesson 4, Lesson Preparation, Required Materials: “Cool-down, compasses, four-function calculators, tracing paper”

The materials reviewed for Math Nation Grade 8 meet expectations for including an assessment system that provides multiple opportunities throughout the grade, course, and/or series to determine students’ learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

Student sample responses are provided for all assessments. Rubrics are provided for scoring restricted constructed response and extended response questions on the Mid-Unit Assessments and End-of-Unit Assessments. Mid-Unit Assessments and End-of-Unit Assessments include notes that provide guidance for teachers to interpret student understanding and make sense of students’ correct/incorrect responses. 

Suggestions to teachers for following up with students are provided throughout the materials via the Check-Your-Readiness, Mid-Unit, and End-of-Unit Teacher Guides, and each lesson provides a Cool-down Guidance that details how to support student learning.

Examples of the assessment system providing multiple opportunities to determine students' learning and sufficient guidance to teachers for interpreting student performance include:

• Course Guide, Assessments Overview states the following: “Rubrics for Evaluating Students Answers Restricted constructed response and extended response items have rubrics that can be used to evaluate the level of student responses. 

  • Restricted Constructed Response

    • Tier 1 response: Work is complete and correct.

    • Tier 2 response: Work shows General conceptual understanding and mastery, with some errors.

    • Tier 3 response: Significant errors in work demonstrate lack of conceptual understanding or mastery. Two or more error types from Tier 2 response can be given as the reason for a Tier 3 response instead of listing combinations.

  • Extended Response

    • Tier 1 response: Work is complete and correct, with complete explanation or justification. 

    • Tier 2 response: Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification. 

    • Tier 3 response: Work shows a developing but incomplete conceptual understanding, with significant errors. 

    • Tier 4 response: Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.”

• Unit 8, End-of-Unit Assessment (A), Question 4, “Plot these numbers on the number line: \sqrt{2}, \sqrt{5}, \sqrt[3]{8}, \sqrt{9}, \sqrt{15}, \sqrt[3]{25} (A number line is shown going start at 0, 1, 2, 3, 4, and 5 are labeled) Solution \sqrt{2}, \sqrt[3]{8}, \sqrt{5}, \sqrt[3]{25}, \sqrt{9}, \sqrt{15} (on the number line) Minimal Tier 1 response: Work is complete and correct. Sample: See number line. Tier 2 response: Work shows general conceptual understanding and mastery, with some errors. Sample errors: some of the irrational numbers are placed in the correct unit interval but not within the correct half of the interval; one or two points are plotted completely incorrectly; all points are correct but at least two are not labeled (so it is not possible to tell which point represents which number). Tier 3 response: Significant errors in work demonstrate lack of conceptual understanding or mastery. Sample errors: three or more points are placed completely incorrectly. ”

Examples of the assessment system providing multiple opportunities to determine students' learning and suggestions to teachers for following up with students include:

• Course Guide, Cool-Down Guidance, states the following: “Each cool-down is placed into one of three support levels: 1. More chances. This is often associated with lessons that are exploring or playing with a new concept. Unfinished learning for these cool-downs is expected and no modifications need to be made for upcoming lessons. 2. Points to emphasize. For cool-downs on this level of support, no major accommodations should be made, but it will help to emphasize related content in upcoming lessons. Monitor the student who have unfinished learning throughout the next few lessons and work with them to become more familiar with parts of the lesson associated with this cool-down. Perhaps add a few minutes to the following class to address related practice problems, directly discuss the cool-down in the launch or synthesis of the warm-up of the next lesson, or strategically select students to share their thinking about related topics in the upcoming lessons. 3. Press pause. This advises a small pause before continuing movement through the curriculum to make sure the base is strong. Often, upcoming lessons rely on student understanding of the ideas from this cool-down, so some time should be used to address any unfinished learning before moving on to the next lesson.”

• Unit 1, Check-Your-Readiness (B), Question 2, “The content assessed in this problem is first encountered in Lesson 8: Rotation Patterns. Students identify parallel and perpendicular lines. If most students struggle with this item, plan to use Lesson 3 Activity 1 to review the term parallel using the isometric grid paper. Lesson 5 Activity 3 provides an opportunity to review the erm perpendicular.” 

• Unit 4, Lesson 4, Cool-down Guidance, “Support Level 1. More Chances. Notes Students will have more opportunities to develop procedural fluency in solving multistep equations. The card sort in the following lesson provides a great opportunity to reinforce concepts from this cool-down.”

The materials reviewed for Math Nation Grade 8 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of the course-level standards and practices across the series.

All assessments regularly demonstrate the full intent of grade-level content and practice standards through a variety of item types such as multiple choice, short answer, extended response prompts, graphing, mistake analysis, and constructed response items. Assessments are to be downloaded as Word documents or PDFs and designed to be printed and administered in-classroom. Examples Include:

• Unit 1, Mid-Unit Assessment (B), Question 6, demonstrates the full intent of 8.G.1 and MP1. “Describe a sequence of transformations that takes Figure Q to Figure P.” A coordinate plane is provided with two figures labeled Q and P. 

• Unit 2, End-of-Unit Assessment (B), Question 6, demonstrates the full intent of 8.EE.6.“All of the points in the picture are on the same line. 1. Find the slope of the line. Explain or show your reasoning. 2. Write an equation for the line. 3. What is the value of c? Explain or show your reasoning. 4. Is the point (0, -2) on this line? Explain how you know.”  A picture is shown of a line with the following points on the line labeled: (2, 4), (c, 10) and (5, 13).

• Unit 6, End-of-Unit Assessment (A), Question 4, demonstrates the full intent of 8.SP.1 and MP4.“1. Draw a scatter plot that shows a negative, linear association and has one clear outlier.  Circle the outlier. 2. Draw a scatter plot  that shows a positive association that is not linear.”

The materials reviewed for Math Nation Grade 8 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level mathematics.

Course Guide, How to Use These Materials, Supporting Students with Disabilities sections states the following: “The philosophical stance that guided the creation of these materials is the belief that with proper structures, accommodations, and supports, all children can learn mathematics. Lessons are designed to maximize access for all students, and include additional suggested supports to meet the varying needs of individual students. While the suggested supports are designed for students with disabilities, they are also appropriate for many children who struggle to access rigorous, grade-level content. Teachers should use their professional judgment about which supports to use and when, based on their knowledge of the individual needs of students in their classroom.” Suggested supports are identified for teachers in the Full Lesson Plan to support learners of all levels. Lesson and activity-level supports, identified as “Support for Students with Disabilities,” are aligned to an area of cognitive functioning and are paired with a suggested strategy aimed to increase access and eliminate barriers. Supports are classified under the following categories: eliminate barriers, processing time, peer tutors, assistive technology, visual aids, graphic organizers, and brain breaks. Examples include:

• Assistive Technology: “Assistive technology can be a vital tool for students with learning disabilities, visual spatial needs, sensory integration, and students with autism. Assistive technology supports suggested in the materials are designed to either enhance or support learning, or to bypass unnecessary barriers. Physical manipulatives help students make connections between concrete ideas and abstract representations. Often, students with disabilities benefit from hands-on activities, which allow them to make sense of the problem at hand and communicate their own mathematical ideas and solutions.” Unit 1, Lesson 3, Full Lesson Plan, 1.3.2 Exploration Activity, “Support for Students with Disabilities…Assistive Technology. Provide access to the digital version of this activity.”

• Graphic Organizers: “Word webs, Venn diagrams, tables, and other metacognitive visual supports provide structures that illustrate relationships between mathematical facts, concepts, words, or ideas. Graphic organizers can be used to support students with organizing thoughts and ideas, planning problem solving approaches, visualizing ideas, sequencing information, or comparing and contrasting ideas.” Unit 2, Lesson 11, Full Lesson Plan, 2.11.3 Exploration Activity, “Support for Students with Disabilities Executive Functioning: Graphic Organizers. Provide a Venn diagram with which to compare the between lines k and l.”

• Visual Aids: “Visual aids such as images, diagrams, vocabulary anchor charts, color coding, or physical demonstrations, are suggested throughout the materials to support conceptual processing and language development. Many students with disabilities have working memory and processing challenges. Keeping visual aids visible on the board allows students to access them as needed, so that they can solve problems independently. Leaving visual aids on the board especially benefits students who struggle with working or short term memory issues.” Unit 5, Lesson 7, Full Lesson Plan, 5.7.1 Warm Up, “Support for Students with Disabilities Visual-Spatial Processing: Visual Aids. Provide handouts of the representations for students to draw on or highlight.”

There are several accessibility options (accessed via the wrench icon in the lower left-hand corner of the screen) available to help students navigate the materials. Examples include:

• Tools Menu allow students to change the language, and access a Demos Scientific and Graphing Calculator.

• Accessibility Menu allows students to change the language, page zoom, font style, background and font color, and enable/disable the following features: text highlighter, notes, screen reader support. 

• UserWay, allows students to adjust the following: Change contrast (4 settings), Highlight links, Enlarge text (5 settings), Adjust text spacing (4 settings), Hide images, Dyslexia Friendly, Enlarge the cursor, show a reading mask, show a reading line, Adjust line height (4 settings), Text align (5 settings), Saturation (4 settings). 

Additionally, differentiated videos explaining course content - varying from review to in-depth levels of explanation - are resources available for each lesson to support students.

The materials reviewed for Math Nation Grade 8 meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.

Course Guide, How to Use These Materials, Are You Ready For More? section states the following: “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 not routine or procedural, and 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.” If individual students would complete these optional activities, then they might be doing more assignments than their classmates.

Examples of opportunities for advanced students to investigate grade-level mathematics content at a higher level of complexity include:

• Unit 3, Lesson 2, 3.2.4 Exploration Extension: Are you Ready for More?, “A giant tortoise travels at 0.17 miles per hour and an arctic hare travels at 37 miles per hour. 1. Draw separate graphs that show the relationship between time elapsed, in hours, and distance traveled, in miles, for both the tortoise and the hare. 2. Would it be helpful to try to put both graphs on the same pair of axes? Why or why not? 3. The tortoise and the hare start out together and after half an hour the hare stops to take a rest. How long does it take the tortoise to catch up?” 

• Unit 4, Lesson 14, 4.14.4 Exploration Extension: Are you Ready for More?, “In rectangle 𝐴𝐵𝐶𝐷, side 𝐴𝐵 is 8 centimeters and side 𝐵𝐶 is 6 centimeters. 𝐹 is a point on 𝐵𝐶 and 𝐸 is a point on 𝐴𝐵. The area of triangle 𝐷𝐹𝐶 is 20 square centimeters, and the area of triangle 𝐷𝐸𝐹 is 16 square centimeters. What is the area of triangle 𝐴𝐸𝐷?”

• Unit 7, Lesson 3, 7.4.4 Exploration Extension: Are you Ready for More?, "2^{12}=4096. How many other whole numbers can you raise to a power and get 4,096? Explain or show your reasoning.”

The materials reviewed for Math Nation Grade 8 meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

The Course Guide, How to Use These Materials section states the following: “The framework for supporting English language learners (ELLs) in this curriculum includes four design principles for promoting mathematical language use and development in curriculum and instruction. The design principles and related routines work to make language development an integral part of planning and delivering instruction while guiding teachers to amplify the most important language that students are expected to bring to bear on the central mathematical ideas of each unit.” The four design principles are, support sense-making, optimize output, cultivate conversation, and maximize meta-awareness. Each design principle has an explanation that goes into more detail about how teachers can use it to support students. The routines are the Mathematical Language Routines (MLRs), the materials state, “The mathematical language routines (MLRs) were selected because they are effective and practical for simultaneously learning mathematical practices, content, and language. A mathematical language routine is a structured but adaptable format for amplifying, assessing, and developing students' language. The routines emphasize uses of language that is meaningful and purposeful, rather than about just getting answers. These routines can be adapted and incorporated across lessons in each unit to fit the mathematical work wherever there are productive opportunities to support students in using and improving their English and disciplinary language use. Each MLR facilitates attention to student language in ways that support in-the-moment teacher-, peer-, and self-assessment for all learners. The feedback enabled by these routines will help students revise and refine not only the way they organize and communicate their own ideas, but also ask questions to clarify their understandings of others' ideas.” These design principles and routines are referenced under Instructional Routines, in the Full Lesson Plan for lesson, to assist teachers with lesson planning. The “Supports for English Language Learners” section within the Full Lesson Plan contains explanations of how to implement the MLRs. 

Examples where the materials provide strategies and supports for students who read, write, and/or speak in a language other than English include:

• Unit 2, Lesson 6, Full Lesson Plan, 2.6.3 Exploration Activity, “Support for English Language Learners Speaking, Listening: MLR 7 Compare and Connect. Ask students to prepare a visual display of their figures that are similar to Figure A. As students investigate each other’s work, ask students to share what transformations are especially clear in the display of similar figures. Listen for and amplify any comments about what might make the transformations clearer in the display. Then encourage students to make connections between the words ‘translation,’ ‘rotation,’ ‘reflection,’ and ‘dilation’ and how they affect the figure. Listen for and amplify language students use to describe what happens to figures under different kinds of transformations. This will foster students’ meta-awareness and support constructive conversations as they compare images of the same figure and make connections between transformations and their effects on figures. Design Principle(s): Cultivate conversation; Maximize meta-awareness”

• Unit 4, Lesson 14, Full Lesson Plan, 4.14.2 Exploration Activity, “Support for English Language Learners Representing, Speaking, Listening: MLR 2 Collect and Display. As students discuss which systems they thought would be easiest to solve and which would be hardest, create a table with the headings ‘least difficult’ and ‘most difficult’ in the two columns. Circulate through the groups and record student language in the appropriate column. Look for phrases such as ‘different variables on the same side,’ ‘variables already isolated,’ and ‘various terms.’ Invite students to share strategies they can use to address the features that make these systems of equations more difficult to solve. This will help students begin to generalize and make sense of the structures of equations for substitution. Design Principle(s): Support sense-making”

• Unit 6, Lesson 3, Full Lesson Plan, 6.3.3 Exploration Activity, “Support for English Language Learners Writing, Speaking: MLR 1 Stronger and Clearer Each Time. Use this routine to give students a structured opportunity to revise and refine their response to the last question. Ask each student to meet with 2–3 other partners in a row for feedback. Provide students with prompts for feedback that will help students strengthen their ideas and clarify their language (e.g., ‘Can you give an example?’, ‘Why do you think…?’, ‘How do you know…?’, etc.). Students can borrow ideas and language from each partner to strengthen their final version. Design Principle(s): Optimize output (for explanation)”

The materials reviewed for Math Nation Grade 8 meet expectations for providing manipulatives, both virtual and physical, that are representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

Virtual and physical manipulatives support student understanding throughout the materials. Examples include:

• Unit 1, Lesson 15, 1.15.3 Exploration Activity, students manipulate paper angles to try and form a triangle. “Your teacher will give you a page with three sets of angles and a blank space. Cut out each set of three angles. Can you make a triangle from each set that has these same three angles?” 

• Unit 3, Lesson 9, 3.9.3 Exploration Activity, students use an applet of a graph to answer questions and write a question about the information represented. “Here is a graph that shows the amount on Han’s fare card for every day of last July. 1. Describe what happened with the amount on Han’s fare card in July. 2. Plot and label 3 different points on the line. 3. Write an equation that represents the amount on the card in July, y, after x days. 4. What value makes sense for the slope of the line that represents the amounts on Han’s fare card in July?” A GeoGebra applet is provided that shows the amount on Han’s fare card, students are able to use the applet to put additional points on the line and draw additional lines.

• Unit 5, Lesson 11, 5.11.2 Exploration Activity, students use an applet to investigate the relationship between the height and volume of water in a cylinder. “Use the applet to investigate the height of water in the cylinder as a function of the water volume. 1. Before you get started, make a prediction about the shape of the graph. 2. Check Reset and set the radius and height of the graduated cylinder to values you choose. 3. Fill the cylinder with different amounts of water and record the data in the table. 4. Create a graph that shows the height of the water as a function of the water volume. 5. Choose a point on the graph and explain its meaning in the context of the situation.” Two applets are provided, one is a Geogebra applet that allows students to change the height and radius of the cylinder and fill it with water. Students can pause and fill at any time and the volume, height and diameter are displayed. The second is a Desmos applet for students to use to create the graph of the function.