High School - Gateway 3

The materials reviewed for enVisionMath A/G/A 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 in order to guide their mathematical development. 

There is a Teacher’s Edition Program Overview specific to each course that provides comprehensive guidance to assist teachers in presenting the student and ancillary materials. It contains four major components: an overview of enVision A/G/A, a User’s Guide, Correlation, and Professional Development.

• The Overview provides the table of contents for the course and a pacing guide for a traditional year-long course and a block/half-year course. The authors provide the program goal and organization and information about their attention to Focus, Coherence, Rigor, and the Math Practices at the end of this section.

• The User’s Guide introduces the program's components and illustrates how to use a “lesson”: Lesson Overview, Explore, Understand and apply, Practice and problem Solving, and Assess and Differentiate. This section also includes additional information that addresses more specific areas such as Mathematical Modeling, STEM, Literacy, and English Language Learners.

• The Correlation section provides connections between each course, the Common Core State Standards, and enVision A/G/A.

• Finally, the Professional Development portion includes research-based articles that are written by the author's program.

Within the Teacher’s Edition, each Lesson is presented in a consistent format, with probing questions to provide multiple entry points to the content, guidance on how to effectively present the materials, and additional examples coded to support struggling learners or extend student thinking. 

Examples of how the instructional materials provide guidance on presenting the materials include:

• In Algebra 1, Topic 4, Lesson 4-3, students solve systems of equations using elimination. The lesson opens with a Critique & Explain, where students compare and contrast the work of two students who solved a system. The teacher’ notes include probing questions  before the activity, “Why might a person choose a particular approach to solve the problem?”, during the activity, “What do you notice about Sadie’s approach?” and after, “Why could there be more than one solution method for solving a system of linear equations?” It also provides sample responses to the questions. The probing questions continue to be provided throughout the lesson, and additional examples can be used along with each example. 

• In Geometry, Topic 4, Lesson 4-4, students prove and apply the SAS and SSS congruence criteria for triangles. In the Explore & Reason activity, students “Make five triangles that have a 5-inch side, a 6-inch side and one 40$$\degree$$

•  angle. A. How many unique triangles can you make? B. How are the unique triangles different from each other?” The teacher's edition includes a section called Habits of Mind, prompting instructors to include questioning that incorporates the mathematical practices. “Make Sense and Persevere How could you organize your work to make sure you have tried every possible combination of the given side lengths and angle measure?” The text also includes guiding questions throughout the lesson to help the instructor lead the conversation in a way that will ensure understanding, such as “What transformation is needed so the triangle can be reflected over one of the congruent segments?” “Only one pair of sides is marked congruent. How can you apply SAS to this problem?” in a problem that incorporates the Reflexive Property. Additionally, there are ideas to further explore and confirm SSS both for struggling learners “using manipulatives” like drinking straws cut to size and to extend thinking “with straightedge and compass constructions.”

• In Algebra 2, Topic 5, Lesson 5-3, students explore graphing radical functions. The lesson opens with an Explore & Reason activity where students use tools to graph a quadratic function and a square root function on the same axis. The Teachers Edition includes probing questions to make the connection between the two graphs, “How can you use the ordered pairs from your first graph to help you graph f(A) = \sqrt{\smash[b]{A}}?”. The text prompts teachers to “Discuss with students the familiar transformations that can occur for various types of parent functions.” There is an additional example to investigate the effect of negative coefficients by graphing “each radical function. 1. f(x) = \sqrt[3]{x} and g(x) = -\sqrt[3]{x}. 2. g(x)=2\sqrt{\smash[b]{x - 1}} and g(x)=-2\sqrt{\smash[b]{x-1}}.” 

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. The Teacher’s Edition includes numerous brief annotations and suggestions at the topic and lesson level organized around multiple mathematics education strategies and initiatives, including the CCSSM Shifts in Instructional Practice (i.e., focus, coherence, rigor), CCSSM practices, STEM projects, and 3-Act Math Tasks, NCTM Mathematics Teaching Practices, Teaching through Problem Solving (i.e., Before-During-After structure), Growth Mindset, and Problem-Based Learning. 

Examples of how the instructional materials provide teacher guidance on how to plan for instruction include:

• In Algebra 1, Topic 5, the Topic Planner provides instructors with a pacing guide for the lessons as well as a list of the vocabulary that will be introduced in each section. The Topic Resources provides a list of the resources that will be used in that lesson and where to find them within the program materials. 

• In Geometry, Topic 4, the Vocabulary Builder section guides instructors through making connections through vocabulary by recommending that “students construct an idea map with the word congruent at the center. Prompt students to recall the definitions of congruent angles and segments they learned in Topic 1. When students have finished their maps, have them trade maps with a partner.” 

• In Algebra 2, Topic 10, Lesson 10-2, the annotations alert the instructor to a “Common Mistake” within multiplying matrices, “Some students may switch the matrix that is assigned to each variable. Have the students write the matrix for G as it is and then rewrite it as a 2 \times 3 matrix. Have students write the matrix for W as it is and then rewrite it as a 3 \times 1 matrix. Explain that because they are now 2 \times 3 and 3 \times 1 matrices, they can be multiplied.” The text also includes detailed instructions to help the teacher plan for English Language Learners who may struggle with the terminology in the lesson, such as “When have you used the word diagonal in your daily life? Can a matrix have more than one diagonal?”

The materials reviewed for enVisionMath A/G/A meet expectations for containing adult-level explanations and examples of the more complex course-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject. 

The materials provide a Topic Overview at the beginning of each topic that provides information pertaining to the math background of the topic in addition to connections to prior and future learning. The overview includes an illustration of how the lessons within the topic emphasize conceptual understanding, procedural skill and fluency, and application. 

Examples of how the materials support teachers to develop their own knowledge of more complex, course-level concepts include:

• The online resources for each course provide teachers with a Professional Development Video for each Topic. “In each Topic Overview Video, an author highlights and gives helpful perspectives on important mathematics concepts and skills in the topic. The videos are quick, focused ‘Watch me first’ experiences to help you plan for the topic.”

• The Algebra 1, Topic 2, Math Background Focus gives a mathematically rigorous description of the topic’s content,“In Lesson 2-2, students learn that there is another form that a linear equation can take called point-slope form. Given (x_1, y_1) as a point on the line, and m for the slope, the question in point-slope form is y - y_1 =m(x - x_1). The most apparent purpose for this form is to write an equation when given a slope and a point on the line. Starting with a graph, this form is useful when the exact location of the y-intercept is not clear.”

• In Geometry, Topic 2, Math Background Rigor, the material provides an adult-level description for how the three aspects of mathematical rigor are addressed in the topic. The Conceptual Understanding section notes that “students apply their understanding of the relationships of angles formed by parallel lines cut by a transversal to prove the Triangle Angle-Sum Theorem. A parallel line is constructed through the vertex opposite the chosen side using one side of the triangle. The resulting parallel lines can be used to prove the Triangle Angle-Sum Theorem or to find missing measurements in the interior or exterior of the triangle.”

• In Algebra 2, Topic 6, Math Practices the material includes an adult-level explanation for how two of the eight Mathematical Practices are addressed in the topic.  “Look for and make use of structure. Look for patterns to determine whether they can use the natural log or the common log to solve an exponential equation. They use structure to determine the annual rate that was used to project the amount of money in an account.”

Examples of how the materials support teachers to develop their own knowledge beyond the current course:

• In Algebra 1, Topic 9, Mathematical Background, Looking Ahead, the materials state, “In Algebra 2, students will continue to use these methods to find the zeros of polynomial equations.” The materials include an image of a cubic function with points on the zeros for emphasis. 

• In Geometry, Topic 5, Mathematical Background, Looking Ahead, the materials state, “Algebra 2 Trigonometry Students will extend the relationship of triangles and circles when they make sense of the trigonometric ratios. Students will relate the ratios of the lengths of the sides of right triangles by representing the hypotenuse of a right triangle as the circle's radius on a coordinate grid.”

• In Algebra 2, Topic 11, Mathematical Background, Looking Ahead, the materials state, “In Statistics, they [students] will extend this knowledge to include finding the variance, the standard score, and the moment of deviation from the mean. Students will also use paired t-tests to determine whether the means of two samples vary considerably and statistical tests to determine the statistical importance of an observation.” The materials include an image of normally distributed SAT Math Scores with the standard deviation illustrated.

The materials reviewed for enVisionMath A/G/A meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series. 

Correlation information is present for the mathematics standards addressed throughout the series at the start of each course, topic, and lesson as well as throughout the lesson material itself. Examples include:

• In Algebra 1, Topic 1, Topic Planner, presents a table of information including the name of each lesson, the essential understanding and content objective, any relative vocabulary for that lesson, and the CCSSM that will be addressed as well as any Mathematical Practice Standards that will be incorporated. 

• In Geometry, Topic 7, Lesson 7-4, Mathematics Overview, in the Teacher’s Edition at the start of each lesson shows the focus content standards (G.SRT.4, G.SRT.5) and practice standards (MP.5, MP.7) for the lesson.

• In Algebra 2, Topic 1, Lesson 1-1, correlation information is listed in white boxes within the margin of the lesson within the student edition.  For example, “Common Core State Standards HSF.IF.B.4, HSF.IF.B.6, HSF.IF.C.7, HSF.IF.B.5, MP.3, MP.4, MP.6”

Explanations of the role of the specific course-level mathematics are present in the context of the series. The Program Overview includes a table titled Common Core State Standards - Mathematics in enVision A|G|A that illustrates each standard and where it occurs throughout the series. Each topic includes a section that explains the role of that topic to the math that came before and will come after. Examples include:

• In Algebra 1, Topic 1, Math Background Coherence, aligns the work of the topic with Grade 7 inequalities (7.EE.4b), Grade 8 properties of equality (8.EE.7a), later Algebra 1 content in Topic 2: Linear Equations (A-CED.2 and S-ID.7) and Topic 4: Systems of linear equations and inequalities (A-REI.6, A-REI.12 and A-CED.3), and Algebra 2 quadratic, exponential, and logarithmic equations (A-REI.1, A-REI.2, A-REI.4 and A-REI.11).

• In Geometry, Topic 1, Math Background Coherence, describes the relationship to Algebra 1 work with identifying patterns (F-IF.3, F-BF.2 and F-LE.2) and properties of real numbers (N-RN.3); later Geometry work with parallel and perpendicular lines (G-GPE.5), triangle congruence (G-CO.5), and relationships in triangles (G-CO.9 and G-CO.10), and Algebra 2 work with trigonometry (F-TF.1).

• In Algebra 2, Topic 1, Math Background Coherence, describes the relationship to Algebra 1 work with solving equations and inequalities algebraically (A-CED.3), transforming linear functions (F.BF.3), solving systems of equations (A-REI.6), and graphing absolute value functions (F-IF.7b) with later Algebra 2 work such as transforming linear and quadratic functions (F-BF.3), solving radical equations and graphing radical functions (A-REI.2) and Limits in Calculus. 

The materials reviewed for enVisionMath A/G/A meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. The Teacher’s Edition Program Overview of each course provides detailed explanations behind the instructional approaches of the program and cites research-based strategies for the layout of the program. Unless otherwise noted, all examples are found in the Teacher’s Edition Program Overview.

Examples where materials explain the instructional approaches of the program and describe research-based strategies include:

• The Program Goal section states, "What were the major goals in developing enVision A|G|A? One major goal of enVision A|G|A was to create a high school mathematics program that reflects the latest research in mathematics education and learning theory and supports all learners on their pathway to college- and career-readiness. To achieve this goal, we developed a brand new program built from the ground up around three foundational principles:

  • A balanced pedagogy. Research has shown that teaching for understanding requires equal attention to helping students develop deep understanding of concepts, fluency with important processes and skills, and the ability to apply these concepts and skills to solve real-world and mathematical problems…

  • A focus on visual learning. Recent research (Park & Brannon, 2013) has found that powerful learning occurs when students use different areas of the brain, specifically the area that governs symbolic thinking and the area that focuses on visual thinking…

  • A focus on effective teaching and learning. Recent research has also shown that students make significant academic gains when they explore ‘worthwhile tasks’ and engage in meaningful mathematical discourse using mathematical language. Research also suggests that teachers need to create learning environments that facilitate and encourage this meaningful discourse. Every lesson in enVision A|G|A opens with a worthwhile task, a student-centered activity that requires students to think critically and construct sound mathematical arguments to defend their reasoning and their solutions. The teacher support was created using the NCTM’s Guiding Principles for School Mathematics, in particular Teaching and Learning…”

• The Instructional Model section states, "The Common Core State Standards for Mathematics represent a major curricular initiative to create a common set of learning expectations for all high school students. In addition, recent research in mathematics instruction highlight the importance of having students actively engage in worthwhile, meaningful tasks. The instructional model for enVision A|G|A is grounded in these two research foci.  An integral part of the instructional model is a focus of the habits of mind that the Standards for Mathematical Practice describe. Throughout every lesson are multiple opportunities to help students develop proficiency with the Math Practices. In addition, each topic features a lesson called Mathematical Modeling in 3 Acts that is designed to engage students in the mathematical modeling process. STEP 1: EXPLORE Introduce concepts and procedures with ‘worthwhile tasks.’  Research shows that conceptual understanding is developed when new mathematics is introduced in the context of solving a real problem in which ideas related to the new content are embedded (Kapur, 2010; Lester and Charles, 2003; Scott, 2014)...STEP 2: UNDERSTAND & APPLY Make the important mathematics explicit with enhanced direct instruction connected to Step 1. The second step, Understand and Apply, is designed to connect students’ thinking about the opening activity to the new ideas of the lesson. These concepts are presented through a series of visually rich example types purposefully designed to promote understanding… STEP 3: PRACTICE & PROBLEM SOLVING: Offer robust and balanced practice to solidify understanding. In Step 3, students embark on a series of carefully sequenced and crafted exercises to apply what they just learned and to practice toward mastery…STEP 4: ASSESS & DIFFERENTIATE: Check for understanding and provide remediation. enVision A|G|A provides quality assessment and differentiation support. enVision A|G|A offers diagnostic, formative, and summative assessments in print and digital formats. The digital assessments offer a wide range of item types that students may encounter in their state-mandated assessment from multiple-response multiple-choice items to rich, multi-part performance tasks…”

• The Professional Development, Teaching for Understanding section states the following: “At the turn of the 21st Century, however, the National Research Council published Adding it Up (NAP, 2001) in which it defined mathematical proficiency as having five interwoven components:

  • Conceptual understanding. Conceptual understanding ‘reflects a student’s ability to reason in settings involving the careful application of concept definitions, relations, or representations of either’¹. With conceptual understanding, students are able to transfer their knowledge to new situations and contexts in order to solve the problem presented. It is this transfer of knowledge that is so vital for success not only in mathematics, but in all disciplines and in the workplace. The authors of Principles and Standards for School Mathematics (NCTM, 2000) summarize it best: ‘Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge.’²… ¹NAEP (2003). What Does the NAEP Mathematics Assessment Measure? Online at nces.ed.gov/nationsreportcard/mathematics/abilities.asp. ²http://www.nctm.org/Standards-and-Positions/Principles-and-Standards/Principles,-Standards,-and-Expectations/ “

• Throughout the Teacher’s Edition Program Overview references to research-based strategies are cited with some reference pages included at the end of some authors' work.

The materials reviewed for enVisionMath A/G/A meet expectations for providing a comprehensive list of supplies needed to support instructional activities.

Examples of where materials include a comprehensive list of supplies needed to support instructional activities include:

• In the online Teacher Resources for each course, a Materials List is provided in table format identifying the required material and the topic(s) or lesson(s) where it will be used. The list includes items such as graph paper, graphing calculators, algebra tiles, protractors, compasses, scissors, and optional materials such as coins, spinners, and colored pencils. 

• The teacher’s edition includes materials in the description of the activities when necessary.

  • Algebra 1, Topic 10, Lesson 10-3, English Language Learners for use with example 3, “Have the students duplicate the first graph from the example on a piece of graph paper…”

  • Geometry, Topic 4, Lesson 4-3, Support Struggling Students for use with example 4, “Each student will need three drinking straws…”

  • Algebra 2, Topic 8, Lesson 8-1, English Language Learners for use with example 5, “Distribute scissors and a piece of unlined square paper to each student…”

The materials reviewed for enVisionMath A/G/A 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. 

All assessments include a digital and paper option and an answer key for correcting students’ work. Most also include a “Skills Review and Practice” assignment for the corresponding assessment item for following up with students. If you take the assessment on the digital platform, the system will auto-adapt intervention assignments based on an individual student’s needs. For those assessment that do not include a “Skills Review and Practice” assignment, additional practice can be assigned via the Standards Practice Workbook.

Examples of the assessment system providing opportunities for teachers to interpret student performance and suggestions for follow-up:

• The Standards Progress Report captures performance on each standard-aligned question for all assessments throughout the year. Standards are colored green if students answered correctly and red if they answered incorrectly, with the student's standards progress bar showing their overall performance. When you click on a standard, you can also view all the relevant standard-aligned resources that you can assign to that student. 

• Algebra 1, Topic 5, Lesson 5-1, Lesson Quiz, Problem 2, “Graph the function g(x)=-\frac{1}{2}|x|.” The materials says the following about the Lesson Quiz, “Use the Lesson Quiz to assess students’ understanding of the mathematics in the lesson…Use the student scores on the Lesson Quiz to prescribe differentiated assignments.” The breakdown of the assignment is as follows: I = Intervention 0-3 points, assignments would be Reteach to Build Understanding, Mathematical Literacy and Vocabulary, and Lesson Virtual Nerd videos, O = On-Level 4 points assignment would be Enrichment, and A = Advanced 5 points assignment would be Enrichment.

• Algebra 2, Topic 1, Topic Readiness Assessment, Problem 1, “Graph the linear inequality 6x-3y>12.” The Item Analysis for Diagnosis and Intervention table identifies the Skills Review and Practice for follow-up with the student as Lesson A16. Lesson A16, is titled Linear Inequalities, and begins with a four-step review of how to graph a  linear inequality. Underneath the review, there are twelve practice problems to graph.

The materials reviewed for enVisionMath A/G/A meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of course-level standards and practices across the series. 

Each topic begins with a Readiness Assessment and ends with both a formal Topic Assessment and a Topic Performance Assessment that assesses the full range of standards from that topic. The Topic Assessments are primarily multiple-choice or short-answer questions, while the Topic Performance Assessment incorporates constructed response items. The assessments are provided as PDFs and are recommended to be taken in class. 

The formative assessments throughout the topics include primarily constructed response questions. They are mapped to both standards and mathematical practices. 

Examples of assessments, including opportunities for students to demonstrate the full intent of course-level standards and practices, include:

• In Algebra 1, Topic 3, Topic Assessment Form A, Question 8 demonstrates the full intent of the standards S-ID.6, MP1 and MP7. “ Part A Each day, Yumiko exercises by first doing sit-ups and then running. Make a scatter plot of the total time she exercises as a function of the distance she runs. Draw a trend line. [Table with distance (mi)/Time (min) pairs: 1.5, 18; 2, 23; 2.5, 28; 3, 34; 3.5, 34; 4, 40, along with a blank graph with Total Distance (mi) on the x-axis and Total Time (min) on the y-axis.] Part B Which sentence describes the correlation of the scatter plot. A. The correlation is positive because the time increases as the distance decreases. B. The correlation is negative because the time decreases as distance increases. C. It is impossible to tell what the correlation is based on the given data. D. There is no correlation between time and distance in this situation.”

• In Geometry, Topic 3, Lesson 3-3, Lesson Quiz, Question 5 demonstrates the full intent of the content standards G-CO.5 and MP2. “How many times does the rotation R_{(120o, P)} need to be applied to a figure to map the figure onto itself?” 

• In Algebra 2, Topic 3, Topic Performance Assessment B, Question 2 demonstrates the full intent of the content standards A-APR.2, MP2, and MP4. “Jamie decides that the container described in the previous section will not be practical to handle because of its shape. He plans to build containers with sides which increase by 1 foot. Let x be the smallest dimension of the container. Part A Write and graph a function V for the volume of the new containers. Part B The volume of the container will be 150 ft^3. Transform the graph of the function V from Part A, so that the x-intercept is the width of the container. Write a function f to represent this graph. How does the graph of this f relate to the graph of the function V in Part A? Part C What are the dimensions of the container to the nearest tenth?”

The materials reviewed for enVisionMath A/G/A meet expectations for providing strategies and support for students in special populations to support their regular and active participation in learning series mathematics. 

At the end of each lesson,, there is a differentiated resources section. These resources are assigned based on how students score on the lesson quiz taken on or offline. If taken online, the resources are automatically assigned as the quiz is automatically scored. Resources are assigned based on the following scale based on the following scale: I = Intervention 0-3 points, O = On-Level 4 points and A = Advanced 5. The types of resources include the following:

• Mathematical Literacy & Vocabulary (I, O) - Helps students develop and reinforce understanding of key terms and concepts.

• Reteach for Understanding (I) - Provides scaffolded reteaching for the key lesson concepts.

• Additional Practice (I, O) - Provides extra practice for each lesson.

• Enrichment (O, A) - Presents engaging problems and activities that extend the lesson concepts.

Other resources offered are personalized study plans to provide targeted remediation for students, as well as support for English Language Learners and struggling readers. Additionally, Virtual Nerd instructional tutorials are, “accessible online, or by scanning the QR codes on the exercise pages,  providing high school students with 24-7 tutorial video support.”

Examples of the materials providing strategies and support for students in special populations include:

• In Algebra 1, Topic 6, Lesson 6-3, students graph an exponential function relative to the spreading of computer viruses. The text provides guidance to instructors for English Language Learners: “Writing (Beginning). A virus is something that has a bad influence on someone or something. Have students write two different definitions for the word virus in their journals. Then, have students write their answers to the following questions in their journals under the correct definition. Q: What are some words you think about when you hear the word virus in relation to a person? Q: What does it mean for a computer to have a virus? ”

• In Geometry, Topic 3, Lesson 3-2, students apply translations to an animation. The text provides guidance to Support Struggling Students: "Students describe a translation as a composition of the horizontal displacement and the vertical displacement. Draw a segment from A(2, 2) to B(4, 3) and its image from A’(-4, -3) to B’(-2, -2) on the board. Q: Can you write the horizontal displacement as a translation? Q: Can you write the vertical displacement as a translation? Q: Can you write the complete translation as a composition of the vertical and horizontal displacements? Q: Does the order of the transformations in the composition matter? Explain. ” 

• In Algebra 2, Topic 12, Lesson 12-1, Reteach to Build Understanding, Problem 3, students calculate the probability of two events happening. “A classmate asks Juan to find the probability of tossing a number cube and getting an even number on the first roll and a 2 on the second roll. Complete the calculation. P(even) = \frac{3}{6}P(2) = \frac{ }{6} P(even and 2) = \frac{1}{ }\dot\frac{ }{6}=\frac{ }{ }

The materials reviewed for enVisionMath A/G/A 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. 

There is an English Language Learner supports and Mathematical Literacy and Vocabulary worksheet provided for every lesson in the series. The student edition and assessment materials are also available in Spanish. The bilingual glossary has text-to-speech for both languages as well as a visual representation of the terms. Additionally, the Multilingual Handbook provides a downloadable glossary in 10 languages, including Cambodian, Cantonese, English, Haitian Creole, Hmong, Korean, Mandarin, Filipino, Spanish, and Vietnamese. The Virtual Nerd instructional tutorials videos offer Spanish captions. The Teacher’s Edition Program Overview also references the many illustrations to help with context for English Language Learners and the connections to relative contexts so students may share personal experiences.  

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

• In Algebra 1, Topic 1, Lesson 1-6, Mathematical Literacy and Vocabulary, tasks students to complete a table with the headings Word or Word Phrase, Description, and Picture or Example for vocabulary including compound inequality and open circle on the number line to support a better understanding of the content in this lesson. Students would fill in the missing information in either the “Description” column or the “Picture or Example” column.

• In the Geometry Visual Glossary, under parallelogram, the materials state, “A parallelogram is a quadrilateral with two pairs of parallel sides. You can choose any side to be the base. An altitude is any segment perpendicular to the line containing the base drawn from the side opposite the base. The height is the length of an altitude.” [Figure of a parallelogram with altitude and base marked, along with markings indicating opposite sides are parallel.] “Un paralelogramo es un cuadrilátero con dos pares de lados paralelos. Se puede escoger cualquier lado como la base. Una altura es un segmento perpendicular a la recta que contiene la base, trazada desde el lado opuesto a la base. La altura, por extensión, es la longitud de una altura.” In the online platform, these definitions can be read aloud in both languages. 

• In Algebra 2, Topic 6, Lesson 6-4, English Language Learners, “Speaking [Beginning] In small groups, have students discuss the meanings of inter- and change, and how they relate to the meaning of interchange. Q: Where do you hear the word interchange used? Q: What does the word interchange mean?” This support is supposed to be used with Example 3, and it has two additional supports, Listening [Developing] and Writing [Expanding].

The materials reviewed for enVisionMath A/G/A meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods. 

The materials include virtual manipulatives that are presented as Desmos activities throughout the series. Examples of how virtual manipulatives are accurate representations of mathematical objects and are connected to written methods, when appropriate, include:

• In Algebra 1, Topic 10, Lesson 10-6, Explore & Reason, students explore how adding a constant to a function output changes the function's domain and range. The interactive materials provide a demo applet so students can see how adding a constant affects the domain and range. The applet shows the graphs, f(x)=x^2 and g(x)=x^2+3 students are tasked with graphing another function using a different constant and recording the change to the domain and range. 

• In Geometry, Topic 5, Lesson 5-4, Explore & Reason, students use pieces of straw to investigate the Triangle Inequality Theorem. “Cut several drinking straws to the sizes shown. [Drawing shows straws with lengths 2, 3, 4, 6, 7, and 10 centimeters] A. Take your two shortest straws and your longest straw. Can they form a triangle? Explain. B. Try different combinations of three straws to form triangles. Which side length combinations work?  Which combinations do not work? C. Look for relationships What do you notice about the relationship between the combined lengths of the two shorter sides and the length of the longest side?”

• In Algebra 2, Topic 9, Modeling in 3 Acts, students determine the best sprinkler placement on a grassy lawn. The interactive materials provide a desmos applet simulating a green 10-by-10 lawn with blue, red and purple circles representing the sprinklers. “Drag points to place each sprinkler. Use the sliders to change the radius of each sprinkler.” Students are able to move the circles around the ‘lawn’ and change the radii using a slide bar while the activity adjusts the area of watered space.