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Report Overview
Summary of Alignment & Usability: enVision Mathematics | Math
Math K-2
The materials reviewed for enVision Mathematics Grades K-2 meet expectations for Alignment to the CCSSM. In Gateway 1, the materials meet expectations for focus and coherence. In Gateway 2, the materials meet expectations for rigor and practice-content connections. In Gateway 3, the materials meet expectations for Usability.
Kindergarten
View Full ReportEdReports reviews determine if a program meets, partially meets, or does not meet expectations for alignment to college and career-ready standards. This rating reflects the overall series average.
Alignment (Gateway 1 & 2)
Materials must meet expectations for standards alignment in order to be reviewed for usability. This rating reflects the overall series average.
Usability (Gateway 3)
1st Grade
View Full ReportEdReports reviews determine if a program meets, partially meets, or does not meet expectations for alignment to college and career-ready standards. This rating reflects the overall series average.
Alignment (Gateway 1 & 2)
Materials must meet expectations for standards alignment in order to be reviewed for usability. This rating reflects the overall series average.
Usability (Gateway 3)
2nd Grade
View Full ReportEdReports reviews determine if a program meets, partially meets, or does not meet expectations for alignment to college and career-ready standards. This rating reflects the overall series average.
Alignment (Gateway 1 & 2)
Materials must meet expectations for standards alignment in order to be reviewed for usability. This rating reflects the overall series average.
Usability (Gateway 3)
Math 3-5
The materials reviewed for enVision Mathematics Grades 3-5 meet expectations for Alignment to the CCSSM. In Gateway 1, the materials meet expectations for focus and coherence. In Gateway 2, the materials meet expectations for rigor and practice-content connections. In Gateway 3, the materials meet expectations for Usability.
3rd Grade
View Full ReportEdReports reviews determine if a program meets, partially meets, or does not meet expectations for alignment to college and career-ready standards. This rating reflects the overall series average.
Alignment (Gateway 1 & 2)
Materials must meet expectations for standards alignment in order to be reviewed for usability. This rating reflects the overall series average.
Usability (Gateway 3)
4th Grade
View Full ReportEdReports reviews determine if a program meets, partially meets, or does not meet expectations for alignment to college and career-ready standards. This rating reflects the overall series average.
Alignment (Gateway 1 & 2)
Materials must meet expectations for standards alignment in order to be reviewed for usability. This rating reflects the overall series average.
Usability (Gateway 3)
5th Grade
View Full ReportEdReports reviews determine if a program meets, partially meets, or does not meet expectations for alignment to college and career-ready standards. This rating reflects the overall series average.
Alignment (Gateway 1 & 2)
Materials must meet expectations for standards alignment in order to be reviewed for usability. This rating reflects the overall series average.
Usability (Gateway 3)
Math 6-8
The materials reviewed for enVision Mathematics Grades 6-8 meet expectations for Alignment to the CCSSM. In Gateway 1, the materials meet expectations for focus and coherence. In Gateway 2, the materials meet expectations for rigor and practice-content connections. In Gateway 3, the materials meet expectations for Usability.
6th Grade
View Full ReportEdReports reviews determine if a program meets, partially meets, or does not meet expectations for alignment to college and career-ready standards. This rating reflects the overall series average.
Alignment (Gateway 1 & 2)
Materials must meet expectations for standards alignment in order to be reviewed for usability. This rating reflects the overall series average.
Usability (Gateway 3)
7th Grade
View Full ReportEdReports reviews determine if a program meets, partially meets, or does not meet expectations for alignment to college and career-ready standards. This rating reflects the overall series average.
Alignment (Gateway 1 & 2)
Materials must meet expectations for standards alignment in order to be reviewed for usability. This rating reflects the overall series average.
Usability (Gateway 3)
8th Grade
View Full ReportEdReports reviews determine if a program meets, partially meets, or does not meet expectations for alignment to college and career-ready standards. This rating reflects the overall series average.
Alignment (Gateway 1 & 2)
Materials must meet expectations for standards alignment in order to be reviewed for usability. This rating reflects the overall series average.
Usability (Gateway 3)
Report for 6th Grade
Alignment Summary
The materials reviewed for enVision Mathematics Grade 6 meet expectations for Alignment to the CCSSM. In Gateway 1, the materials meet expectations for focus and coherence, and in Gateway 2, the materials meet expectations for rigor and practice-content connections.
6th Grade
Alignment (Gateway 1 & 2)
Usability (Gateway 3)
Overview of Gateway 1
Focus & Coherence
The materials reviewed for enVision Mathematics Grade 6 meet expectations for focus and coherence. For focus, the materials assess grade-level content and provide all students extensive work with grade-level problems to meet the full intent of grade-level standards. For coherence, the materials are coherent and consistent with the CCSSM.
Gateway 1
v1.5
Criterion 1.1: Focus
Materials assess grade-level content and give all students extensive work with grade-level problems to meet the full intent of grade-level standards.
The materials reviewed for enVision Mathematics Grade 6 meet expectations for focus as they assess grade-level content and provide all students extensive work with grade-level problems to meet the full intent of grade-level standards.
Indicator 1A
Materials assess the grade-level content and, if applicable, content from earlier grades.
The materials reviewed for enVision Mathematics Grade 6 meet expectations for assessing grade-level content and, if applicable, content from earlier grades.
The materials contain diagnostic, formative, and summative assessments. Each Topic includes a Topic Readiness Assessment, Lesson Quizzes, Mid-Topic Checkpoint, Mid-Topic Performance Task, Mid-Topic Assessment, Topic Performance Task, and Topic Assessment. Even-numbered Topics include a Cumulative/Benchmark Assessment. In addition, teacher resources include a Grade Level Readiness Assessment and Progress Monitoring Assessments. Assessments can be administered online or printed in paper/pencil format. No above-grade-level assessment items are present.
Examples of grade-level assessment items aligned to standards include:
Topic 6, Lesson 6-3 Quiz, Problem 4, “Javier is 175% heavier than his brother. If Javier’s brother weighs 80 pounds, how much does Javier weigh?” (6.RP.3c)
Topic 7, Mid-Topic Checkpoint, Problem 3, “An earring has the shape of a rhombus. The height is 5.2 mm and the area of the earring is 39 mm. What is the length of each side of the earring?” (6.G.1)
Topic 8, Performance Task Form A, Problem 3, “Individual members of three teams raced through a maze. The box plot shows the results of the Red Team. The Red Team decides to practice for the next competition. Their goal is to get their mean time to 80 but also keep the variability low. Assess whether you think the goal is reasonable, or whether it should be modified. If it should be modified, offer your own goal. Justify your answer.” A box plot is provided that shows the Maze Completion Times (seconds) for the Red Team. (6.SP.3 and 6.SP.5c)
Topics 1 - 6, Cumulative/Benchmark Assessment, Problem 5, “Last month, Tara worked 16.5 hours the first week, 19 hours the second week, 23 hours the third week, and 15.75 hours the fourth week. She plans to work more hours this month than last month. Write an inequality to represent the number of hours, h, Tara plans to work this month.” (6.NS.3 and 6.EE.8)
Indicator 1B
Materials give all students extensive work with grade-level problems to meet the full intent of grade-level standards.
The materials reviewed for enVision Mathematics Grade 6 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards.
The “Solve & Discuss It!” section presents students with high-interest problems that embed new mathematical ideas, connect prior knowledge, and provide multiple entry points. Example problems provide guided instruction and formalize the mathematics of the lesson frequently using multiple representations. The “Try It!” sections provide problems that can be used as formative assessments following example problems and the “Convince Me!” sections provide problems that connect back to the essential understanding of the lesson. “Do You Understand?/Do You Know How?” problems have students answer the Essential Question and determine students’ understanding of the concept and skill application.
Examples of extensive work with grade-level problems to meet the full intent of grade-level standards include:
In Topic 1, Lesson 1-5, Solve & Discuss It!, students use a model and an algorithm to divide fractions by fractions, “A granola bar was cut into 6 equal pieces. Someone ate part of the granola bar so that of the original bar remains. How many parts are left? Use the picture to draw a model to represent and find ÷ .” There is an additional example available online and/or in the teacher materials, “Kate has gallon of popcorn. She puts the popcorn in gallon bags. How many bags can she fill? Problem 27 in the Practice & Problem Solving portion of that same lesson furthers this engagement. “Find ÷ . Then draw a picture and write an explanation describing how to get the answer.” In Lesson 1-6, Practice & Problem Solving, students divide mixed numbers by rewriting them as fractions and then using their skills of dividing fractions by fractions from the previous lesson. In Problem 27, they find the width of a rectangle when given the area as 257 in and the length as 10 in. In Lesson 1-7, Practice & Problem Solving, Problem 9, students apply this concept to a real-world scenario where they are portioning supplies. “Students put 2 pounds of trail mix into bags that each weight pound. They bring of the bags of trail mix on a hiking trip. Can you determine how many bags of trail mix are left by completing just one step? Explain.” Students engage in extensive work with grade-level problems to meet the full intent of 6.NS.1 (Interpret and compute quotients of fractions by fractions).
In Topic 4, Lesson 4-5, students compare and contrast the process of solving an equation with fractions to solving an equation with whole numbers in the Do You Understand? section of this lesson. In the Do You Know How?, Problems 6-8, students solve equations that contain fractions, mixed numbers, and decimals using inverse relationships and properties of equality, “6. t - = 25 7. = 8. 13.27 = t - 24.45.” In Problem 23 in Practice & Problem Solving, students are writing an equation to find the cost of waterpark tickets. “Mr. Marlon buys these tickets for his family to visit the waterpark. The total cost is $210. Write and solve an equation to find the cost of each ticket.” Students are further engaging in this standard in Problems 29 and 30 within the context of a swimming pool. Problem 29, “Helen is filling the pool shown for her little brother. She can carry 1 gallons of water each trip. Write and solve an equation to find how many trips Helen needs to make.” A picture is shown of the pool with a caption attached to it that says it, “Holds 10 gallons”. Problem 30, “After the pool was full, Helen’s little brother and his friend splashed g gallons of water out of the pool. There are 7 gallons still left in the pool. Write and solve an equation to find how much water was splashed out of the pool.” Students engage in extensive work with grade-level problems to meet the full intent of 6.EE.7 (Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q, and x are all nonnegative rational numbers).
In Topic 5, Lesson 5-3, Try It! following Example 2, students use ratio tables to compare ratios and solve ratio problems, “Tank 3 has a ratio of 3 guppies for every 4 angelfish. Complete the ratio table to find the number of angelfish in Tank 3 with 12 guppies. Using the information in Example 2 and the table at the right, which tank with guppies has more fish?” In Practice & Problem Solving, Problem 11 further engages students in this standard, “The ratio tables at the right show the comparison of books to games for sale at Bert’s Store and at Gloria’s Store. Complete the ratio tables. Which store has the greater ratio of books to games? Explain.” In the Lesson 5-3 Quiz, Problem 5, students determine the color of a sample of paint based on its ratio, “The ratio of blue paint to red paint in color A is 2:5. The ratio of blue paint to red paint in paint color B is 3:5. Tell whether each ratio in the table represents paint color A or paint color B.” Students engage in extensive work with grade-level problems to meet the full intent of 6.RP.3 (Use ratio and rate reasoning to solve real-world and mathematical problems).
In Topic 8, Lesson 8-5, Try It!, students summarize numerical data in relation to a given context, “Jonah’s team scored 36, 37, 38, 38, 41, 46, 47, 47, and 48 points in the last nine games. Find the IQR and range of the points Jonah’s team scored in its last nine games. Are these good measures for describing the points scored?” Within the Practice & Problem Solving section, students use additional measures to summarize numerical data. Problems 11 and 12, “What are the mean and the MAD? [data presented in a dot plot]” and “Describe the variability of the data.” Students engage in extensive work with grade-level problems to meet the full intent of 6.SP.5 (Summarize numerical data sets in relation to their context, such as by: a. reporting the number of observations. b. describing the nature of the attribute under investigation, including how it was measured and its units of measurement. c. giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. d. relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered).
Criterion 1.2: Coherence
Each grade’s materials are coherent and consistent with the Standards.
The materials reviewed for enVision Mathematics Grade 6 meet expectations for coherence. The materials: address the major clusters of the grade, have supporting content connected to major work, make connections between clusters and domains, and have content from prior and future grades connected to grade-level work.
Indicator 1C
When implemented as designed, the majority of the materials address the major clusters of each grade.
The materials reviewed for enVision Mathematics Grade 6 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each grade.
The approximate number of Topics devoted to major work of the grade (including assessments and supporting work connected to the major work) is 6.5 out of 8, which is approximately 81%.
The number of lessons (content-focused lessons, 3-Act Mathematical Modeling tasks, projects, Topic Reviews, and assessments) devoted to the major work of the grade (including supporting work connected to the major work) is 75 out of 93, which is approximately 81%.
The number of days devoted to major work (including assessments and supporting work connected to the major work) is 161 out of 194, which is approximately 83%.
A lesson-level analysis is most representative of the instructional materials as the lessons include major work, supporting work connected to major work, and the assessments embedded within each Topic. As a result, approximately 81% of the instructional materials focus on major work of the grade
Indicator 1D
Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.
The materials reviewed for enVision Mathematics Grade 6 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.
Materials are designed so that supporting standards/clusters are connected to the major standards/clusters of the grade. Examples of connections include:
In Topic 2, Lesson 2-6, Practice & Problem Solving, Problem 15, students have to graph points in all four coordinates then determine the area of the space included. “Mr. Janas is building a pool in his backyard. He sketches a rectangular pool on a coordinate plane. The vertices of the pool are A(-5, 7), B(1, 7), C(1, -1), and D(-5, -1). If each unit represents 1 yard, how much area of the backyard is needed for the pool?” This connects the supporting work of 6.G.3 (Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.) to the major work of 6.NS.7 (Understand ordering and the absolute value of rational numbers) and 6.NS.8 (Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include the use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.).
In Topic 3, Lesson 3-2, Practice & Problem Solving, Problem 33, students use the greatest common factors and the distributive property to find a sum. “Gena has 28 trading cards, Sam has 91 trading cards, and Tiffany has 49 trading cards. Use the GCF and the Distributive Property to find the total number of trading cards Gena, Sam, and Tiffany have.” This connects the supporting work of 6.NS.4 (Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor.) to the major work of 6.EE.3 (Apply the properties of operations to generate equivalent expressions).
In Topic 7, Lesson 7-8, Practice & Problem Solving, Problem 18, students compare the volume of two boxes with fractional dimensions. “Sandy has two boxes with the dimensions shown. [A table of values is given with the length, width, and height of Box A and Box B] She wants to use the box with the greater volume to ship a gift to her friend. Which box should Sandy use? Explain.” This connects the supporting work of 6.G.2 (Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas and to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.) to the major work of 6.EE.2 (Write, read, and evaluate expressions in which letters stand for numbers).
In Topic 8, Lesson 8-3, Do You Know How?, Problem 7, students use understanding of number lines to represent data using box plots, “Sarah’s scores on tests were 79, 75, 82, 90, 73, 82, 78, 85, and 78. Draw a box plot that shows the distribution of Sarah’s test scores.” This connects the supporting work of 6.SP.4 (Display numerical data in plots on a number line, including dot plots, histograms, and box plots.) to the major work of 6.NS.6 (Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates).
Indicator 1E
Materials include problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade.
The materials reviewed for enVision Mathematics Grade 6 meet expectations for including problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade.
Examples from the materials include:
In Topic 3, Lesson 3-4, Practice & Problems Solving, Problem 22, students write an algebraic expression and identify the parts in order to represent a real-world problem with a variable to represent an unknown number, “Last month, a truck driver made 5 round-trips to Los Angeles and some round-trips to San Diego. Write an expression that shows how many miles he drove in all. Identify and describe the part of the expression that shows how many miles he drove and trips he made to San Diego.” A table is provided that shows the round-trip distance in miles from Sacramento to San Jose, Los Angeles, and San Diego respectively. This connects the major work of 6.EE.A (Apply and extend previous understandings of arithmetic to algebraic expressions) to the major work of 6.EE.6 (Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set).
In Topic 6, Lesson 6-5, Do You Know How?, Problem 11, students solve a real-world problem by finding the percent, “The original price of a computer game is $45. The price is marked down by $18. What percent of the original price is the markdown?” This connects the major work of 6.RP.A (Understand ratio concepts and use ratio reasoning to solve problems) to the major work of 6.EE.A (Apply and extend previous understandings of arithmetic to algebraic expressions).
In Topic 7, Lesson 7-6, Practice & Problem Solving, Problem 22, students compare the surface area of a footrest to the amount of fabric that has been purchased, “Margaret wants to cover a footrest in the shape of a rectangular prism with cotton fabric. The footrest is 18 inches by 12 inches by 10 inches. Margaret has 1 square yard of fabric. Can she completely cover the footrest? Explain.” This connects the supporting work of 6.NS.B (Compute fluently with multi-digit numbers and find common factors and multiples) with the supporting work of 6.G.A (Solve real-world and mathematical problems involving area, surface area, and volume).
In Topic 8, Lesson 8-7, Practice & Problem Solving, Problems 7 and 8, students analyze a set of data presented comparing the number of home runs by the nine players on a baseball team. Problem 7, “Describe the overall shape of the data.” and Problem 8, “Make a generalization about the data distribution.” A table is provided that shows the number of home runs hit by players on the team. This connects the supporting work of 6.SP.A (Develop understanding of statistical variability) to the supporting work 6.SP.B (Summarize and describe distributions).
Indicator 1F
Content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades.
The materials reviewed for enVision Mathematics Grade 6 meet expectations that content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades.
The materials identify content from prior and future grade levels and use it to support the progressions of the grade-level standards. According to the Teacher’s Edition Program Overview, “Connections to content in previous grades and in future grades are highlighted in the Coherence page of the Topic Overview in the Teacher’s Edition.” These sections are labeled Look Back and Look Ahead.
Examples of connections to future grades include:
Topic 1, Topic Overview, Math Background Coherence, “Topic 1 How is content connected within Topic 1?... Multiply and Divide Fractions In Lesson 1-3, students use area models, number lines, and equations to multiply with fractions and mixed numbers. In Lesson 1-4, students use their multiplication skills as they learn to divide with unit fractions by multiplying by the reciprocal of the divisor. In Lesson 1-5, students extend their understanding of dividing with unit fractions to include dividing a whole number by a fraction, a fraction by a whole number, and a fraction by a fraction. Students progress from using models to using an algorithm. They divide fractions by multiplying the dividend by the reciprocal of the divisor. In Lesson 1-6, students apply their understanding of dividing fractions to divide with mixed numbers by renaming the mixed numbers as fractions.” Looking Ahead, “How does Topic 1 connect to what students will learn later? … Grade 7 Rational Numbers Students will apply their understanding of fraction computation to proportional relationships and percents.”
Topic 5, Topic Overview, Math Background Coherence, Topic 5 How is content connected within Topic 5?...Rates In Lesson 5-5, students learn about a special type of ratio called a rate. In Lessons 5-6 and 5-7, students use their understanding of rates and their experience using tables to create equivalent rates, to compare rates, and to solve unit rate problems. In Lessons 5-8, 5-9, and 5-10, students use ratio reasoning and unit rates when converting measurements both within and between measurement systems.” Looking ahead, “How does Topic 5 connect to what students will learn later?… Grade 7 Unit Rates of Fractions Students will compute unit rates associated with ratios of fractions. Pi Students will apply their understanding of the ratio between the circumference and diameter of a circle when they solve problems involving the area and circumference of a circle. Proportions Students will recognize and represent proportional relationships between quantities. They will also use proportional relationships to solve multistep ratio and percent problems.”
Topic 7, Topic Overview, Math Background Coherence, Topic 7 How is content connected within Topic 7? Area of a Triangle The formula for the area of a triangle is developed in Lesson 7-2 and is used throughout the rest of Topic 7. It is used to find the areas of trapezoids and kites in Lesson 7-3 and of other polygons in Lesson 7-4. It is also used to find the surface area of prisms in Lesson 7-6 and of pyramids in Lesson 7-7.” Looking ahead, “How does Topic 5 connect to what students will learn later?… Grade 7 Solve Measurement Problems Students will solve mathematical and real-world problems involving area, surface area, and volume of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. ”
The materials relate grade-level concepts explicitly to prior knowledge from earlier grades. Each Lesson Overview contains a Coherence section that connects learning to prior grades. Examples include:
In Topic 2, Lesson 2-4, Lesson Overview, Coherence, students, “extend their knowledge to plot ordered pairs with integer and rational coordinates in all four quadrants of a coordinate plane and to reflect points across both axes.” In Grade 5, students “represent real-world and mathematical problems by graphing points in the first quadrant of the coordinate plane.”
In Topic 3, Lesson 3-3, Lesson Overview, Coherence, students, “extend their work with evaluating expressions to include fractions, decimals, and exponents, as well as parentheses within brackets.” In Grade 5, students “learned how to evaluate numerical expressions that contain parentheses or brackets”.
In Topic 5, Lesson 5-1, Lesson Overview, Coherence, students, “develop an understanding of ratios as comparison of two quantities and learn to express ratios in three ways” and “extend their knowledge of models as they use bar diagrams and double number line diagrams to represent ratio relationships.” In Grade 5, students “analyzed patterns and relationships” and “used models to represent fractional relationships.”
Indicator 1G
In order to foster coherence between grades, materials can be completed within a regular school year with little to no modification.
The materials reviewed for enVision Mathematics Grade 6 foster coherence between grades and can be completed within a regular school year with little to no modification. As designed, the instructional materials can be completed in 170-194 days.
According to the Pacing Guide in the Teacher’s Edition, Program Overview: Grade 6, “Teachers are encouraged to spend 2 days on each content-focused lesson, giving students time to build deep understanding of the concepts presented, 1 to 2 days for the 3-Act Mathematical Modeling lesson, and 1 day for the enVisions STEM project and/or Pick a Project. This pacing allows for 2 days for each Topic Review and Topic Assessment, plus an additional 2 to 4 days per topic to be spent on remediation, fluency practice, differentiation, and other assessments.”
There are 8 Topics with 61 content-focused lessons for a total of 122 instructional days.
Each of the 8 Topics contains a 3-Act Mathematical Modeling Lesson for a total of 8-16 instructional days.
Each of the 8 Topics contains a STEM Project/Pick a Project for a total of 8 instruction days.
Each of the 8 Topics contains a Topic Review and Topic Assessment for a total of 16 instructional days.
Materials allow 16-32 additional instructional days for remediation, fluency practice, differentiation, and other assessments.
Overview of Gateway 2
Rigor & the Mathematical Practices
The materials reviewed for enVision Mathematics Grade 6 meet expectations for rigor and balance and practice-content connections. The materials reflect the balances in the Standards and help students develop conceptual understanding, procedural skill and fluency, and application. The materials make meaningful connections between the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs).
Gateway 2
v1.5
Criterion 2.1: Rigor and Balance
Materials reflect the balances in the Standards and help students meet the Standards’ rigorous expectations, by giving appropriate attention to: developing students’ conceptual understanding; procedural skill and fluency; and engaging applications.
The materials reviewed for enVision Mathematics Grade 6 meet expectations for rigor. The materials develop conceptual understanding of key mathematical concepts, give attention throughout the year to procedural skill and fluency, spend sufficient time working with engaging applications of mathematics, and do not always treat the three aspects of rigor together or separately.
Indicator 2A
Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.
The materials reviewed for enVision Mathematics Grade 6 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.
Materials include problems and questions that develop conceptual understanding throughout the grade level. According to the Teacher Resource Program Overview, “Problem-Based Learning The Solve & Discuss It in Step 1 of the lesson helps students connect what they know to new ideas embedded in the problem. When students make these connections, conceptual understanding takes seed. Visual Learning In Step 2 of the instructional model, teachers use the Visual Learning Bridge, either in print or online, to make important lesson concepts explicit by connecting them to students’ thinking and solutions from Step 1.” Examples from the materials include:
Topic 1, Lesson 1-3, Solve & Discuss It!, students use a model to connect multiplying a fraction by a fraction. “The art teacher gave each student half of a sheet of paper. Then she asked the students to color one fourth of their pieces of paper. What part of the original sheet did the students color? Solve this problem any way you choose.” (6.NS.1)
Topic 4, Lesson 4-5, Explore It!, students write and solve equations with rational numbers. “The cost of T-shirts for four different soccer teams are shown below. A. Lorna is on Team A. Ben is on another team. They paid a total of $21.25 for both team T-shirts. Write an equation to represent the cost of Ben’s shirt. B. Dario also plays soccer and he says that, based on the price of Ben’s T-shirt, Ben is on Team B. Is Dario correct? Explain.” (6.EE.7)
Topic 7, Lesson 7-1, Solve & Discuss It!, students develop conceptual understanding of how to find the area of a parallelogram by decomposing the parallelogram and then composing shapes into a rectangle. “Sofia drew the grid below and plotted the points A, B, C, and D. Connect points A to B, B to C, C to D, and D to A. Then find the area of the shape and explain how you found it. Using the same grid, move points B and C four units to the right. Connect the points to make a new parallelogram ABCD. What is the area of this shape?” (6.EE.2c and 6.G.1)
Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Practice & Problem Solving exercises found in the student materials provide opportunities for students to demonstrate conceptual understanding. Try It! provides problems that can be used as a formative assessment of conceptual understanding following Example problems. Do You Understand?/Do You Know How? Problems have students answer the Essential Question and determine students’ understanding of the concept. Examples from the materials include:
Topic 1, Lesson 1-6, Practice & Problem Solving, Problem 31, students independently apply conceptual understanding of multiplying and dividing mixed numbers to explain the solution to an equation. “Higher Order Thinking If 9 × = 9÷, then what does n equal? Explain.” (6.NS.1)
Topic 5, Lesson 5-5, Do You Understand?, Problem 3, students independently demonstrate conceptual understanding of problems involving rate and unit rate. “Reasoning A bathroom shower streams 5 gallons of water in 2 minutes. a. Find the unit rate for gallons per minute and describe it in words. b. Find the unit rate for minutes per gallon and describe it in words.” (6.RP.2 and 6.RP.3)
Topic 6, Lesson 6-1, Explain It!, students create a visual representation of two pizzas in order to compare them and support their argument. “Tom made a vegetable pizza and a pepperoni pizza. He cut the vegetable pizza into 5 equal slices and the pepperoni pizza into 10 equal slices. Tom’s friends ate 2 slices of vegetable pizza and 4 slices of pepperoni pizza. A. Draw lines on each rectangle to represent the equal slices. B. Construct Arguments Tom Says his friends ate the same amount of vegetable pizza as pepperoni pizza. How could that be true?”
Indicator 2B
Materials give attention throughout the year to individual standards that set an expectation for procedural skill and fluency.
The materials reviewed for enVision Mathematics Grade 6 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.
The materials develop procedural skill and fluency throughout the grade level. According to the Teacher Resource Program Overview, “Students develop skill fluency when the procedures make sense to them. Students develop these skills in conjunction with understanding through careful learning progressions.” Try It! And Do You Know How? Provide opportunities for students to build procedural fluency from conceptual understanding. Examples include:
Topic 1, Lesson 1-2, Try It!, students divide multi-digit decimals using the standard algorithm. “Divide. a. 65 ÷ 8 b. 14.4 ÷ 8 c. 128.8 ÷ 1.4” (6.NS.3)
Topic 4, Lesson 4-7, Try It!, students graph solutions of inequalities. “Graph all of the solutions of x < 8…” (6.EE.8)
Topic 6, Lesson 6-1, Do You Know How?, Problem 11, students use ratio and rate reasoning to find the percentage of line a point represents. “Find the percent of the line segment that point D represents in Example 2.” (6.RP.3)
The materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade level. Practice & Problem Solving exercises found in the student materials provide opportunities for students to independently demonstrate procedural skill and fluency. Additionally, at the end of each Topic is a Concepts and Skills Review which engages students in fluency activities. Examples include:
Topic 2, Lesson 2-2, Practice & Problem Solving, Problem 15, students write the number based on the position of the corresponding point on the number line. “Write the number positioned at each point. 15. A [-3.25]” (6.NS.6 and 6.NS.7)
Topic 3, Lesson 3-1, Practice & Problem Solving, Problem 17, students identify the exponent for each expression. “Write the exponent for each expression. 17. 9×9×9×9” (6.EE.1)
Topic 6, Lesson 6-2, Practice & Problem Solving, Problem 12, students write a given number in two other forms of notation, fraction, decimal, or percent based other than the given notation. “Write each number in equivalent forms using the two other forms of notation: fraction, decimal, or percent. 12.7%” (6.RP.3c)
Indicator 2C
Materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.
The materials reviewed for enVision Mathematics Grade 6 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which mathematics is applied.
The materials include multiple opportunities for students to independently engage in routine and non-routine application of mathematical skills and knowledge of the grade level. According to the Teacher Resource Program Overview, “3-Act Mathematical Modeling Lessons In each topic, students encounter a 3-Act Mathematical Modeling lesson, a rich, real-world situation for which students look to apply not just math content, but math practices to solve the problem presented.” Additionally, each Topic provides a STEM project that presents a situation that addresses real social, economic, and environmental issues, along with applied practice problems for each lesson. For example:
Topic 4, Lesson 4-1, Practice & Problem Solving, Problem 25, students apply their understanding of solving equations to answer whether an equation can be used with the given answers. “Alisa’s family planted 7 palm trees in their yard. The park down the street has 147 palm trees. Alisa guessed that the park has either 11 or 31 times as many palm trees as her yard has. Is either of Alisa’s guesses correct? Use the equation 7n = 147 to justify your answer. (6.EE.5)
Topic 7, STEM Project, Pack It, students determine how the volume of the packaging relates to the volume of the food items being packed. "Food packaging engineers consider many elements related to both form and function when designing packaging. How do engineers make decisions about package designs as they consider constraints, such as limited dimensions or materials? You and your classmates will use the engineering design process to explore and propose food packaging that satisfies certain criteria.” (6.G.2 and 6.G.4)
Topic 8, 3-Act Mathematical Modeling: Vocal Ranges, Question 12, students use informal arguments and statistical reasoning to decide who should win a singing competition. "Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?" (6.SP.2, 6.SP.3, and 6.SP.5)
The materials provide opportunities for students to independently demonstrate the use of mathematics flexibly in a variety of contexts. Pick a Project is found in each Topic and students select from a group of projects that provide open-ended rich tasks that enhance mathematical thinking and provide choice. Additionally, Practice & Problem Solving exercises found in the student materials provide opportunities for students to independently demonstrate mathematical flexibility in a variety of contexts. For example:
Topic 1, Lesson 1-5, Practice & Problem Solving, Problem 26, students use models to divide fractions by fractions, “Be Precise A large bag contains pound of granola. How many pound bags can be filled with this amount of granola? How much granola is left over?” (6.NS.1)
Topic 4, Pick a Project 4C, students make a model of a staircase using tables and equations, “Think about what you need in order to make a model of a staircase. Design a staircase following these rules: The staircase must follow a linear pattern. Use identical blocks to model the staircase. Keep track of the number of blocks you need for each step. Make a table of data for the number of blocks used for any number of steps. Write an equation to represent your staircase pattern. Present your model, table, and equation to your teacher.” (6.EE.9)
Topic 5, Lesson 5-6, Practice & Problem Solving, Problem 20, students apply unit rates to a real-world situation. “Reasoning Which container of milk would you buy? Explain.” Students are given the price of a half-gallon and a gallon of milk. (6.RP.3b)
Indicator 2D
The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade.
The materials reviewed for enVision Mathematics Grade 6 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade.
All three aspects of rigor are present independently throughout the program materials. Examples, where materials attend to conceptual understanding, procedural skill and fluency, and application, include:
Topic 2, Lesson 2-2, Concepts and Skills Review, Problem 4, students use procedural skill and fluency to compare two numbers using greater than, less than or equal to. “Use <, >, or = to compare. 4. 0.25 ___ ” (6.NS.7a)
Topic 4, Lesson 4-1, Practice & Problem Solving, Problem 21, students apply their understanding of solving an equation as a process of answering a question. “Construct Arguments Gerard spent $5.12 for a drink and a sandwich. His drink cost $1.30. Did he have a ham sandwich for $3.54, a tuna sandwich for $3.82, or a turkey sandwich for $3.92? Use the equation s +1.30 = 5.12 to justify your answer. (6.EE.5)
Topic 8, Lesson 8-6, Convince Me!, students develop conceptual understanding when they determine whether the mean, median, or mode best describes the data in a set. “Gary says that he usually scores 98 on his weekly quiz. What measure of center did Gary use? Explain.” A number line is given with dots of data with the mean, median, and mode labeled. (6.SP.5)
Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. Examples include:
Topic 3, Mid-Topic Performance Task, Part B, students develop conceptual understanding, procedural skill and fluency, and application as they find what the least number of cups and napkins Monique would have to purchase to have an equal number. “Monique and Raoul are helping teachers make gift bags and gather supplies for a student celebration day at Pineville Middle School… Monique wants to have an equal number of cups and napkins. What is the least number of packages of cups and the least number of packages of napkins Monique should buy to have an equal number of cups and napkins? Justify our answer.” (6.NS.4)
Topic 5, Assessment Form B, Problem 4, student develop procedural skill and fluency and apply their knowledge about ratios in the real-world context of basketball. “A varsity boys’ basketball team has a ratio of seniors to juniors that is 7:4. Part A if the team is made up only of juniors and seniors, what is the ratio of seniors to total team members? Part B The varsity girls’ basketball team has a ratio of seniors to juniors that is 3 to 2. If each team has 12 juniors, which team has more seniors?” (6.RP.1 and 6.RP.3)
Topic 7, Lesson 7-1, Convince Me!, students develop conceptual understanding and procedural skill and fluency as they calculate and compare the areas of two shapes. “Compare the area of this parallelogram to the area of a rectangle with a length of 7 cm and a width of 4.5 cm. Explain.” (6.EE.2c and 6.G.1)
Criterion 2.2: Math Practices
Materials meaningfully connect the Standards for Mathematical Content and Standards for Mathematical Practice (MPs).
The materials reviewed for enVision Mathematics Grade 6 meet expectations for practice-content connections. The materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs).
Indicator 2E
Materials support the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.
The materials reviewed for enVision Mathematics Grade 6 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them, and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.
Some examples where the materials support the intentional development of MP1 are:
Topic 3, Lesson 3-7, Solve & Discuss It!, students make sense of problems by using prior knowledge and applying properties of operations to rewrite algebraic expressions through simplifying. “Write an expression equivalent to x + 5 + 2x + 2 by combining as many terms as you can. Solve this problem any way you choose.”
Topic 6, Lesson 6-6, Practice & Problem Solving, Problem 20, students make sense of the relationship between the given quantity and percent to solve the problem. “Make Sense and Persevere Sydney completed 60% of the math problems assigned for homework. She has 4 more problems to finish. How many math problems were assigned for homework?”
Topic 7, Lesson 7-6, Solve & Discuss It!, students analyze a multistep problem involving the surface area of prisms and consider different ways to find solutions. “Marianne orders a pack of shipping boxes shaped like cubes. When they arrive, she finds flat pieces of cardboard as shown. What is the least amount of cardboard needed to make each box? Explain how you know. Solve this problem any way you choose.”
Some examples where the materials support the intentional development of MP2 are:
Topic 1, Lesson 1-2, Practice & Problem Solving, Problem 37, students use proportional reasoning to find a fair division of the money earned. “You and a friend are paid $38.25 for doing yard work. You worked 2.5 hours and your friend worked 2 hours. You split the money according to the amount of time each of you worked. How much is your share of the money? Explain.”
Topic 5, Lesson 5-3, Solve & Discuss It!, students use quantitative reasoning to determine the meaning of the quantities in problems and determine what needs to be done to find a solution. “Scott is making a snack mix using almonds and raisins. For every 2 cups of almonds in the snack mix, there are 3 cups of raisins. Ariel is making a snack mix that has 3 cups of almonds for every 5 cups of sunflower seeds. If Scott and Ariel each use 6 cups of almonds to make a bag of snack mix, who will make a larger batch?”
Topic 8, Lesson 8-3, Practice & Problem Solving, Problem 17, students use reasoning to determine the importance of ordering to find the median. “Reasoning The price per share of Electric Company’s stock during 9 days, rounded to the nearest dollar, was as follows: $16, $17, $16, $16, $18, $18, $21, $22, $19. Use a box plot to determine how much greater the third quartile’s price per share was than the first quartile’s price per share.”
Indicator 2F
Materials support the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.
The materials reviewed for enVision Mathematics Grade 6 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.
Student materials consistently prompt students to construct viable arguments. These opportunities are found in the following activities: Solve & Discuss It!, Explain It!, Explore It!, Practice & Problem Solving, Do You Understand?, and Performance Tasks. Examples include:
Topic 1, Lesson 1-3, Do You Understand?, Problem 3, students use their understanding of fraction multiplication to construct arguments to support their response. “Construct Arguments Why is adding and different from multiplying the two fractions?”
Topic 4, Lesson 4-2, Practice & Problem Solving, Problem 20, students use their understanding of properties of equality to write equivalent equations to construct arguments as they form the basis for the procedure used to solve algebraic equations. “Construct Arguments John wrote that 5 + 5 = 10. Then he wrote that 5 + 5 + n = 10 + n. Are the equations John wrote equivalent? Explain.”
Topic 8, Lesson 8-7, Explain It!, students use their understanding of data distributions to construct arguments. “George tosses two six-sided number cubes 20 times. He records his results in a dot plot. A. Describe the shape of the data distribution. B. Critique Reasoning George says that he expects to roll a sum of 11 on his next roll. Do you agree? Justify your reasoning. Focus on math practices Construct Arguments Suppose George tossed the number cubes 20 more times and added the data to his dot plot. Would you expect the shape of the distribution to be different? Construct an argument that supports your reasoning.”
Student materials consistently prompt students to analyze the arguments of others. These opportunities are found in the following activities: Solve & Discuss It!, Explain It!, Explore It!, Practice & Problem Solving, Do You Understand?, and Performance Tasks. Examples include:
Topic 1, Lesson 1-2, Practice & Problem Solving, Problem 33, students analyze the arguments of others as they divide whole numbers and decimals and apply these skills to solve mathematical problems. “Critique Reasoning Henrieta divided 0.80 by 20 as shown. Is her work correct? If not, explain why and give a correct response.”
Topic 2, Lesson 2-1, Explain It!, students analyze the arguments of others as they explain differences between integers. “Sal recorded the outdoor temperature as -4℉ at 7:30 A.M. At noon, it was 22℉. Sal said the temperature changed by 18℉ because 22 - 4 = 18. A. Critique Reasoning Is Sal right or wrong? Explain. B. Construct Arguments What was the total temperature change from 7:30 A.M. until noon? Use the thermometer to help justify your solution.”
Topic 7, Lesson 7-6, Practice & Problem Solving, Problem 14, students analyze the arguments of others as they explain how to find the surface area of a cube. “Critique Reasoning Jacob says that the surface area of the cube is less than 1,000cm. Do you agree with Jacob? Explain.” An image of a cube is shown with a side length of 10 cm.
Indicator 2G
Materials support the intentional development of MP4: Model with mathematics; and MP5: Use appropriate tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.
The materials reviewed for enVision Mathematics Grade 6 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Use appropriate tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.
The materials allow for the intentional development of MP4 to meet its full intent in connection to grade-level content. Examples of this include:
Topic 2, Lesson 2-3, Practice & Problem Solving, Problem 40, students write and solve equations to represent real-life situations. “Model with Math Find the distance from Alberto’s horseshoe to Rebecca’s horseshoe. Explain.” A diagram is given showing the distances of two horseshoes from the stake.
Topic 3, 3-Act Mathematical Modeling: The Field Trip, students write an algebraic expression to model costs on a field trip and decide how much money to take. “How much cash should [the teacher] bring [on the field trip]? Model with Math Represent the situation using mathematics. Use your representation to answer the Main Question…What is your answer to the Main Question? Does it differ from your prediction? Explain.”
Topic 5, Lesson 5-6, Practice & Problem Solving, Problem 19, students use unit rates to model the relationships between quantities presented in real-world problems, as well as identifying important quantities and using them to complete a double number line diagram to model ratio relationships. “Model with Math Katrina and Becca exchanged 270 text messages in 45 minutes. An equal number of texts was sent each minute. The girls can send 90 more text messages before they are charged additional fees. Complete the double number line diagram. At this rate, for how many more minutes can the girls exchange texts before they are charged extra?”
The materials allow for the intentional development of MP5 to meet its full intent in connection to grade-level content. Examples of this include:
Topic 4, Lesson 4-7, Practice & Problem Solving, Problem 29, students use a number line to write and represent solutions to inequalities. “Reasoning Graph the inequalities x > 2 and x < 2 on the same number line. What value, if any, is not a solution of either inequality? Explain.”
Topic 5, Lesson 5-2, Try It!, students use mathematical tools to help find equivalent ratios. “Which of the following ratios are equivalent to 16:20? 2:3, 4:5, 18:22, 20:25” Students have access to a ratio table and other mathematical tools to help identify equivalent ratios.
Topic 6, Lesson 6-5, Do You Understand?, Problem 5, students explain how to use a calculator to produce a desired result. “Use Appropriate Tools How can you use a calculator to find the percent of 180 is 108?”
Indicator 2H
Materials attend to the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.
The materials reviewed for enVision Mathematics Grade 6 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.
Students are encouraged to attend to the specialized language of mathematics throughout the materials. A chart in the Topic Planner lists the vocabulary being introduced for each lesson in the Topic. As new words are introduced in a Lesson they are highlighted in yellow and students are encouraged to utilize the Vocabulary Glossary in the back of the text (with an animated version online in both English and Spanish) to find both definitions and examples where relevant. Lesson Practice includes questions that reinforce vocabulary comprehension and the teacher's side notes provide specific information about what math language and vocabulary are pertinent for each section.
Examples where students are attending to the full intent of MP6 and/or attend to the specialized language of mathematics include:
Topic 1, Topic Review, Use Vocabulary in Writing, students attend to the specialized language of mathematics as they use mathematical terms in their explanations. “Explain how to use multiplication to find the value of ÷ . Use the words multiplication, divisor, quotient, and reciprocal in your explanation.”
Topic 4, Lesson 4-8, Try It!, students attend to precision as they use information from the problem to determine which variable is dependent and illustrate how one is dependent on the other. “A baker used a certain number of cups of batter, b, to make p pancakes. Which variable, p, pancakes or b, batter is the dependent variable? Explain. Convince Me! If the baker doubles the number of cups of batter used, b, what would you expect to happen to the number of pancakes made, p? Explain.”
Topic 7, Lesson 7-3, Try It!, students attend to precision as they answer questions about the shapes that they previously found the areas of. “When you decompose the trapezoid in Part A of the Try It! into two triangles and a rectangle, are the triangles identical? Explain. What is the height of the two large, identical triangles that compose the kite in Part B of the Try It!?”
Topic 8, Mid-Topic Checkpoint, Problem 1, students attend to the specialized language of mathematics as they select the answer that describes the definition of mean. “Vocabulary Which of the following describes the mean of a data set? (A) The data value that occurs most often (B) The middle data value (C) The sum of the data value divided by the total number of data values (D) The difference of the greatest and least data values”
Indicator 2I
Materials support the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.
The materials reviewed for enVision Mathematics Grade 6 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards.
Students are encouraged to look for and make use of structure as they work throughout the materials, both with the instructor's guidance and independently. Examples of where there is intentional development of MP7 include:
Topic 3, Lesson 3-6, Practice & Problem Solving, Problem 28, students use structure and properties of operations to determine the equivalence of expressions. “Use Structure Write an algebraic expression to represent the area of the rectangular rug. Then use properties of operations to write an equivalent expression.” Dimensions provided in illustration are l = 2(x - 1) and w = 5.
Topic 7, Lesson 7-6, Try It!, students analyze the structure of prisms to find the surface area. “Find the surface area of each prism. [Drawing of a cube and a triangular prism with edge lengths with appropriate lengths given.]”
Topic 8, Lesson 8-2, Practice & Problem Solving, Problem 14, students use the structure of a data set to analyze statistical measures. “Look for Relationships Does increasing the 3 to 6 change the mode? If so, how?” Students are given a data set of states traveled to or lived in.
Students look for and express regularity in repeated reasoning as they are engaged in the course materials. Examples of intentional development of MP8 include:
Topic 3, Lesson 3-2, Practice & Problem Solving, Problem 3, students use repeated reasoning from examples to answer what is the greatest common factor of two prime numbers. “Generalize Why is the GCF of two prime numbers always 1?”
Topic 6, Lesson 6-2, Practice & Problem Solving, Problem 24, students use repeated reasoning to recognize a pattern and generalize what is the same between fractions that are equivalent to 100%. “Generalize What are the attributes of fractions that are equivalent to 100%?”
Topic 7, Lesson 7-5, Practice & Problem Solving, Problems 20 and 21, students analyze the patterns in a table to create an equation. “Look for Relationships The Swiss mathematician Leonhard Euler and the French mathematician Rene Descartes both discovered a pattern in the number of edges, vertices, and faces of polyhedrons. Complete the table. Describe a pattern in the table. Higher Order Thinking Write an equation that relates the number of edges, E, to the number of faces, F, and vertices, V.”
Overview of Gateway 3
Usability
The materials reviewed for enVision Mathematics Grade 6 meet expectations for Usability. The materials meet expectations for Criterion 1, Teacher Supports, and Criterion 2, Assessment, and partially meet expectations for Criterion 3, Student Supports.
Gateway 3
v1.5
Criterion 3.1: Teacher Supports
The program includes opportunities for teachers to effectively plan and utilize materials with integrity and to further develop their own understanding of the content.
The materials reviewed for enVision Mathematics Grade 6 meet expectations for Teacher Supports. The materials: provide teacher guidance with useful annotations and suggestions for enacting the materials, contain adult-level explanations and examples of the more complex grade-level concepts beyond the current grade so that teachers can improve their own knowledge of the subject, include standards correlation information that explains the role of the standards in the context of the overall series, provide explanations of the instructional approaches of the program and identification of the research-based strategies, and provide a comprehensive list of supplies needed to support instructional activities.
Indicator 3A
Materials provide 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.
The materials reviewed for enVision Mathematics Grade 6 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.
The Teacher’s Edition Program Overview provides comprehensive guidance to assist teachers in presenting the student and ancillary materials. It contains four major components: Overview of enVision Mathematics, User’s Guide, Correlation, and Content Guide.
The Overview provides the table of contents for the course as well as a pacing guide. The authors provide the Program Goal and Organization, in addition to information about their attention to Focus, Coherence, Rigor, the Math Practices, and Assessment.
The User’s Guide introduces the components of the program and then proceeds to illustrate how to use a ‘lesson’: Lesson Overview, Problem-Based Learning, Visual Learning, and Assess and Differentiate. In this section, there is additional information that addresses more specific areas such as STEM, Pick a Project, Building Literacy in Mathematics, and Supporting English Language Learners.
The Correlation provides the correlation for the grade.
The Content Guide portion directs teachers to resources such as the Scope and Sequence, Glossary, and Index.
Within the Teacher’s Edition, each Lesson is presented in a consistent format that opens with a Lesson Overview, followed by probing questions to provide multiple entry points to the content, error intervention, support for English Language Learners, Response to Intervention, Enrichment and ends with multiple Differentiated Interventions.
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, and Problem-Based Learning. Examples of these annotations and suggestions from the Teacher’s Edition include:
Topic 1, Lesson 1-1, Solve & Discuss It!, “Purpose Students engage in productive struggle to connect representing multiplying a whole number by a decimal on decimal grids to evaluating operations with decimal numbers during the Visual Learning Bridge. Before Whole Class 1 Introduce the Problem Provide additional decimal grids or decimal cube manipulatives as needed. 2 Check for Understanding of the Problem Activate prior knowledge by asking: What does each square of a decimal grid represent? What does an entire decimal grid represent?”
Topic 3, Lesson 3-3, Do You Know How?, Problem 9, evaluate each expression “6 + 4 × 5 ÷ 3 - 8 × 1.5” Teacher guidance: “Prevent Misconceptions ITEM 9 Make sure students understand that multiplication and division together are done from left to right (not multiplication, then division.) Note: Although finding 8 × 1.5 before dividing 20 by 2 would not change the value of this particular expression, make sure students do not misinterpret the rule by saying that they must do it first thinking that multiplication comes before division. Q: Because there are no parentheses or exponents, what operation should you do first? [4 × 5]”
Topic 5, Lesson 5-1, Practice & Problem Solving, Problem 19, “Reasoning Rita’s class has 14 girls and 16 boys. How does the ratio 14:30 describe Rita’s class?” Teacher guidance: “Error Intervention ITEM 19 Reasoning Students may think that the ratio 14:30 is incorrect because they may forget that ratios can compare a part to the whole. Q: What are the two ways that ratios can compare quantities? [Part to part and part to whole] Q: How can you find the total number of students in the class? [Add the number of girls, 14, to the number of boys, 16.]”
Indicator 3B
Materials contain adult-level explanations and examples of the more complex grade-level/course-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject.
The materials reviewed for enVision Mathematics Grade 6 meet expectations for containing adult-level explanations and examples of the more complex grade concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject.
The materials provide professional development videos at two levels to help teachers improve their knowledge of the grade they are teaching.
“Topic-level Professional Development videos available online. In each Topic Overview Video, an author highlights and gives helpful perspectives on important mathematics concepts and skills in the topic. The video is a quick, focused ‘Watch me first’ experience as you start your planning for the topic.
Lesson-level Professional Development videos available online. These Listen and Look For videos, available for some lessons in the topic, provide important information about the lesson.”
The Teacher’s Edition Program Overview, Professional Development section, states the “Advanced Concepts for the Teacher provides examples and adult-level explanations of more advanced mathematical concepts related to the topic. This professional development feature provides the teacher opportunities to improve his or her personal knowledge and build understanding of the mathematics in each topic. The explanations and examples in this section also support the teacher’s understanding of the underlying mathematical progressions.”
An example of an Advanced Concept for the Teacher:
Topic 1, Topic Overview, Advanced Concepts for the Teacher, “Multiplying Decimals The standard algorithm for multiplying decimals extends the algorithm for multiplying multi-digit whole numbers by combining it with properties of multiplication and exponents. However, the manner in which decimal multiplication is typically represented presents some apparent contradictions. Consider the product of 4.32 and 1.8.[an example is provided]...This same principle applies to multiplying numbers in scientific notation involves explicit tracking of powers of 10, rather than transient adding and removing of ‘decimal places’, so no apparent contradictions arise in those standards algorithms.”
The Topic Overview, Math Background Coherence, and Look Ahead sections, provide adult-level explanations and examples of concepts beyond the current grade as they relate what students are learning currently to future learning.
An example of how the materials support teachers to develop their own knowledge beyond the current grade:
Topic 4, Topic Overview, Math Background Coherence, Look Ahead, the materials state, “Grade 7 Proportional Relationships Students will use tables and equations to represent proportional relationships and to graph proportional relationships on a coordinate plane. Solve Simple Equations and Inequalities Students will fluently solve algebraic word problems with relationships in the form px + q = r. Students will also represent and solve inequalities in the forms px + q < r and px + q > r.”
Indicator 3C
Materials include standards correlation information that explains the role of the standards in the context of the overall series.
The materials reviewed for enVision Mathematics Grade 6 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.
Standards correlation information is indicated in the Teacher’s Edition Program Overview, the Topic Planner, the Lesson Overview, and throughout each lesson. Examples include:
The Teacher’s Edition Program Overview, Correlation to Grade 6 Common Core Standards, organizes standards by their Domain and Major Cluster and indicates those lessons and activities within the Student’s Edition and Teacher’s Edition that align with the standard. Lessons and activities with the most in-depth coverage of a standard are distinguished by boldface. The Correlation document also includes the Mathematical Practices. Although the application of the mathematical practices can be found throughout the program, the document indicates examples of lessons and activities within the Student’s Edition and Teacher’s Edition that align with each math practice.
The Teacher’s Edition Program Overview, Scope & Sequence organizes standards by their Domain, Major Cluster, and specific component. The document indicates those topics that align with the specific component of the standard.
The Teacher’s Edition, Topic Planner indicates the standards and Mathematical Practices that align to each lesson.
The Teacher’s Edition, Math Background Coherence provides information that summarizes the content connections across grades. Examples of where explanations of the role of the specific grade-level mathematics are present in the context of the series include:
Topic 2, Topic Overview, Math Background Coherence, the materials highlight two of the learnings within the topics: “Rational Numbers” and “Coordinate Plane” with a description provided for each learning including which lesson(s) cover the learnings. The “Look Ahead” section asks the question, “How does Topic 2 connect to what students will learn later?” and provides a Grade 6 and 7 connection, “Later in Grade 6...Graph Equations In Topic 4, students will graph equations on the coordinate plane...Grade 7 Operations with Rational Numbers Students will add, subtract, multiply, and divide both positive and negative rational numbers. They will solve multistep problems involving operations on rational numbers, renaming numbers as appropriate.”
Topic 3, Topic Overview, Math Background Coherence, the materials highlight two of the learnings within the topics: “Exponents and Expressions” and “Equivalent Expressions” with a description provided for each learning including which lesson(s) cover the learnings. The “Look Back” section asks the question, “How does Topic 3 connect to what students will learn earlier?” and provides a Grade 5 connection, “Grade 5 Patterns with Exponents and Powers of 10 Students used whole-number exponents to express powers of 10. [an example is provided] Evaluate, Write, and Interpret Numerical Expressions Students used the order of operations to simplify numerical expressions that contain parentheses…”
Topic 6, Topic Overview, Math Background Coherence, the materials highlight two of the learnings within the topics: “Concept of Percent” and “Find the Percent, the Whole, or a Part” with a description provided for each learning including which lesson(s) cover the learnings. The “Look Ahead” section asks the question, “How does Topic 6 connect to what students will learn later?” and provides a Grade 6 and 7 connection, “Later in Grade 6 Describe Data In Topic 8, students will use percentages when they summarize data distributions. Students will draw box plots to represent where 25%, 50%, and 75% of the data occur. Grade 7 Percent Problems Students will analyze and solve percent problems. They will solve percent increase and percent error problems. They will also solve mark-up and markdown problems as well as simple interest problems.”
Indicator 3D
Materials provide strategies for informing all stakeholders, including students, parents, or caregivers about the program and suggestions for how they can help support student progress and achievement.
The materials reviewed for enVision Mathematics Grade 6 provide strategies for informing all stakeholders, including students, parents, or caregivers about the program and suggestions for how they can help support student progress and achievement.
Each material has a Family Engagement Letter, which can be found in the Teacher Resources section. The Family Engagement Letter is available in both English and Spanish and provides a QR code that brings you to the Family Engagement Section. The materials state the following:
“Welcome Thank you for working with your student’s teacher and with us, the authors of enVision Mathematics, to advance your student’s learning. This is important to us, and we know it is to you. enVision Mathematics was specifically designed to implement the Common Core State Standards for Mathematics and to foster your student’s success. enVision Mathematics was developed to help students see the math. And the program includes resources to help families see the math as well.”
These resources are divided into the following areas:
Overview of Resources “enVision Mathematics offers a variety of digital resources to help your student succeed.”
Content and Standards “enVision Mathematics was specifically developed for the Common Core State Standards for Mathematics. Each lesson is correlated to one or more of the content standards and one or more of the Standards for Mathematical Practice (MP Standards).”
Topic/Lesson Support “enVision Mathematics provides topic and lesson-level support. Look for an overview of each Topic’s content, lesson objectives, and suggestions for helping with homework.”
Indicator 3E
Materials provide explanations of the instructional approaches of the program and identification of the research-based strategies.
The materials reviewed for enVision Mathematics Grade 6 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 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 Goals section states the following: “The major goal in developing enVision Mathematics was to create a middle grades program that embodies the philosophy and pedagogy of the enVision series and was adapted for the middle school teacher and learner…enVision Mathematics embraces time-proven research principles for teaching mathematics with understanding. One understands an idea in mathematics when one can connect that idea to previously learned ideas (Hiebert et al., 1997). So, understanding is based on making connections, and enVision Mathematics was developed on this principle.”
The Instructional Model section states the following: “Over the past twenty years, there have been numerous research studies measuring the effectiveness of problem-based learning, a key part of the core instructional approach used in enVision Mathematics. These studies have found that students taught partly or fully through problem-based learning showed greater gains in learning (Grant & Branch, 2005; Horton et al., 2006; Johnston, 2004; Jones & Kalinowski, 2007; Ljung & Blackwell, 1996; McMiller, Lee, Saroop, Green, & Johnson, 2006; Toolin, 2004). However, the interaction of problem-based learning, which fosters informal mathematical learning, and more explicit visual instruction that formalizes mathematical concepts with visual representations leads to the greatest gains for students (Barron et al., 1998; Boaler, 1997, 1998). The enVision Mathematics instructional model is built on the interaction between these two instructional approaches. STEP 1 PROBLEM-BASED LEARNING Introduce concepts and procedures with a problem-solving experience. 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). Conceptual understanding results because the process of solving a problem that involves a new concept or procedure requires students to make connections of prior knowledge to the new concept or procedure. The process of making connections between ideas builds understanding. In enVision Mathematics, this problem-solving experience is called Solve & Discuss It. STEP 2 VISUAL LEARNING Make the important mathematics explicit with enhanced direct instruction connected to Step 1. The important mathematics is the new concept or procedure students should understand. Quite often the important mathematics will come naturally from the classroom discussion around students’ thinking and solutions for the Solve & Discuss It! task. Regardless of whether the important mathematics comes from discussing students’ thinking and work, understanding the important mathematics is further enhanced when teachers use an engaging and purposeful classroom conversation to explicitly present and discuss an additional problem related to the new concept or procedure…”
Other research includes the following:
Resendez, M.; M. Azin; and A. Strobel. A study on the effects of Pearson’s 2009 enVisionMATH program. PRES Associates, 2009.
What Works Clearinghouse. enVisionMATH, Institute of Education Sciences, January 2013.
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.
Indicator 3F
Materials provide a comprehensive list of supplies needed to support instructional activities.
The materials reviewed for enVision Mathematics Grade 6 meet expectations for providing a comprehensive list of supplies needed to support instructional activities.
In the online Teacher Resources for each grade, a Materials List is provided in table format identifying the required materials and the topic(s) where they will be used. Example includes:
The table indicates that Topic 1 will require the following materials: “Blank paper, Decimal cubes, Fraction strips, Fraction tiles...”
The table indicates that Topic 4 will require the following materials: “Algebra tiles, Base-ten blocks, Counters, Cubes/unit cubes, Different-sized boxes...”
The table indicates that Topic 8 will require the following materials: “Graphing calculator/calculator, Index cards, Number cubes and other polyhedral game pieces (optional)...”
Indicator 3G
This is not an assessed indicator in Mathematics.
Indicator 3H
This is not an assessed indicator in Mathematics.
Criterion 3.2: Assessment
The program includes a system of assessments identifying how materials provide tools, guidance, and support for teachers to collect, interpret, and act on data about student progress towards the standards.
The materials reviewed for enVision Mathematics Grade 6 meet expectations for Assessment. The materials include an assessment system that provides multiple opportunities throughout the courses to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. The materials also provide assessments that include opportunities for students to demonstrate the full intent of course-level standards and practices. The materials partially include assessment information in the materials to indicate which standards are assessed.
Indicator 3I
Assessment information is included in the materials to indicate which standards are assessed.
The materials reviewed for enVision Mathematics Grade 6 partially meet expectations for having assessment information included in the materials to indicate which standards are assessed. The materials do not identify practices for most of the assessment items.
The materials identify the following assessments in the Teacher’s Edition Program Overview:
Progress Monitoring Assessments are used at the start, middle, and end of the year to, “Diagnose and assess students’ understanding of and proficiency with concepts and skills taught throughout the school year with A, B, and C parallel assessments; results can be used to prescribe intervention.”
Diagnostic Assessments are to be given at the start of the year and the start of a topic; they consist of a Beginning-of-the-Year Assessment, Topic Readiness Assessment, and “Review What You Know.”
Formative Assessments are incorporated throughout the lesson in the form of “SCOUT Observational Assessment”, “Try It!”, “Convince Me!”, “Do You UNDERSTAND?” and “Do You Know HOW?” to check for understanding or a need to supplement instruction, or in the form of Lesson Quizzes to assess students’ conceptual understanding and procedural fluency with lesson content. Mid-Topic Checkpoint given at the midpoint of a topic, assesses students’ understanding of concepts and skills presented in lessons.
Summative Assessments, including Topic Assessments and Cumulative/Benchmark Assessments, are provided in multiple editable forms to assess student understanding after each topic and/or group of topics as well as at the end of the course.
The Teacher’s Edition maps content standards to items from Diagnostic and Summative Assessments and identifies Standards for Mathematical Practices only when the assessment is within the lesson. The standards are not listed in the student materials. Examples of how the materials identify the standards include:
Topic 3, Performance Task Form A, Problem 1, “Ali is planning a cookout for family and friends. There will be 24 people at her cookout. Ali is renting tables for the cookout and wants to seat an equal number of people at each table. She needs to decide how many tables to get. How could she arrange the seating so that a reasonable and equal number of people sit at each table. Explain.” Item Analysis for Diagnosis and Intervention indicates Standards, 6.NS.B.4, MP.1, and MP.2.
Topic 5, Assessment Form A, Problem 2, “José is using 12 brown tiles and 8 white tiles to design a section of an outdoor patio. Which ratio compares the number of brown tiles to the total number of tiles in one section? (A) 12:8 (B) 8:12 (C) 12:20 (D) 8:20” Item Analysis for Diagnosis and Intervention indicates Standard, 6.RP.A.1.
Topic 7, Lesson 7-7, Do You Understand?, Problem 3, “Make Sense and Persevere In the formula SA = 4T, for the surface area of a triangular pyramid in which the faces are equilateral triangles, what does the variable T represent?” The Lesson Overview indicates Standards, 6.G.A.4, 6.EE.A.2a, 6.EE.2c, 6.EE.B.6, MP.2, and MP.5. In addition, the label on this problem indicates that it addresses MP.1 as well.
Topics 1-8, Cumulative/Benchmark Assessment, Problem 7, “Which of the following is a statistical question? (A) How tall is Mr. Leung? (B) What are the ages of all your cousins? (C) What is the formula for the volume of a cube? (D) What is the school’s address?” Item Analysis for Diagnosis and Intervention indicates Standard, 6.SP.A.1.
Indicator 3J
Assessment system 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.
The materials reviewed for enVision Mathematics Grade 6 meet expectations for including an assessment system that provides multiple opportunities throughout the grade to determine students’ learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.
The assessment system provides multiple opportunities to determine student’s learning throughout the lessons and topics. Answer keys and scoring guides are provided. In addition, teachers are given recommendations for Math Diagnosis and Intervention System (MDIS) lessons based on student scores. If assessments are given on the digital platform, students are automatically placed into intervention based on their responses.
Examples include:
Topic 1, Lesson 1-2, Lesson Quiz, “Use the student scores on the Lesson Quiz to prescribe differentiated assignments.” I Intervention 0-3 points, O On-Level 4 points, A Advanced 5 points.” The materials provide follow-up activities—to be assigned at the teacher’s discretion—to students at each indicated level: Reteach to Build Understanding I, Additional Vocabulary Support I O, Build Mathematical Literacy I O, Enrichment O A, Math Tools and Games I O A, and Pick a Project and STEM Project I O A. For example, Problem 3, “The librarian purchased 22 copies of a best-selling book for $385.66. How much did each copy of the book cost?”
Topic 3, Assessment Form A, Problem 4, “The same digits are used for the expression 3 and 4. Explain how to compare the values of the expressions.” The accompanying Scoring Guide gives the following recommendations based on the score: Greater than 85% /Assign the corresponding MDIS for items answered incorrectly. Use Enrichment activities with the student. 70% - 85% / Assign the corresponding MDIS for items answered incorrectly. You may also assign Reteach to Build Understanding and Virtual Nerd Video assets for the lessons correlated to the items the student answered incorrectly. Less Than 70% / Assign the corresponding MDIS for items answered incorrectly. Assign appropriate intervention lessons available online. You may also assign Reteach to Build Understanding, Additional Vocabulary Support, Build Mathematical Literacy, and Virtual Nerd Video assets for the lessons correlated to the items the student answered incorrectly. Item Analysis for Diagnosis and Intervention indicates Points 1, DOK 2, MDIS L2, Standard 6.EE.A.1.
Topics 1-4, Cumulative/Benchmark Assessment, Problem 24, “Find the quotient. 1,107 ÷ 27” The accompanying Scoring Guide gives the following recommendations based on the score: Greater than 85% /Assign the corresponding MDIS for items answered incorrectly. 70% - 85% / Assign the corresponding MDIS for items answered incorrectly. Monitor the student during Step 1 and Try It! parts of the lessons for personalized remediation needs. Less Than 70% / Assign the corresponding MDIS for items answered incorrectly. Assign appropriate remediation activities available online. Item Analysis for Diagnosis and Intervention indicates DOK 1, MDIS L31, Standard 6.NS.B.2.
Topic 6, Performance Task Form A, Problem 2, “Ed wants to borrow $20,000 from a bank to open a small gym. Three banks charge different interest rates. 2. To help decide the best loan option, Ed wants to know the percent profit he will make each month. Part A The cost to run the gym each month is $5,000. Find Ed’s total monthly expenses for each loan option. Enter this in Table B. Part B Ed expects to average $7,500 in sales per month the first year. Find Ed’s average monthly profit. Explain how to find the monthly profit. Then complete Table B. Part C Complete Table B for the estimated percent profit. Explain how you found the estimated percent profit.” The Scoring Rubric indicates 1 possible point for part A (correct answers), 2 possible points for parts B and C (2 points: Correct answers and explanation, 1 point: Correct answers or explanation). The Item Analysis for Diagnosis and Remediation indicates for 2A, DOK 2, MDIS M16, Standards 6.RP.A.3c, MP.2, for 2B, DOK 2, MDIS M16, Standards 6.RP.A.3c, MP.2, and for 2C, DOK 2, MDIS M41, Standards 6.RP.A.3c, MP.3.
Indicator 3K
Assessments include opportunities for students to demonstrate the full intent of grade-level/course-level standards and practices across the series.
The materials reviewed for enVision Mathematics Grade 6 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series.
The materials provide formative and summative assessments throughout the grade as print and digital resources. As detailed in the Assessment Sourcebook, the formative assessments—Try It! and Convince Me!, Do You Understand? and Do You Know How?, and Lesson Quiz—occur during and/or at the end of a lesson. The summative assessments—Topic Assessment (Form A and Form B), Topic Performance Task (Form A and Form B), and Cumulative/Benchmark Assessments—occur at the end of a topic, group of topics, and at the end of the year. The four Cumulative/Benchmark Assessments address Topics 1-2, 1-4, 1-6, and 1-8.
Try It! and Convince Me! “Assess students’ understanding of concepts and skills presented in each example; results can be used to modify instruction as needed.”
Do You Understand? and Do You Know How? “Assess students’ conceptual understanding and procedural fluency with lesson content; results can be used to review or revisit content.”
Lesson Quiz “Assess students’ conceptual understanding and procedural fluency with lesson content; results can be used to prescribe differentiated instruction.”
Topic Assessment, Form A and Form B “Assess students’ conceptual understanding and procedural fluency with topic content. Additional Topic Assessments are available with ExamView CD-ROM.”
Topic Performance Task, Form A and Form B “Assess students’ ability to apply concepts learned and proficiency with math practices.”
Cumulative/Benchmark Assessments “Assess students’ understanding of and proficiency with concepts and skills taught throughout the school year.”
The formative and summative assessments allow students to demonstrate their conceptual understanding, procedural fluency, and ability to make applications through a variety of item types. Examples include:
Order; Categorize
Graphing
Multiple choice
Fill-in-the-blank
Multi-part items
Selected response (e.g., single-response and multiple-response)
Constructed response (i.e., short or extended responses)
Technology-enhanced items (e.g., drag and drop, drop-down menus, matching)
Indicator 3L
Assessments offer accommodations that allow students to demonstrate their knowledge and skills without changing the content of the assessment.
The materials reviewed for enVision Mathematics Grade 6 partially provide assessments which offer accommodations that allow students to determine their knowledge and skills without changing the content of the assessment.
Text-to-speech accommodation in English is available for online assessments. Spanish versions of the assessments are available in print only. In the digital format, students have access to Desmos Graphing, Geometry, and Scientific Calculators, English/Spanish Glossary, and additional Math tools.
According to the Teacher’s Edition Program Overview, “Online assessments can be customized as needed.” Assessments can be edited by the teacher, and as a result, they have the potential to alter course-level expectations because they are teacher-created items.
Criterion 3.3: Student Supports
The program includes materials designed for each student’s regular and active participation in grade-level/grade-band/series content.
The materials reviewed for enVision Mathematics Grade 6 partially meet expectations for Student Supports. The materials provide: strategies and supports for students in special populations and for students who read, write, and/or speak in a language other than English to support their regular and active participation in learning grade-level mathematics; and 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 partially provide extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.
Indicator 3M
Materials provide strategies and supports for students in special populations to support their regular and active participation in learning grade-level/series mathematics.
The materials reviewed for enVision Mathematics Grade 6 meet expectations for providing strategies and support for students in special populations to support their regular and active participation in learning grade-level mathematics.
At the end of each lesson, there is a differentiated intervention 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:
Reteach to Build Understanding (I) - Provides scaffolded reteaching for the key lesson concepts.
Additional Vocabulary Support (I, O) - Helps students develop and reinforce understanding of key terms and concepts.
Build Mathematical Literacy (I, O) - Provides support for struggling readers to build mathematical literacy.
Enrichment (O, A) - Presents engaging problems and activities that extend the lesson concepts.
Math Tools and Games (I, O, A) - Offers additional activities and games to build understanding and fluency.
Pick a Project and STEM Project (I, O, A) - Provides an additional opportunity for students to demonstrate understanding of key mathematical concepts.
Other resources offered are personalized study plans to provide targeted remediation for students, as well as support for English Language Learners and Enrichment. Additionally, Virtual Nerd tutorials are available for every lesson and can be accessed online.
Examples of the materials providing strategies and support for students in special populations include:
Topic 3, Lesson 3-4, RtI, “Use With Example 3 Some students may need additional help identifying parts of an expression. Provide students with strips of paper and scissors. Have them write expressions with at least three terms on the strips and then trade the strips with a partner. Q: How many terms does the expression on your strip have? Q: Make a cut in your strip after each operation to separate the terms. How many pieces do you have? Students should check this answer against their answer to the first question, Discuss any discrepancies.”
Topic 5, Lesson 5-2, RtI, “Error Intervention ITEM 19 Use Appropriate Tools Students may not understand how they can use a table to find equivalent ratios. Q: As you go across each row, what does each of the numbers represent? Q: If you look at the column with 3 at the top, what do 6 and 15 in that column represent?”
Indicator 3N
Materials provide extensions and/or opportunities for students to engage with grade-level/course-level mathematics at higher levels of complexity.
The materials reviewed for enVision Mathematics Grade 6 partially meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.
The materials provide a Differentiated Intervention section within each lesson, which has resources intended for more advanced students such as an Enrichment worksheet, Math Tools and Games, and Pick a Project and STEM Project. These assignments can be auto-assigned based on formative assessment scores in the online platform. Additionally, each lesson also has Enrichment activities that accompany certain problems. However, there is no guidance on how to use these materials in the classroom in a way that would ensure advanced learners would not be completing more assignments than their peers.
Examples include:
Topic 2, Lesson 2-3, Enrichment, “USE WITH EXAMPLE 1 TRY IT! Challenge advanced students to apply their knowledge to another situation. State the following question or write it on the board. The students recorded changes in the water level for two more weeks. During Week 4, the water level rose, and during Week 5, the water level fell. If both of these changes were less than the amount of change during the first three weeks, what could have been the change in water level for each of these weeks? Q: Would you use positive or negative numbers to show a rise in water level? Q: Would you use positive or negative numbers to show a fall in water level?”
Topic 4, Lesson 4-3, Differentiated Intervention, Enrichment, “Some situations are modeled by equations in which more than one number is added to or subtracted from the variable. Although solving these equations takes more than one step, you can still solve them by applying the properties of equality. For each situation, define a variable for the unknown quantity. Write and solve an equation to answer the question. 1. Brody and Carrie like to hike in the mountains. From the trailhead, they climb 103 feet until they reach a waterfall. On the next part of the trail, they descend 84 feet. To reach their destination, Brody and Carrie must them climb another 194 feet. Their destination is 734 feet above sea level. At what elevation is the trailhead?...”
Indicator 3O
Materials provide varied approaches to learning tasks over time and variety in how students are expected to demonstrate their learning with opportunities for students to monitor their learning.
The materials reviewed for enVision Mathematics Grade 6 partially provide varied approaches to learning tasks over time and variety in how students are expected to demonstrate their learning with opportunities for students to monitor their learning. The materials provide a variety of approaches for students to learn the content over time but provide limited opportunities for students to monitor their learning.
Students engage with problem-solving in a variety of ways within a consistent lesson structure. The Teacher’s Edition Program Overview indicates that the lesson structure incorporates both Problem-based Learning and Visual Learning within the 5Es instruction framework: Engage, Explore, Explain, Elaborate, and Evaluate. Examples of how the lesson structure allows for varied approaches to learning tasks and variety in how students demonstrate their learning include:
Problem-based Learning
Engage and Explore: Solve & Discuss It! begins the lesson instruction by asking students to solve a problem that embeds new ideas. Students will use concrete materials or pictorial representations and may solve these problems any way they choose.
Visual Learning
Explain: Visual Learning Bridge, “provides explicit instruction that connects students’ work in Solve & Share, Explore It, and Explain It to new ideas taught in the lesson. The Visual Learning Bridge at times shows pictures of concrete materials, drawings of concrete materials, and/or diagrams that are representations of mathematical concepts.” Try It!, “offers a formative assessment opportunity after each example.” Convince Me!, “connects back to the Essential Understanding of the lesson.”
Elaborate: Key Concept, “includes guiding questions to monitor students’ understanding.” Do You Understand?, always includes having students answer the lesson's Essential Question, and focus on determining students’ understanding of lesson concepts. Do You Know How? focuses on determining students’ understanding of concepts and skill application. Practice & Problem Solving builds proficiency as students work on their own, and Higher Order Thinking exercises are always included.
Evaluate: A Lesson Quiz, is available for print or digital administration, based on the quiz score differentiated intervention will be assigned to students in one of three levels (intervention, on-level, or advanced).
Indicator 3P
Materials provide opportunities for teachers to use a variety of grouping strategies.
The materials reviewed for enVision Mathematics Grade 6 provide some opportunities for teachers to use a variety of grouping strategies. The Teacher’s Edition routinely suggests using groups for different activities. While suggestions for the timing and size of groups are explicit within a structured instructional routine; suggestions do not always address how to form specific groups based on the needs of individual students. Examples of how the materials provide opportunities for teachers to use grouping in instruction include:
Teacher’s Edition
Pick a Project, “Grouping You might have students work independently, with a partner, or in small groups…Project Sharing Invite students to share their completed projects with a partner, a small group, or with the whole class.”
Solve & Discuss It! guidance indicates, “Before - Whole Class, … During - Small Group, … and After - Whole Class.”
Program Overview
Let’s Investigate, Student Engagement, “Students typically work in groups to solve the problems together using manipulatives, patterns, visuals, and prior mathematical knowledge. Students prepare and present their work and participate in rich classroom conversations about their work and others’ work.”
STEM Project, “You may choose to require the entire class to pursue the same design problem or allow smaller groups of students to choose which design project to pursue.”
Tips for Facilitating Problem-Based Learning, “Foster communication. Have students share their thinking with a partner, small group, or the whole class.”
Indicator 3Q
Materials provide 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 materials reviewed for enVision Mathematics Grade 6 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 Teacher’s Edition Program Overview, Supporting English Language Learners section, list the following strategies and supports:
“Daily ELL instruction is provided in the Teacher’s Edition.
Levels of English language proficiency are indicated, and they align with the following levels identified in WIDA (World-Class Instructional Design and Assessment): Entering, Emerging, Developing, Expanding, Bridging.
ELL Principles are based on Jim Cummins’ work frame.
Visual Learning Animation Plus provides stepped-out animation to help lower language barriers to learning. Questions that are read aloud also appear on screen to help English language learners connect oral and written language.
Visual Learning Example often has visual models to help give meaning to math language. Instruction is stepped out to organize important ideas visually.
Animated Glossary is always available to students and teachers while using digital resources. The glossary is in English and Spanish to help students connect Spanish math terms they may know to English equivalents.
Pictures with a purpose appear in lesson practice to help communicate information related to math concepts or to real-world problems.”
Examples where the materials provide strategies and supports for students who read, write, and/or speak in a language other than English include:
Topic 3, Lesson 3-1, English Language Learners (Use with Example 1), “ENTERING Ask students to review Example 1. Q: What words are new to you? Circle them. Write the words with simple definitions. Refer to them as you discuss the examples. Q: The word base is a homonym. It has more than one meaning. Where have you heard the word base before? Students may suggest bases in baseball or base, the safe place when playing tag, or the bottom of something like a statue.”
Topic 4, Lesson 4-7, English Language Learners (Use with Example 1), “DEVELOPING Draw a number line from 0 to 10. Write the inequality x < 6. Point to 6 in the inequality, and draw an open circle at 6 on the number line. Ask students to share with partners the direction indicated by < 6. Q: What conclusions can be made about the solutions? Have students respond with sentences using inequality vocabulary. Repeat in a similar manner for x > 6.”
Topic 6, Lesson 6-1, English Language Learners (Use with Example 2), “ENTERING Help students read the labels on the line segments in Example 2. Q: What word does the symbol % stand for? Q: What does the abbreviation in. represent? Q: What do the letters above the line segments represent on the line segments?”
Indicator 3R
Materials provide a balance of images or information about people, representing various demographic and physical characteristics.
The materials reviewed for enVision Mathematics Grade 6 provide a balance of images or information about people, representing various demographic and physical characteristics.
Materials represent a variety of genders, races, ethnicities, and physical characteristics. All are indicated with no bias and represent different populations. When images of people are used, they represent different races and portray people from many ethnicities positively and respectfully, with no demographic bias for who achieves success in the context of problems. Lessons include a variety of names that are representative of various demographics. Examples include:
Topic 2, Lesson 2-1, Additional Practice, Problem 30, “Roberto and Jeanne played a difficult computer game. Roberto’s final score was -60 points, and Jeanne’s final score was -160 points. Use <, >, or = to compare the scores, then find the player who had the higher final score.”
Topic 3, Lesson 3-1, Practice & Problem Solving, Problem 28, “Jia is tiling a floor. The floor is a square with side length 12 feet. Jia wants the tiles to be squares with side length 2 feet. How many tiles does Jia need to cover the entire floor?...”
Topic 7, Lesson 7-3, Practice & Problem Solving, Problem 16, “Marique is making a large table in the shape of a trapezoid. She needs to calculate the area of the table. The longest side of the table is twice as long as the table’s width. Find the area of the table by decomposing the trapezoid into familiar shapes. Show your work.”
Indicator 3S
Materials provide guidance to encourage teachers to draw upon student home language to facilitate learning.
The materials reviewed for enVision Mathematics Grade 6 partially provide guidance to encourage teachers to draw upon student home language to facilitate learning.
The materials include a Language Support Handbook and Spanish versions of the Interactive Student Edition, all online and print instructional resources (e.g., Glossary), and the Family Engagement materials (which entails an overview of Resources, Content and Standards, and Topic/Lesson Supports).
The Language Support Handbook clarifies the philosophy about drawing upon students' home language to facilitate learning: “ … Over the years, new language is meaningful when it is connected to a variety of experiences, objects, pictures, abstract ideas, and previously-learned language. … For meaningful learning, help students connect new ideas and languages to a variety of experiences, objects, pictures, abstract ideas, and previously-learned language. … Provide language support as needed, while giving all students full access to rich experiences that facilitate meaningful, engaging learning. Make math class a place that continues to nurture children’s natural love of learning.”
The Language Support Handbook provides Professional Reading: Language Support in Mathematics, Academic Vocabulary Resources, and Language Support Activities. Professional Reading focuses on supporting access to mathematical thinking; supporting productive struggle in mathematics; supporting reading, writing, speaking, and representing; supporting vocabulary and language in mathematics; supporting classroom conversations in mathematics; and scaffolding without overscaffolding. Additional Resources include WIDA proficiency level descriptors, types of math problems involving operations, academic vocabulary activities, academic vocabulary in six languages, and the Language Demands in Mathematics Lessons (LDML) Tool.
Materials can be accessed in different languages by highlighting any text in the Student Edition (not available in the interactive version) and pressing the translate button. The highlighted text will be translated with text only or with text and text-to-speech (audio support) depending on the language availability in the settings. All translations are done by Google and students are also able to control the speed of the voice. Available languages include but are not limited to the following: Afrikaans (audio support), Belarusian, Bosnian, Chinese Traditional (audio support), Finnish (audio support), Galician (audio support), Greek (audio support), Haitian Creole, Portuguese (audio support), Spanish (audio support)...etc.
While Language Supports are regularly embedded within lessons and support mathematical language development, they do not include specific suggestions for drawing on a student’s home language.
Indicator 3T
Materials provide guidance to encourage teachers to draw upon student cultural and social backgrounds to facilitate learning.
The materials reviewed for enVision Mathematics Grade 6 partially provide guidance to encourage teachers to draw upon student cultural and social backgrounds to facilitate learning.
The Teacher’s Edition Program Overview, states the following about Pick a Project, “Student Choice Pick a Project offers students the opportunity to explore areas of interest and complete projects of their choosing. This kind of student choice has special benefits related to differentiation, motivation, and open-ended rich tasks…Varied contexts in the projects let students choose contexts related to everyday life as well as contexts with cross-curricular connections to social studies, science, art, and literacy.” Some of the project choices in the Pick a Project give students opportunities to draw upon their cultural and social background.
Examples of the materials drawing upon students’ cultural and/or social backgrounds to facilitate learning include:
Topic 1, Pick a Project, the project choices are the following: Project 1A What is the most challenging board game you have ever played? Project: Make Your Own Board Game, Project 1B What is your favorite party food? Project: Plan The Menu For a School Fundraiser, Project 1C If you planted a grade, what would be in it? Project: Design a Vegetable and Herb Garden, and Project 1D How much food does a tiger eat? Project: Present a Proposal For A Tiger Exhibit.
Topic 8, Pick a Project, the project choices are the following: Project 8A If you recorded a video blog, what would it be about? Project: Explore Video Blogs, Project 8B How many different breakfast cereals have you tasted? Project: Investigate Cereals, Project 8C How far could you bike before needing to rest? Project: Analyze a Time Trial, and Project 8D If you could change something at your school, what would it be? Project: Survey Your School.
Indicator 3U
Materials provide supports for different reading levels to ensure accessibility for students.
The materials reviewed for enVision Mathematics Grade 6 provide supports for different reading levels to ensure accessibility for students.
The Teacher’s Edition Program Overview, Building Literacy in Mathematics section provides “Vocabulary and writing support as well as reading connections!” for four different parts, “Literacy Support at the Start of Topics”, “Literacy Support in Lessons”, “Literacy Support at the End of Lessons”, and “Literacy Support at the End of Topics”.
The following are examples where materials provide supports for different reading levels to ensure accessibility to students:
Examples of the supports that are offered in the “Literacy Support at the Start of Topics” section include the following:
“…Include vocabulary, close reading, study and organizational supports.
Language Development activity in the Student’s Edition to support and reinforce vocabulary and language development
Language Support Handbook provides lists of pertinent math terminology needed to understand and communicate the math ideas of the topic; provides language support activities to reinforce math vocabulary learned throughout the topic; supports the development and use of academic vocabulary.”
Examples of the supports that are offered in the “Literacy Support in Lessons” section include the following:
“...Glossary in the Student’s Edition A glossary at the back of the Volume 1 Student’s Edition can be used for reference at any time.
Language Support Handbook The Language Support Handbook includes various activities that promote the development and use of precise mathematical language. Each activity engages students by focusing on one or more modalities: reading, writing, listening, speaking, and representing. These activities encourage student-teacher interaction and participation.”
Examples of the supports that are offered in the “Literacy Support at the End of Lessons” section include the following:
“Additional Vocabulary Activities offers vocabulary development support for all students, especially English Language Learners and struggling readers.
Build Mathematical Literacy provides structured support to help students build literacy strategies for mathematics.”
Examples of the supports that are offered in the “Literacy Support at the End of Topics” section include the following:
“Vocabulary Review At the end of each topic is a page of Vocabulary Review. It includes questions to reinforce understanding of the vocabulary used in the topic and asks students to use vocabulary in writing.
Animated Glossary An animated glossary is always available to students and teachers online or through the eTexts.
Vocabulary Game Online The Game Center online includes a vocabulary game that students can access anytime.”
An example of student support:
Topic 5, Lesson 5-10, Build Mathematical Literacy, students are provided with questions to help understand the problem. “Read the problem below. Answer the questions to help understand the problem. At the State Fair, a person must be at least 138 centimeters tall to ride the roller coaster. Billy wants to ride the coaster. He is 4 feet 7 inches tall. Is Billy tall enough to ride the coaster? Explain. 1. Circle Billy’s height. Underline the minimum height to ride the roller coaster. 2. To solve the problem, will you convert metric units to customary units or convert customary units to metric units? Explain. …5. How many steps must be completed so that you can compare Billy’s height to the minimum height to ride the roller coaster? Describe the steps in order.”
Throughout the materials, students can enable a text-to-speech feature in both the interactive and non-interactive student editions.
Indicator 3V
Manipulatives, both virtual and physical, are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.
The materials for enVision Mathematics Grade 6 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 Teacher’s Edition Program Overview, Concrete, Representational, Abstract section states the following: “Digital interactivities Digital interactivities can simulate work with concrete models and can let students interact with pictorial representations. Using the digital Math Tools, students can move counters around on the screen, arrange fraction strips, manipulate geometric figures, and more. Many of the interactivities in the Visual Learning Animation Plus provide those same opportunities. Physical Manipulatives Physical manipulatives, including algebra tiles, counters, cubes, geoboards, and anglegs, provide opportunities for students to engage in concrete modeling when developing abstract thinking with mathematical concepts. A recommended set of manipulatives is available for each grade…Digital versions of the manipulatives are also available online.”
Examples of how manipulatives, both virtual and physical, are representations of the mathematical objects they represent and, when appropriate are connected to written methods, include:
Topic 1, Lesson 1-4, Explore It!, students are given number lines (or Teaching Tool 6) to represent a problem based on a diagram. “Students are competing in a 4-kilometer relay race. There are 10 runners. A. Use the number line to represent the data for the race. B. Use multiplication or division to describe your work on the number line.”
Topic 4, Lesson 4-6, Solve & Discuss It!, students are given number lines (or Teaching Tool 6) to assist in solving a problem that has to do with breaking records. “The record time for the girls’ 50-meter freestyle swimming competition is 24.49 seconds. Camilla has been training and wants to swim the 50-meter freestyle in less time. What are some possible times Camilla would have to swim to break the current record? Solve this problem any way you choose.”
Topic 7, Student Interactive Lesson 7-2, Solve & Discuss It!, students use an applet to form a triangle and find the area. “Connect point A to B, B to C, C to D, and D to A. Then draw a diagonal line connecting opposite vertices in the figure and find the area of each triangle formed.”
Criterion 3.4: Intentional Design
The program includes a visual design that is engaging and references or integrates digital technology, when applicable, with guidance for teachers.
The materials reviewed for enVision Mathematics Grade 6 integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the grade-level standards, and the materials partially include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other. The materials have a visual design that supports students in engaging thoughtfully with the subject, and is neither distracting nor chaotic, and the materials provide teacher guidance for the use of embedded technology to support and enhance student learning.
Indicator 3W
Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the grade-level/series standards, when applicable.
The materials reviewed for enVision Mathematics Grade 6 integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the grade-level/series standards, when applicable. Built-in tools are integrated into the “Interactive Student Edition”. Students also have access to other tools that can be found on the main page of the website. These tools found under the“Tools” section include Desmos Graphing Calculator, Desmos Geometry Tool, Desmos Scientific Calculator, and Math Tools that contain 2D and 3D Geometric Constructor, Algebra Tiles, Graphing Utility, and Number Line.
Examples of how the materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the grade-level standard include:
Topic 1, Lesson 1-2, Solve & Discuss It!, “Some friends went to lunch and split the bill equally. If each person paid $6.75, how many people went to lunch? Use a diagram or equation to explain your thinking. Solve this problem any way you choose.” The option is given for the students to play a recording of someone reading the problem. Students use the tools from DrawPad to present their work.
Topic 4, Lesson 4-2, Practice & Problem Solving, Problem 18, “A scale balanced with 1 blue x-block and 20 green blocks on the left side and 40 green blocks on the right side. A student bumped into the scale and knocked some blocks off so that only 1 blue x-block and 3 green blocks remained on the left side. How many blocks do you need to remove from the right side to make the scale balance?” Students can access a virtual pan balance to solve the problem.
Topic 7, Lesson 7-1, Interactive Additional Practice, Problem 10, “ A rhombus has a base of 5.2 meters and a height of 4.5 meters. The rhombus is divided into two identical triangles. What is the area of each triangle?” Students have access to the DrawPad where they can create and label shapes to solve the problem.
Indicator 3X
Materials include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other, when applicable.
The materials reviewed for enVision Mathematics Grade 6 partially include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other, when applicable. The materials include digital technology that provides opportunities for student-to-teacher, and student-to-student collaboration but opportunities for teacher-to-teacher collaboration are not provided.
The digital system allows students and teachers to collaborate by commenting on assignments. The Savvas Realize help page states the following: “Realize Reader Comments Using the Realize Assignment Viewer, you can provide your student with feedback in their Realize Reader assignments by adding a comment to a highlight, annotation, or inline Notebook prompt response. When you or your student adds a comment, a comment thread is created that enables you to continue to communicate with each other in context.”
The digital system allows students to collaborate with other students and teachers through the Discussion Forums. The Savvas Realize help page states the following: “Discussion Forum Discussions enable you to facilitate class and group discussions on important academic and social topics. Students can reflect on learning, share ideas and opinions, or ask and answer questions. You can create, monitor, and reply to discussions, and students can participate in discussions you create. In addition, you can choose whether or not to score discussions.”
Indicator 3Y
The visual design (whether in print or digital) supports students in engaging thoughtfully with the subject, and is neither distracting nor chaotic.
The materials reviewed for enVision Mathematics Grade 6 have a visual design (whether print or digital) that supports students in engaging thoughtfully with the subject, and is neither distracting nor chaotic.
There is a consistent design within topics and lessons that supports student understanding of mathematics. Examples include:
Each topic begins with the Topic Overview, Math Practices, Topic Readiness Assessment, Topic Opener, STEM Project, Get Ready!, and Pick a Project. A Mid-Topic CheckPoint, Mid-Topic Performance Task, and 3-Act Math is included at the midpoint of each topic and all topics end with a Topic Review, Topic Assessment, Topic Performance Task, and Cumulative/Benchmark Assessment (if applicable).
Each lesson follows a common format:
Lesson Overview which includes the Mathematics Overview, Language Support, and Math Anytime
Step 1: Problem-Based Learning focuses on Solve & Discuss It!
Step 2: Visual Learning consists of the Essential Question, Examples, Try IT!, Convince Me!, Key Concepts, Do You Understand/Do You Know How?, and Practice & Problem Solving
Step 3: Assess & Differentiate consists of the Lesson Quiz, Video Tutorials, Additional Practice, and Differentiated Interventions
Student materials include appropriate font size and placement of direction. There is ample space in the printable Student Task Statements, Assessment PDFs, and workbooks for students to capture calculations and write answers.
When images, graphics, or models are included, they clearly communicate information supporting student understanding of topics, texts, or concepts.
Indicator 3Z
Materials provide teacher guidance for the use of embedded technology to support and enhance student learning, when applicable.
The materials reviewed for enVision Mathematics Grade 6 provide teacher guidance for the use of embedded technology to support and enhance student learning, when applicable. The materials provide teachers with multiple easy access points for technology and with specific guidance provided in the supplementary handouts.
Examples of teacher guidance for the use of embedded technology include:
Examples from the “Let’s Go Digital!” Handout,
Tools “Open the Tools menu anytime to find a variety of interactive tools that you and your students can use. Check out the Game Center and Math Tools.”
Planning a Topic “…Then, review the Today’s Challenge problems. Notice that each problem of the five-day challenge uses the same data with increasing difficulty each day. Consider displaying the problem at the beginning of the day and having students use the DrawPad tools to respond...”
Teaching a Lesson “...Start each lesson with the problem-based Solve & Share task. Display the problem from your computer and use the DrawPad tools to model your students’ ideas...”
An example from the Assessment Handout, “Additional Assessment Options On Savvas Realize, you can customize assessments to meet your instructional needs. To explore these options, click Customize under the assessment name. You can modify the title, the description, and whether the test should count toward mastery. To add questions, click Add items from the test bank and search the bank of test items by standard or keyword. You can also add your own assessments. Select Create Content menu to upload files, add links, or build your own tests. Finally, check out ExamView test generator in the Tools menu.”
All of the above-mentioned handouts are also available as Tutorial Videos.
An example from the Savvas Realize help page, “Remove Students from a Realize Class You can remove students from a Realize class using the instructions in this topic. To remove a student that was imported from Google Classroom, see Remove Students Imported From Google Classroom. 1. Click Classes on the top menu bar, then select the class. 2. Click Students & groups on the left. 3. Click the 3-dot menu next to the student you want to remove, then click Remove Student.” Pictures are included with some steps to provide additional guidance.