6th Grade - Gateway 1

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that they assess grade-level content.

Each Topic contains diagnostic, formative, and summative assessments. Summative assessments include: Topic Assessments Forms A and B, Topic Performance Tasks Forms A and B, and Cumulative/Benchmark Assessments. Assessments can be administered online or printed for paper/pencil format. No above grade-level assessment questions are present. Examples of grade-level assessment aligned to standards include: 

• Topic 1, Assessment Form A, Question 6, “Raven is making pillows. Each pillow requires $$\frac{3}{5}$$ yard of fabric. Raven has $$6\frac{2}{3}$$ yards of fabric. Find the number of pillows Raven can make. A. 11 pillows, B. 10 pillows, C. 5 pillows, D. 4 pillows.” (6.NS.1)
• Topic 3, Performance Task Form B, Question 3, Part A, “Mr. Jones is going to build a garden in back of the restaurant to have fresh produce available. The garden will be rectangular, with a length of 2x + 3 feet and a width of x feet. Part A: Fencing material costs $3 per foot including delivery. Write an expression to show the amount of fencing that Mr. Jones will need and then write an expression for the cost of the fence.” (6.EE.2 and 6.EE.6)
• Topic 6, Assessment Form B, Question 5, “A survey found that 78% of high school freshmen have Internet access at home. Of the 754 freshmen at one high school, about how many would be expected to have Internet access at home? Explain.” (6.RP.3)
• Topics 1-6, Cumulative /Benchmark Assessment, Question 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.EE.8 and 6.NS.3)
• Topic 7, Assessment Form A, Question 1, “Curtis is making a triangular frame with a base of 12 feet. The perpendicular distance from the base of the frame to its vertex is 6 feet. What is the area of the frame? A. 144 ft$$^2$$, B. 72 ft$$^2$$, C. 36 ft$$^2$$, D. 18 ft$$^2$$.” (6.G.1 and 6.EE.2)
• Topic 8, Performance Task Form A, Question 3, “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.” (6.SP.3 and 6.SP.5)

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations for spending a majority of instructional time on major work of the grade. For example:

• 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 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.

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that supporting work enhances focus and coherence simultaneously by engaging students in the major work of the grade.

Materials are designed so supporting standards/clusters are connected to the major standards/clusters of the grade. Examples from the Teacher Resource include:

• Lesson 2-6, Represent Polygons on the Coordinate Plane, Visual Learning, Example 2, students graph polygons on a four quadrant coordinate plane and find distances using absolute value, “A rancher maps the coordinates for a holding pen for his cows. How much fencing does the rancher need to enclose the cows’ holding pen?” This example 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 to the major work of 6.NS.7, understand ordering and 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.
• Lesson 3-2, Find Greatest Common Factor and Least Common Multiple, Visual Learning, Examples 2, 3 and 4, students find common factors and multiples of two whole numbers while using the properties of operations, such as the Distributive Property, to generate equivalent algebraic expressions. Example 3, “Use the GCF and the Distributive Property to find the sum of 18 and 24. Can you use the Distributive Property to rewrite the sum of any two numbers? Explain. Why do you want to identify the common multiples?” These examples connect 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 to the major work of 6.EE.3, apply the properties of operations to generate equivalent expressions.
• Lesson 7.5, Represent Solid Figures Using Nets, Visual Learning, Example 3, students use understanding of the coordinate plane to draw nets of three-dimensional figures, “How can you draw a net of a rectangular prism that has a height of 2 units and bases that are 4 units long and 2 units wide?” This example connects the supporting work of 6.G.4, represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures to the major work 6.NS.8, solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane.
• Lesson 7.8, Find Volume with Fractional Edge Lengths, Visual Learning, Example 2, students write algebraic expressions to find volume, “Sean bought the fish tank shown. What is the volume of Sean’s fish tank?” This example connects the supporting work of 6.G.2, find the volume of a right rectangular prism with fractional edge lengths to the major work of 6.EE.2, write, read, and evaluate expressions in which letters stand for numbers.
• Lesson 8-3, Display Data in Box Plots, Visual Learning, Do You Know How?, Item 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 question 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.

Note: In the Cross-Cluster Connection in the Teacher Resource a connection between the supporting work of 6.SP.A and major work of 6.EE.C is identified. This connection is not supported by student work. The two instances where this occurs include:

• Lesson 8-1, Lesson Overview, Coherence, Cross-Cluster Connection, “Prior work with recognizing and analyzing quantitative relationships between dependent and independent variables in Lesson 4-8 (6.EE.C) connects to students recognizing a statistical question and understanding statistical variability (6.SP.A). Student work with independent and dependent variables is not present.
• Lesson 8-2, Lesson Overview, Coherence, Cross-Cluster Connection, “Prior work with recognizing and analyzing quantitative relationships between dependent and independent variables in lesson 4-8 (6.EE.C) connects to students recognizing how to summarize a data set by its measure of center of variability (6.SP.A). Student work with independent and dependent variables is not present.

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that the amount of content designated for one grade-level is viable for one year. As designed, the instructional materials can be completed in 170-194 days.

According to the Pacing Guide in the Teacher Resource, Program Overview, “enVision Mathematics 6-8 was designed to provide students rich opportunities to build understanding of important new mathematical concepts, develop fluency with key skills necessary for success in algebra, and to gain proficiency with the habits of mind and thinking dispositions of proficient mathematical students. To achieve these goals, the program includes content-focused lessons, 3-Act Mathematical Modeling lessons, STEM projects, and Pick a Project. All of these instructional activities are integral to helping students achieve success, and the pacing of the program reflects this. 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 assessment.” For example:

• 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.

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations for the materials being consistent with the progressions in the Standards.

The instructional materials clearly identify content from prior and future grade-levels and use it to support the progressions of the grade-level standards. According to the Teacher Resource, 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 from the Teacher Resource include:

• Topic 1 Overview, Use Positive Rational Numbers, Math Background, Coherence, “In Grade 5, students added, subtracted, and multiplied decimals through hundredths. They used place-value strategies to divide decimals by whole numbers and decimals. They estimated quotients to place the first digit in the quotient and to determine whether their calculations were reasonable. The work students do in this Topic connects directly to Topic 5: Understand and Use Ratio and Rate as students solve problems involving rates, Topic 6: Understand and Use Percent as students relate decimals to percents, Topic 7: Solve Area, Surface Area, and Volume Problems as students compute areas and volumes of figures and summarizes in Topic 8: Display, Describe, and summarize data when students summarize data. Beyond sixth grade, students extend their understanding of ratios and rates to investigate proportional relationships in seventh grade. In Grade 7, students will apply their understanding of fraction computation to proportional relationships and percents.” 
• Topic 2 Overview, Integers and Rational Numbers, Math Background, Coherence, “In Grade 5, students extended value to the thousandths place. They graphed decimals on a number line to help them compare and round decimals. They also extended their ability to do computations with rational numbers to include adding, subtracting, multiplying, and dividing decimals and fractions. Students learned about the coordinate plane and graphed points in the first quadrant to solve real-world and mathematical problems. In Grade 7, students will add, subtract, multiply, and divide both positive and negative rational numbers. They will solve multistep problems involving operations of rational numbers, renaming numbers as appropriate.”
• Topic 5 Overview, Understand and Use Ratio and Rate, Math Background, Coherence, “In Grade 5, students used equivalent fractions to add and subtract fractions and mixed numbers with unlike denominators, and to multiply with fractions. They also learned to divide two whole numbers and get a quotient expressed as either a fraction or mixed number. Students learned to convert measurements within a given measurement system by using multiplication and division. Students learned to graph points in the first quadrant of the coordinate plane. In Topic 6, students will understand percent as a rate in which the comparison is to 100. They will use this understanding to relate fractions, decimals, and percents to solve problems. In Topic 8, students will use percents as they summarize data distributions. In Grade 7, students will compute unit rates associated with ratios of fractions. Students will apply their understanding of the ratio between the circumference and diameter of a circle when they solve problems involving area and circumference of a circle. Students will recognize and represent proportional relationships between quantities. They will also use proportional relationships to solve multistep ratio and percent problems.” 

The instructional materials attend to the full intent of the grade-level standards by giving all students extensive work with grade-level problems. The Solve & Discuss It! section presents students with high-interest problems that embed new math ideas, connect prior knowledge to new learning and provide multiple entry points. Example problems are highly visual, provide guided instruction and formalize the mathematics of the lesson. Try It! provides problems that can be used as formative assessment following Example problems and Convince Me! provides 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 from the Teacher Resource include:

• Lesson 1-5 Divide Fractions By Fractions, 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 $$\frac{2}{3}$$ of the original bar remains. How many $$\frac{1}{6}$$ parts are left? Use the picture to draw a model to represent and find $$\frac{2}{3}$$ ÷ $$\frac{1}{6}$$.” (6.NS.1)
• Lesson 3-6, Generate Equivalent Expressions, Visual Learning, Example 1, students use properties of operations to generate equivalent algebraic expressions, “Which of the expressions below are equivalent? 8x - 4, 4x, and 4(2x - 1).” (6.EE.3, and 6.EE.4)
• Lesson 4-5, Write and Solve Equations with Rational Numbers, Do You Know How?, Items 6-8, students write and solve equations that contain fractions, mixed numbers, and decimals using inverse relationships and properties of equality, “t - $$\frac{2}{3}$$ = 25$$\frac{3}{4}$$; $$\frac{f}{2}$$=$$\frac{5}{8}$$; 13.27 = t - 24.45.” (6.EE.7) 
• Lesson 5-3, Visual Learning, Example 2, Try It, 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?” (6.RP.3)
• Lesson 7-8, Find Volume with Fractional Edge Lengths, Visual Learning, Example 1, Convince Me!, students find the volume of a rectangular prism built with appropriate unit fraction edge length cubes, “Suppose that the length of the rectangular prism in the Try It! were 3 $$\frac{1}{2}$$ inches instead of 2 $$\frac{1}{2}$$ inches. How many cubes would there be in the prism? What would be the volume of the prism?” (6.G.2)
• Lesson 8-5, Summarize Data Using Measures of Variability, Visual Learning, Example 3, 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?” (6.SP.5)

The instructional 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:

• Lesson 2-4, Represent Rational Numbers on the Coordinate Plane, Lesson Overview, Coherence, “Students will be able to identify and graph points with rational coordinates on the coordinate plane and reflect points with rational coordinates across both axes.” (6.NS.6b, 6.NS.6c) “In Grade 5, students represent real-world and mathematical problems by graphing points in the first quadrant of the coordinate plane.”
• Lesson 5-1, Understand Ratios, Lesson Overview, Coherence, “Students will be able to use ratios to describe the relationship between two quantities and use bar diagrams and double number line diagrams to model ratio relationships.” (6.RP.1, 6.RP.3). “In Grade 5, students analyzed patterns and relationships and used models to represent fractional relationships.”
• Lesson 7-1 Find Areas of Parallelograms and Rhombuses, Lesson Overview, Coherence, “Students will be able to use a formula to find the areas of parallelograms and rhombuses and find the base or height of a parallelogram or rhombus when the area and the height or base are known.” (6.G.1, 6.EE.2) “In Grades 4 and 5, students used the formula for the area of a rectangle to solve problems.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials foster coherence through connections at a single grade, where appropriate and required by the Standards.

Materials include learning objectives that are visibly shaped by CCSSM cluster headings. Topics are divided into Lessons focused on domains. Grade 6 standards are clearly identified in each Topic Planner found in the Topic Overview. Additionally, each lesson identifies the Content Standards in the Mathematics Overview. Examples from the Teacher Resource include:

• Lesson 1-3, Multiply Fractions, Lesson Overview, Mathematics Objective, “Use models to multiply fractions. Multiply the numerators and then the denominators to find the product of two fractions. Multiply mixed numbers.” (6.NS.1)
• Lesson 3-5, Evaluate Algebraic Expressions, Lesson Overview, Mathematics Objective, “Evaluate algebraic expressions, including those with whole numbers, decimals, and fractions.” (6.EE.2c, 6.EE.6)
• Lesson 5-5, Understand Rates and Unit Rates, Lesson Overview, Mathematics Objective, “Use rates to describe ratios in which the terms have different units. Use rates and unit rates to solve problems.” (6.RP.2, 6.RP.3a, 6.RP.3b)
• Lesson 7-3, Find Areas of Trapezoids and Kites, Lesson Overview, Mathematics Objective, “Find the areas of trapezoids. Find the areas of kites.” (6.G.1, 6.EE.2c) 
• Lesson 8-5, Summarize Data Using Measures of Variability, Lesson Overview, Mathematics Objective, “Display numerical data in plots on a number line, including dot plots, histograms, and box plots.” (6.SP.4)

Materials include problems and activities that connect two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important. Examples from the Teacher Resource include:

• Lesson 2-6, Represent Polygons on the Coordinate Plane, Solve and Discuss It!, students graph points on a coordinate plane and use the side lengths to find the perimeter of polygons, “Draw a polygon with vertices at A(-1, 6), B(-7, 6), C(-7, -3), and D(-1, -3). Then find the perimeter of the polygon.” This question connects the work of 6.NS.C, solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane to 6.G.A, draw polygons in the coordinate plane given coordinates for the vertices.
• Lesson 3-4, Write Algebraic Expressions, Practice and Problems Solving, Item 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.” This question connects the work of 6.EE.A, apply and extend previous understandings of arithmetic to algebraic expressions to 6.EE.B, reason about and solve one-variable equations and inequalities.     
• Lesson 6-5, Find the Percent of a Number, Do You Know How?, Item 11, students find the percent from real-world problems that involve one variable, “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 question connects the work of 6.RP to the work of 6.EE. 
• Lesson 7-4, Find Areas of Polygons, Visual Learning, Example 3, students use their understanding of integers to represent and find the area of polygons on the coordinate plane, “The floor plan for a new stage at a school is sketched on a coordinate plane. A flooring expert recommends bamboo flooring for the stage floor. How much bamboo flooring, in square meters does the school need?” This example connects the work of 6.G to the work of 6.NS.
• Lesson 8-7, Summarize Data Distributions, Visual Learning, Example 2, students begin to understand spread and variability of data sets, “The fat content, in grams, was measured for one slice ($$\frac{1}{8}$$ pizza) of 24 different 12-inch pizzas. The data are displayed in the dot plot. How can the data be used to describe the fat content of a slice of pizza?” This example connects the work of 6.SP.A, develop understanding of statistical variability to the work 6.SP.B, summarize and describe distributions.