6th Grade - Gateway 1

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$$^2$$. 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)

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 \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}.”  There is an additional example available online and/or in the teacher materials, “Kate has \frac{2}{3} gallon of popcorn. She puts the popcorn in \frac{1}{6} gallon bags. How many bags can she fill? Problem 27 in the Practice & Problem Solving portion of that same lesson furthers this engagement. “Find \frac{3}{4}  ÷ \frac{2}{3} . 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 \frac{1}{4} in$$^2$$ and the length as 10 \frac{1}{2}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$$\frac{1}{4}$$ pounds of trail mix into bags that each weight \frac{3}{8} pound. They bring \frac{2}{3} 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 - \frac{2}{3} = 25$$\frac{3}{4}$$ 7. \frac{f}{2}= \frac{5}{8} 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 \frac{7}{8} 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 \frac{1}{2} 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 \frac{7}{8} 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).

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

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 V = l w h and V = b h 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).

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).

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.”