7th Grade - Gateway 1

The materials reviewed for enVision Mathematics Grade 7 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 1, Assessment Form A, Problem 2, “Four out of nine dogs weigh less than 20 pounds. What is the decimal equivalent for the number of dogs weighing under 20 pounds? A) 0.$$\overline{2}$$ B) 0.24 C) 0.$$\overline{4}$$ D) 0.49. (7.NS.2d)

• Topic 2, Assessment Form A, Problem 3, “The graph shows how many bottles a machine fills in a certain number of seconds. Part A: What is the constant of proportionality, and what does it mean in this situation? Part B: Choose one ordered pair on the graph. What does it represent in this situation?” (7.RP.2b and 7.RP.2d)

• Topics 1 - 4, Cumulative/Benchmark Assessment, Problem 20, “The temperature of chicken soup is 192.7° F. As it cools, the temperature of the soup decreases 2.3° F per minute. Part A What is the temperature in degrees Fahrenheit of the soup after 25 minutes? Part B How many minutes will it take for the soup to cool to 100.7° F?” (7.EE.3)

• Topics 1 - 8, Progress Monitoring Assessment: Form C, Problem 14, “Of all the seventh graders, 60% bought a school lunch yesterday. Ten trials of a simulation are conducted and the data are recorded below. 52461, 65709, 58324, 06381, 94381, 84947, 23046, 33789, 57802, 70633 The numbers 0 through 6 represent students who bought a school lunch yesterday and the numbers 7 through 9 represent students who did not. Based on the simulated data, what is the probability that 3 or more of a group of 5 students randomly selected will buy the same school lunch the time it is offered?” (7.SP.8c)

The materials reviewed for enVision Mathematics Grade 7 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 extend their knowledge of positive and negative rational numbers to adding and subtracting with rational numbers and apply their knowledge to solve real-world problems, “Malik hikes Castle Trail from point A to point B. The elevation at Point A is below sea level. What are the possible beginning and ending elevations of Malik’s hike?” They are given an image of a hiking trail with a starting point below sea level and ending on top of a mountain. In the Practice & Problem Solving, Problem 11, students simplify three expressions using the same numbers with different signs, “a. 50 \frac{1}{2} + (-12.3) b. -50 \frac{1}{2} + (-12.3)  c. -50 \frac{1}{2} + 12.3” and in Problem 16 students develop an addition expression from a horizontal number line diagram, “Write an addition expression that is represented by the number line.” Students engage in extensive work with grade-level problems to meet the full intent of 7.NS.1 (Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical line diagram).

• Topic 4 engages students in generating equivalent expressions that can allow for easier interpretation in context. In Lesson 4-4, Practice & Problem Solving, Problem 15, students use the distributive property to rewrite an expression after applying a discount to the purchase of two items, “A grocery store has a 13%-off sale on all bread. You decide to purchase 6 loaves of bread. Let b be the original price of a loaf of bread. Expand the expression 6(b - 0.13b). Once the expression is expanded, what do the terms represent?” Lesson 4-5, Do You Know How?, Problem 3, students find common factors of linear expressions using the distributive property and recognize factoring is the opposite of expanding expressions, “Sahil is putting together supply kits and has 36 packs of x pencils, 12 packs of y crayons, and 24 erasers. a. Write an expression to show the total number of items. b. Use factoring to show how many kits Sahil can make while putting every type of item in each kit. c. Use the factored expression to find the number of each item in each kit.” Lesson 4-6, extends this engagement to include adding expressions. In Practice & Problem Solving, Problem 13, students find the perimeter of a triangle for a mural, “An art class is making a mural for the school that has a triangle drawn in the middle. The length of the bottom of the triangle is x. Another side is 1 more than three times the length of the bottom of the triangle. The last side is 2 more than the bottom of the triangle. Write and simplify an expression for the perimeter of the triangle.” Students engage in extensive work with grade-level problems to meet the full intent of 7.EE.2 (Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related).

• In Topic 6, students compare sets of data using measures of center and variability. In Lesson 6-3, Practice & Problem Solving, Problem 10, students are given two box plots illustrating the average high temperatures of two cities from January to December. “The box plots show the daily average high temperatures of two cities from January to December. Which city should you live in if you want a greater variability in temperature? Explain.” In Lesson 6-4, Practice & Problem Solving, Problem 7, students compare the vertical leap heights of basketball players, “The dot plot shows a random sample of vertical leap heights of basketball players in two different basketball camps. Compare the mean values of the dot plots. Round to the nearest tenth. The mean values tell you that participants in Camp __ jump higher in general.” In Problem 11, students make comparative inverses about two populations of fish, “The dot plots show the weights of a random sample of fish from two lakes. Which comparative inference about the fish in the two lakes is most likely correct? (A) There is about the same variation in weight between small and large fish in both lakes. (B) There is less variation in weights between small and large fish in South Lake than between small and large fish in Round Lake. (C) There is less variation in weight between small and large fish in Round Lake than between small and large fish in South Lake. (D) There is greater variability in the weights of fish in Round Lake” Two dot plots are provided one labeled “Sample from Round Lake” and the other labeled “Sample from South Lake”. Students engage in extensive work with grade-level problems to meet the full intent of 7.SP.4 (Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations).

• In Topic 8, Lessons 8-5 and 8-6 engage students in using the formulas for the circumference and area of a circle to solve problems. In Lesson 8-5, Practice & Problem Solving, Problem 16 students work backwards from the formula for circumference to find the diameter. “A unicycle wheel makes five rotations. The unicycle travels 37.94 feet. Find the diameter of the wheel in inches. Use 3.14 for \pi. Round to the nearest tenth of an inch.” In Problem 19, students find the perimeter of a rectangle with a semicircle on each end. “The diagram shows a track composed of a rectangle with a semicircle on each end. The area of the rectangle is 7,200 square meters. What is the perimeter, in meters, of the track? Use 3.14 for \pi.” In Lesson 8-6, Practice & Problem Solving, Problem 12, students find the area of a circular sidewalk, “A circular flower bed is 20 meters in diameter and has a circular sidewalk around it that is 3 meters wide. Find the area of the sidewalk in square meters. Use 3.14 for \pi. Round to the nearest whole number.” In Problem 18, students use the circumference to find the area of a hubcap. “The circumference of a hubcap of a tire is 81.58 centimeters. Find the area, in square centimeters, of this hubcap. Use 3.14 as an approximation for \pi. Round your answer to the nearest whole centimeter.” Students engage in extensive work with grade-level problems to meet the full intent of 7.G.4 (Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and the area of a circle).

The materials reviewed for enVision Mathematics Grade 7 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 5.5 out of 8, which is approximately 69%.

• 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 67 out of 89, which is approximately 75%.

• The number of days devoted to major work (including assessments and supporting work connected to the major work) is 144 out of 186, which is approximately 77%.

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 75% of the instructional materials focus on major work of the grade.

The materials reviewed for enVision Mathematics Grade 7 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 6, Lesson 6-2, Do You Know How?, Problem 6, students use proportional relationships and equations to make predictions based on given data, “In the dot plot above, 3 of 20 players made all 5 baskets. Based on this data, how many players out of 300 players will make all 5 baskets.” This connects the supporting work of 7.SP.2 (Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.) to the major work of 7.RP.2 (Recognize and represent proportional relationships between quantities). 

• In Topic 7, Lesson 7-4, Practice & Problem Solving, Problem 9, students find ratios and percentages of the experimental probability. “An arts and crafts store has a crate that contains glass, wood, and brass beads. Friends take turns choosing a bead without looking, recording the bead type, and returning the bead to the crate. The table shows the results of 300 selections. a. Write a probability model for choosing a bead. b. Based on the frequencies in the table, estimate the number of each type of bead that will be chosen if the friends select a total of 450 beads from the crate.” This connects the supporting work of 7.SP.C (Investigate chance processes and develop, use, and evaluate probability models) to the major work of 7.RP.A (Analyze proportional relationships and use them to solve real-world and mathematical problems).

• In Topic 8, Lesson 8-4, Practice & Problem Solving, Problem 9, students find the measure of angles using angle relationships and recognize the relationship between different angles formed by intersecting lines and rays. “Find the value of x” An image of two intersecting lines and an additional ray creating complementary, supplementary, and vertical angle pairs is provided. This connects the supporting work of 7.G.5 (Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure) to the major work of 7.EE.3 (Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form, using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies).

• In Topic 8, Lesson 8-6, Lesson 8-6 Quiz, Problem 1, students use an understanding of operations on fractions to solve a problem involving circles. “A cake has a circumference of 25 \frac{1}{7} inches. What is the area of the cake? Use \frac{22}{7} for \pi. Round to the nearest hundredth.” This connects the supporting work of 7.G.4 (Know the formulas for the area and circumference of a circle and use them to solve problems…) to the major work of 7.NS.2 (Apply and extend previous understandings of multiplication and division of fractions to multiply and divide rational numbers).

The materials reviewed for enVision Mathematics Grade 7 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 4, Lesson 4-7, Practice & Problem Solving, Problem 19, students rewrite an expression without parentheses. “Use the expression \frac{1}{4}p-(1-$$\frac{1}{3}$$p). a. Rewrite the expression without parentheses. Simplify. Show your work. b. Use a different method to write the expression without parentheses. Do not simplify.” This connects the major work of 7.NS.A (Apply and extend previous understandings of operations with fractions) to the major work of 7.EE.A (Use properties of operations to generate equivalent expressions).

• In Topic 5, Lesson 5-1, Practice & Problem Solving, Problem 7, students create a linear equation from an initial cost and unit rate. “A farmer ships oranges in wooden crates. Suppose each orange weighs the same amount. The total weight of the crate with g oranges is 24.5 pounds. Write an equation that represents the relationship between the weight of the crate and the number of oranges it contains.” Students are provided a diagram that shows the weight of the empty crate is 15 pounds and the weight of a single orange is 0.38 pounds. This connects the major work of 7.NS.A (Apply and extend previous understandings of operations with fractions) to the major work of 7.EE.B (Solve real-life and mathematical problems using numerical and algebraic expressions and equations).

• In Topic 7, Lesson 7-7, Practice & Problem Solving, Problem 13, students calculate and analyze the probability that a person surveyed would work for a small business. “About 50% of the people surveyed in a certain county work for a small business. A random number generator was used to simulate the results of the next four people surveyed. The numbers 0 to 4 represent people who work for a small business, and the numbers 5 - 9 represent people who do not work for a small business. [there is an array of 20 4-digit numbers] Part A Based on the simulated results shown above, what is the probability that at least one of the next four people surveyed works for a small business? Part B How would the design of the simulation changed if the percent of people who work for a small business was 70%?” This connects the supporting work from 7.SP.A (Use random sampling to draw inferences about a population) to the supporting work from 7.SP.C (Investigate chance processes and develop, use, and evaluate probability models).

• In Topic 8, Lesson 8-3, Practice & Problem Solving, Problem 17, students are given three characteristics for two different triangles and are tasked with drawing two triangles that fit those conditions. “Two triangles have side lengths of 12 units and 15 units and the non-included angle of 45^o. Draw two different triangles with these conditions.” This example connects the supporting work of 7.G.A (Draw construct, and describe geometrical figures and describe the relationships between them) to the supporting work of 7.G.B (Solve real-life and mathematical problems involving angle measure, area, surface area, and volume).

The materials reviewed for enVision Mathematics Grade 7 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?... Rational Numbers  In Lesson 1-2, students convert fractions to decimals and learn how to classify decimals as either terminating or repeating. In Lesson 1-5, students review the different outcomes that result from adding or subtracting rational numbers with different signs. In Lesson 1-7, students solve problems by multiplying rational numbers. In Lesson 1-9, students extend their knowledge of multiplication with rational numbers to solve problems involving division with rational numbers. In Lesson 1-10, students use properties of operations to solve problems involving rational numbers.” Looking Ahead, “How does Topic 1 connect to what students will learn later?... Grade 8 Equations In Grade 8, students will continue to use operations with positive and negative integers and rational numbers to solve equations by using inverse relationships and operation rules. Systems of Equations In Grade 8, students will use operations with positive and negative integers and rational numbers to find solutions for a system of linear equations using algebraic methods. Radicals In Grade 8, students will begin to examine the use of square and cube roots and will discover that square roots of negative numbers cannot be found in the real number system.” 

• Topic 2, Topic Overview, Math Background Coherence, “Topic 2 How is content connected within Topic 2?... Proportional Relationships In Lesson 2-3, students use equivalent ratios to determine whether relationships are proportional. They write and solve proportions to answer questions about situations involving proportional relationships. In Lesson 2-4, they use the constant of proportionality to write equations that describe proportional relationships. In Lesson 2-5, students graph proportional relationships. In Lesson 2-6, they think about how quantities are related and make decisions about using proportional reasoning in problem-solving contexts.” Looking Ahead, “How does Topic 2 connect to what students will learn later?... Grade 8 Proportional Relationships In Grade 8 students will understand the connections among proportional relationships, lines, and linear equations. Students will also graph proportional relationships and compare proportional relationships represented in different ways.”

• Topic 5, Topic Overview, Math Background Coherence, “Topic 5 How is content connected within Topic 5?... Evaluate Models To incorporate different perspectives and ensure conceptual understanding, students are asked to work with tables, graphs, bar diagrams, and number lines throughout the topic. The goal is to demonstrate how the models can be used to help write, solve, and check students’ work as they write and solve equations and inequalities.” Looking Ahead, “How does Topic 5 connect to what students will learn later?... Grade 8 Evaluate Models In Grade 8, students continue to make connections between models and equations including proportional relationships, lines, and linear equations.” 

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 3, Lesson 3-2, Lesson Overview, Coherence, students, “represent percent problems using proportions” and “use bar diagrams and proportions to solve percent problems.” In Grade 6, students, “expressed ratios as part-to-part or as part-to-whole” and “used ratios to solve problems.”

• In Topic 5, Lesson 5-4, Lesson Overview, Coherence, students, “solve inequalities using the Addition and Subtraction Properties of Inequality” and “represent the solution sets of inequalities on number lines.” In Grade 6, students, “wrote inequalities of the form x > c or x < c to represent a real-world situation” and “recognized that inequalities have infinitely many solutions.”

• In Topic 6, Lesson 6-1, Lesson Overview, Coherence, students, “learn to differentiate between a population and a sample” and “learn how to generate random, representative samples.”  In Grade 6, students, “learned that statistical questions include, and account for variability in the data as part of the answers.”