5th Grade - Gateway 1

The materials reviewed for enVision Mathematics Grade 5 meet expectations for assessing grade-level content and, if applicable, content from earlier grades. Probability, statistical distributions, similarities, transformations, and congruence do not appear in the assessments.

The series is divided into topics that include a Topic Assessment, available for online and/or paper and pencil delivery, and a Topic Performance Task. Additional assessments include a Grade 5 Readiness Test; Basic-Facts Timed Tests; four Cumulative/Benchmark Assessments addressing Topics 1–4, 1–8, 1–12, and 1–16; and Progress Monitoring Assessments A–C. Assessments can be found in the digital teacher interface and the Assessment Sourcebook online or in print. The materials include an ExamView Test Generator allowing teachers to build customized tests.

Examples of items that assess grade-level content include:

• Topic 2, Assessment, Problem 9, “Ricardo bought a pair of shoes for $55.60 and a hat for $9.78. How much did he spend in all? If he paid with 4 twenty-dollar bills, how much change did he get?” (5.NBT.7)

• Topic 8, Assessment, Problem 7, “Katsuro ran 3\frac{1}{6} miles each day for a week. How far did he run in all? Give an estimate, and then find the actual amount. Show your work.” (5.NF.4)

• Topic 14, Performance Task, Problem 1, “Mrs. Serrano is organizing a treasure hunt for her math class. She has started a treasure map of the field outside of the school, with the school at the origin. 1. So far Mrs. Serrano has decided on locations for a ‘+ 10’ certificate that can used on a quiz grade and a ‘100’ certificate for a homework assignment. Part A What ordered pair represents the location of the ‘+ 10’ certificate? Explain how you know. Part B Mrs. Serrano is planning to put a Tardy Pass at (9, 10). Graph this point on the coordinate grid and label it T. Explain how you located the point (9,10) in the coordinate grid, using the words origin, x-axis, x-coordinate, y-axis, and y-coordinate.” Students are provided with the first quadrant of the coordinate plane with the x-axis labeled yards going east from 0 to 10 and the y-axis labeled yards going north from 0 to 10. On the coordinate plane school, + 10, and 100 are labeled. (5.G.2) 

• Topics 1–16, Cumulative/Benchmark Assessment, Problem 6, “Maylin is mailing a package that has the size shown below. What is the volume of the package? Write the expression you used to find the volume.” Students are provided a picture of a cube that has a length of a side and the area of the base given. (5.MD.5)

The materials reviewed for enVision Mathematics Grade 5 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards.

The materials attend to the full intent of the grade-level standards by giving all students extensive work with grade-level problems. All Topics include a topic project, and every other topic incorporates a 3-Act Mathematical Modeling Task. During the Solve and Share, Visual Learning Bridge, and Convince Me!, students explore ways to solve problems using multiple representations and prompts to reason and explain their thinking. Guided Practice provides students the opportunity to solve problems and check for understanding. During Independent Practice, students work with problems in various formats to integrate and extend concepts and skills. The Problem Solving section includes additional practice problems for each of the lessons. Examples of extensive work with grade-level problems to meet the full intent of grade-level standards include:

• In Topic 7, Lessons 7-1, 7-3, and 7-6, students engage in extensive work with grade-level problems to meet the full intent of 5.NF.2 (Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers). In Lesson 7-1, Solve & Share, students use visual models of fractions with unlike denominators to estimate a sum. “Jack needs about 1\frac{1}{2} yards of string. He has three pieces of string that are different lengths. Without finding the exact amount, which two pieces should he choose to get closest to 1\frac{1}{2} yards of string? Solve this problem any way you choose.” Pictured are three different colored balls of yarn with pieces of length \frac{1}{2} yard, \frac{1}{3} yard, and \frac{7}{8} yard. In Lesson 7-3, Convince Me!, students use number sense to analyze the information given in the problem to explain why equivalent fractions that use different numbers in the numerator and denominator can have the same value. The stimulus states, “ Alex rode his scooter from his house to the park. Later, he rode from the park to baseball practice. How far did Alex ride?” Pictured is a map with three landmarks—a baseball field, a park, and a house—and two paths labeled with the lengths \frac{1}{3} mile and \frac{1}{2} mile. Students respond to the prompt, “In the example above, would you get the same sum if you used 12 as the common denominator? Explain.” In Lesson 7-6, Assessment Practice, Problem 26, students solve a word problem by estimating the difference between two mixed numbers with unlike denominators. “Annie has  13\frac{1}{12} yards of string. She uses 1\frac{9}{10} yards to fix her backpack. About how much string does she have left?” Students choose from the following choices: (A) 11 yards, (B) 12 yards, (C) 14 yards, and (D) 15 yards. 

• In Topic 11, Lessons 11-2, 11-3 and 11-5, students engage in extensive work with grade-level problems to meet the full intent of 5.MD.4 (Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units). In Topic 11, Today’s Challenge, Problem 6, students explain how to find the volume of a crate, including how to use the formula for the volume of a rectangular prism. Pictured is a placard displaying the dimensions of a crate in inches and centimeters. In Lesson 11-2, Daily Review, Problems 1 and 3, students engage with unit cubes to model a rectangular prism with given dimensions and to find the volume of a complex solid. In Problem 1, students determine how many unit cubes are needed to model a rectangular prism that is “6 units long × 4 units wide × 2 units high.” In Problem 3, students count the unit cubes to find the volume of a complex solid. Pictured is a complex solid consisting of two rectangular prisms. In Independent Practice, Problems 7, 8, and 10, students incorporate square measurements into volume calculations. Pictured are rectangular prisms with a shaded base; base area measurements are given in m², cm², and ft², respectively. Throughout Lesson 11-3, students calculate volume of complex solids. In Guided Practice, Problem 4, students find the volume of a solid that can be separated in more than one way. Given side lengths are in centimeters. 

• In Topic 13, Lessons 13-1 and 13-4 , students engage in extensive work with grade-level problems to meet the full intent of 5.OA.1 (Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols). In Lesson 13-1, Solve & Share, given two choices, students determine which answer is correct for an expression. “Jordan and Annika are working on 15 + 12 ÷ 3 + 5. Jordan says the answer is 14 and Annika says the answer is 24. Who is right?” In Independent Practice, Problems 10-21, students evaluate numerical expressions that include zero to three pairs of grouping symbols; for example, “14. 2 + [4 + (5 ✕ 6)].” In Lesson 13-4, Problem Solving, Performance Task, Problems 9 and 10, students reason about real-world calculations that require grouping symbols. “Math Supplies Ms. Kim is ordering sets of place-value blocks for the 3rd, 4th, and 5th graders. She wants one set for each student, and there are 6 sets of blocks in a carton. How many cartons should Ms. Kim order?” Pictured is a data table that indicates the number of students in each grades 3-6. “9. Model with Math Write an expression to represent the number of cartons Ms. Kim needs to order. You can use a diagram to help. 10. Construct Arguments Did you use grouping symbols in your expression? If so, explain why they are needed.” In Reteach to Build Understanding, Problem 6, students write a numerical expression that includes grouping symbols for calculations. “ A large elementary school has 4 fifth-grade classes and 3 fourth-grade classes. The fifth-grade classes have 28, 29, 30, and 31 students. The fourth-grade classes have 27, 28, and 29 students. Write a numerical expression to find how many more fifth graders there are than fourth graders.” 

• In Topic 14, Lessons 14-2 - 14-4, students engage in extensive work with grade-level problems to meet the full intent of 5.G.2 (Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation). In Lesson 14-2, Problem Solving, Problem 21, students explain how to find the horizontal distance between two points. “Higher Order Thinking Point C is located at (10, 3) and Point D is located at (4, 3). What is the horizontal distance between the two points? Explain.” In Lesson 14-3, Problem Solving, Problems 7 and 8, students graph points on a coordinate grid, draw a line to show the pattern, and then extend the pattern to solve.  Pictured are a reading log and a grid with labeled intervals and axes. Directions, “In 7 and 8, use the table at the right. 7. Graph the points in the table on the grid at the right. Then draw a line through the points. 8. Look for Relationships If the pattern continues, how many pages will have been read after 6 hours? Extend your graph to solve.” In Lesson 14-3, Assessment Practice, Problems 11 and 12, students indicate what points on a graph represent. The graph, titled “A Crawling Ant,” shows total distance (m) for corresponding values of time (sec). “11. What does the point (15, 4) represent on the graph at the right? 12. What does the point (20, 5) represent on the graph?” In Lesson 14-4, Daily Review, Problem 5, students state what a point represents within a scenario. “Tyrese has finished two paintings. He decides to paint 1 picture every day. The graph shows the relationship between the number of days and the number of paintings Tyrese has painted. 5. What does the point (5, 7) represent?”

The materials reviewed for enVision Mathematics Grade 5 meet expectations for that, when implemented as designed the majority of the materials address the major clusters of each grade. The materials devote at least 65% of instructional time to the major clusters of the grade.

• The approximate number of Topics devoted to major work of the grade (including assessments and supporting work connected to the major work) is 12 out of 16, which is approximately 75%.

• The number of lessons devoted to major work of the grade (including assessments and supporting work connected to the major work) is 92 out of 108, which is approximately 85%.

• The number of days devoted to major work (including assessments and supporting work connected to the major work) is 122 out of 148, which is approximately 82%. 

A lesson-level analysis is most representative of the materials since the lessons include major work, supporting work connected to major work, and assessments embedded within each topic. As a result, approximately 85% of the materials focus on the major work of the grade.

The materials reviewed for enVision Mathematics Grade 5 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. These connections are listed for teachers within the Teacher’s Edition, Lesson Overview, Coherence, Cross-Cluster Connections on a document titled “Lessons and Standards” found within the Course Guide tab for each unit. Connections are also listed in a document titled “Scope and Sequence.” Examples of connections include:

• Topic 10, Lesson 10-2 connects the supporting work of 5.MD.2 (Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots.) to the major work of 5.NF.2 (Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers). In Practice Buddy: Additional Practice, Problems 1 and 2, students select a line plot from amongst four options to represent a set of data and perform the calculation 3-2\frac{1}{8} . “1. John and his mother measure the lengths of fabric scraps in the sewing room. Make a line plot of their data. 2. What is the difference between the longest and shortest fabric​ scrap?”

• Topic 12, Lesson 12-6 connects the supporting work of 5.MD.1 (Convert among different-sized standard measurement units within a given measurement system, and use these conversions in solving multi-step, real world problems.) to the major work of 5.NBT.A (Understand the place value system). In Enrichment, Problem 4, students convert between standard measurement units in solving a multi-step, real-world problem. “Mr. Black bought three 2-kilogram jars of peanut butter for school snacks. He spread the peanut butter on bagels to feed 60 students. How many grams of peanut butter did he use for each bagel?

• Topic 13, Lesson 13-3 connects the supporting work of 5.OA.A (Write and interpret numerical expressions) to the major work of 5.NBT.B (Perform operations with multi-digit whole numbers and with decimals to hundredths). In Practice Buddy: Additional Practice, Problem 11, students select an expression that correctly records calculations with numbers. “Tyler bought a​ television, a game​ console, and a controller. Tyler used a ​$50 coupon to make the purchase. He wrote (1,235.00 + 332.50 + 38.95) − 50 to show how he can calculate the final​ cost, not including sales tax. Write an expression that can be used to find the total price of the items he bought before sales tax and the coupon. Choose the correct answer below. A. 1,235.00 + 332.50 + 50 B. 1,235.00 + 332.50 + 38.95 C. 1,235.00 + 38.95 + 50 D. 50 + 332.50 + 38.95”

The materials reviewed for enVision Mathematics Grade 5 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.

There are connections from supporting work to supporting work and major work to major work throughout the grade-level materials, when appropriate. These connections are listed for teachers in the Topic Overview, Scope and Sequence, and Teacher Guides within each topic. Examples include:

• In Topic 4, Lesson 4-1, Reteach to Build Understanding, Problem 3, students apply their understanding of the place value system to multiply decimals by powers of 10. “3. Use patterns to find the products.” The materials prompt for sequences of products: 3.15 times multiples of 10 from 1 to 1,000 represented in standard form and exponential form. In Topic 6, Lesson 6-1, Problem Solving, Problems 26–28, students use data from three events to subtract and multiply numbers to the hundredths place. “For 26–28, use the table that shows the winning times at the Pacific Middle School swim meet. 27. The winning time for the 100-yard freestyle was twice the time for the 50-yard freestyle. What was the winning time for the 100-yard freestyle? 28. What was the difference between the winning 100-yard freestyle time and the winning butterfly time?” The materials show a table that indicates the times for the 50-yard freestyle, 100-yard backstroke, and 100-yard butterfly. This connects the major work of 5.NBT.A (Understand the place value system) to the major work of 5.NBT.B (Perform operations with multi-digit whole numbers and with decimals to hundredths). 

• In Topic 10, Lesson 10-3, Independent Practice, Problems 5 and 6, students multiply, add, and subtract fractions (halves, quarters, and eighths). “In 5 and 6, use the line plot Allie made to show the lengths of string she cut for her art project. 5. Write an equation for the total amount of string. 6. What is the difference in length between the longest and the shortest lengths of string?” The materials show a line plot representing a collection of Lengths of Strings from 12\frac{1}{2} inches to 13\frac{1}{8} inches. This connects the major work of 5.NF.A (Use equivalent fractions as a strategy to add and subtract fractions) to the major work of 5.NF.B (Apply and extend previous understanding of multiplication and division to multiply and divide fractions). 

• In Topic 11, Lesson 11-4, Solve & Share, students solve problems involving volumes of rectangular prisms. “A school has two wings, each of which is a rectangular prism. The school district is planning to install air conditioning in the school and needs to know its volume. What is the volume of the school? Solve this problem any way you choose.” The materials show a 3D model of a school with dimensions 50m \times 50m \times 10m and 75m \times 57m \times 14m. This connects the major work of 5.NBT.5 (Fluently multiply multi-digit whole numbers using the standard algorithm) to the major work of 5.MD.C (Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition). 

• In Topic 15, Lesson 15-3, Independent Practice, Problems 5–8, students use the verbal rule ”add 4.” Problem 5 states, “Megan and Scott go fishing while at camp. Megan catches 3 fish in the first hour and 4 fish each hour after that. Scott catches 5 fish in the first hour and 4 fish each hour after that. Complete the table to show the total number of fish each has caught after each hour.” The materials show a table entitled, “Total Fish Caught.” There are three columns indicating Hours, Megan, Scott; the first row is complete “1, 3, 5.” Students complete the fish caught by Megan and Scott for 2, 3, and 4 hours. “6. What ordered pair represents the total number of fish they each caught after 4 hours? … 8. Graph the ordered pairs of the total number of fish each has caught after each hour.” This connects the supporting work of 5.OA.B (Analyze patterns and relationships) to the supporting work of 5.G.A (Graph points on the coordinate plane to solve real-world and mathematical problems).

The materials reviewed for enVision Mathematics Grade 5 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. 

Prior and Future connections are identified within the Teacher Edition Math Background: Focus, Math Background: Coherence, and Lesson Overview. Examples of connections to future grades include:

• Topic 3, Lessons 3-3 - 3-7 connect 5.NBT.5 (Fluently multiply multi-digit whole numbers using the standard algorithm) to the work of future grades. In Lessons 3-3 through 3-7, students “carefully develop understanding and fluency with using the standard multiplication algorithm for whole numbers. Each lesson builds on the previous one, starting with multiplying 1-digit and 2-digit numbers and continuing to multiplying 3-digit by 2-digit numbers.” In Grade 6, “students will be expected to fluently multiply decimals using the standard algorithm.”

• Topic 10, Lessons 10-2 and 10-3 connect 5.MD.2 (Make a line plot to display a data set of measurements in fractions of a unit. Use operations on fractions for this grade to solve problems involving information presented in line plots) to the work of future grades. In Lesson 10-2, students “organize and display data in a line plot.” In Lesson 10-3, students “solve problems using data in a line plot.” (Data consists of whole numbers, fractions, and mixed numbers.) In Grade 6, “students will use line plots (also called dot plots) to display data.  Students will use these data displays to interpret data distributions.”

• Topic 12 connects 5.MD.1 (Convert among different-sized standard measurement units within a given measurement system, and use these conversions in solving multi-step, real world problems) to the work of future grades. In Topic 12, students use “multiplication and division to convert measurements of length, capacity, weight, and mass within either the customary or metric measurement system, on converting units of time, and on solving problems involving measurement conversions.” In Grade 6, “students will use ratio reasoning to convert units. Using ratio reasoning draws on procedures students have learned for multiplying and dividing with fractions.”

Examples of connections to prior knowledge include:

• Topic 5 connects 5.NBT.6 (Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models) to the work of previous grades. In Grade 4, students “used strategies and properties to divide whole numbers, with 1-digit divisors.” In this topic, students learn different approaches “to estimate quotients and compute quotients of whole numbers with 2-digit divisors … All of the strategies involve breaking apart the dividend and using the Distributive Property.”

• Topic 8, Lessons 8-1 – 8-2 connects 5.NF.4 (Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction) to the work of previous grades. In Grade 4, Topic 9, “students learned to add fractions with common denominators. Students also learned how to decompose fractions and use this skill to change between improper fractions and mixed numbers.” In these lessons, students “use models to develop conceptual understanding of multiplying a fraction by a whole number and a whole number by a fraction.”

• Topic 15 connects 5.OA.3 (Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane.) to the work of previous grades. In Grade 4, “students generated a single number or shape pattern that followed a given rule. They identified features of the pattern that were not apparent in the rule itself.” In this topic, “students use rules to extend two patterns and look for a relationship between corresponding terms.” In addition, students “use graphs to represent the patterns, to help them find relationships, and to solve problems.”