4th Grade - Gateway 1

The materials reviewed for enVision Mathematics Grade 4 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 4 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 7, Assessment, Problem 13, “Jared says all even numbers less than 20 are composite. Find an even number less than 20 that is NOT composite. Explain why the number is not composite.” (4.OA.4)

• Topic 9, Online Assessment, Problem 1, “For each row of the table, choose the equivalent expression.” The materials provide students with a table showing \frac{5}{12}+\frac{4}{12}, (\frac{1}{12}+\frac{3}{12})+\frac{3}{12}), \frac{1}{12}+\frac{2}{12}+\frac{3}{12}, and \frac{15}{12}-\frac{7}{12} as the columns and \frac{4}{12}+\frac{4}{12}, \frac{1}{12}+(\frac{3}{12}+\frac{3}{12}), \frac{4}{12}+\frac{5}{12}, and \frac{2}{12}+\frac{2}{12}+\frac{2}{12} as the rows. (4.NF.1)

• Topic 12, Performance Task, Problem 2, “Analyze the amount of money that the students raised. Use the Walking to Raise Money for Animals table. Part A How much more money did Yuna raise than Ali? Draw bills and coins to show your work. Part B Hayley got the same amount of money for each mile she walked. How much did she get for each mile? Draw bills and coins to show your work.” (4.MD.2)

• Topics 1–4, Online Cumulative/Benchmark Assessment, Problem 14, “Which of the following shows how to find 3 \times 819? Which property was used? A) 819 \times 3 = 2,457; Associative Property B) 3 \times (8 + 1 + 9) = 3 \times 17 = 51; Commutative Property C) 3 \times (800 + 10 + 9) = 2,400 + 30 + 27 = 2,457; Distributive Property D) 3 \times (800 - 90 - 1) = 2,400 - 270 - 3 =2,127; Distributive Property ” (4.NBT.5)

The materials reviewed for enVision Mathematics Grade 4 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 6, Lessons 6-3 and 6-5, students engage in extensive work with grade-level problems to meet the full intent of 4.OA.3 (Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding). In Lesson 6-3, Solve & Share, students use multiplication and subtraction to solve a multistep word problem. “Last year, 18 people went on a family camping trip. This year, three times as many people went. How many more people went this year than last year? Complete the bar diagram and show how you solve.” Pictured are labeled bar diagrams that represent the situation. Lesson 6-5, Convince Me!, students reason about the remainder answer for a division problem. “Does the answer of 11 rows make sense for the problem above? Explain.” In Independent Practice, Problem 3, students write equations to solve multi-step problems and use estimation to determine if their answer is reasonable. “Vanya bought 5 medium packages of buttons and 3 small packages of buttons. What was the total number of buttons Vanya bought?” Pictured is a table showing the number of beads and buttons in small, medium, and large packages. 

• In Topic 9, Lessons 9-1, 9-2, 9-4, and 9-5, students engage in extensive work with grade-level problems to meet the full intent of 4.NF.3 (Understand a fraction a/b with a > 1 as a sum of fractions 1/b). In Lesson 9-1, Independent Practice, Problem 8, students find the sum of fractions with common denominators. "\frac{1}{6}+\frac{2}{6}+\frac{3}{6} and provides the image of the fractions on a number line. In Lesson 9-2, Enrichment, Problem 2, students identify two pets whose votes add to \frac{8}{10}. Shown is a chart representing pet names and the corresponding fraction of student votes. In Lesson 9-4, Practice Buddy: Additional Practice, Problem 2, students use fraction strips to find the difference "\frac{9}{10}-\frac{4}{10}”  In Lesson 9-5, Problem Solving, Problem 20, students explain the similarities of subtracting fractions and subtraction. “Explain how subtracting \frac{4}{5}-\frac{3}{5} involves subtracting 4-3.”

• In Topic 11, Lessons 11-1 - 11-3, students engage in extensive work with grade-level problems to meet the full intent of 4.MD.4 (Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots). In Lesson 11-1, Solve & Share, students use a line plot to answer word problem questions. “Emily went fishing. She plotted the lengths of 12 fish caught on the line plot shown. What was the length of the longest fish caught? What was the length of the shortest fish caught? Solve this problem any way you choose.” Provided is a line plot that displays fish lengths in \frac{1}{4}-increments. In Guided Practice, Problems 3-5, students interpret a line plot that displays giraffe heights in \frac{1}{2}-increments. “3. How many giraffes are 14 feet tall? 4. What is the most common height? 5. How tall is the tallest giraffe?” In Lesson 11-2, Problem Solving, Problems 8 and 9, students make a line plot to display daily distances in \frac{1}{8}-increments and write a question that requires addition or subtraction of the given data. “8. Trisha measured how far her snail moved each day for 5 days. Make a line plot of Trisha’s data. 9. Higher Order Thinking Write a question that would require addition or subtraction to solve using Trisha’s data. What is the answer?” In Lesson 11-3, Reteach to Build Understanding, Problems 3 and 4, students use information presented in a line plot to solve problems involving addition and subtraction. “Audra measured the rainfall in her town for 16 days. The line plot shows the amount of rainfall she recorded each day. Each dot represents one day that it rained. … 3. Find the total number of inches of rainfall Audra recorded. Show your work.  4. How many more days had 3 inches of rain than 1\frac{2}{4} inches of rain?” The materials display a line plot or rainfall data in \frac{1}{4}-inch increments.

• In Topic 16, Lessons 16-2 and 16-3, students engage in extensive work with grade-level problems to meet the full intent of 4.G.2 (Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles). In Lesson 16-2, Solve & Share, students sort 7 triangles into two or more groups according to a self-determined criteria. “Sort the triangles shown into two or more groups. Explain how you sorted them. Solve this problem any way you choose.” Pictured are triangles, each outlined in a different color and able to be categorized by side lengths or interior angle measure. In Assessment Practice, Problem 15, students categorize six triangles as acute, obtuse, or right. In Lesson 16-3, Visual Learning Bridge, students learn how to classify quadrilaterals according to their angles or the line segments that make their sides (one pair or two pairs of parallel sides). In Reteach to Build Understanding, Problem 2, students classify four quadrilaterals by completing sentences. “Figures ___ , ___ , and ___ have opposite sides that are parallel, so they are parallelograms. There are more specific names for two of these figures. Figure A has 4 right angles and 2 pairs of parallel sides. Opposite sides have the same length, so it is a ___ . Figure D has all sides the same length, so it is a ___ .”

The materials reviewed for enVision Mathematics Grade 4 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 13 out of 16, which is approximately 81%.

• The number of lessons devoted to major work of the grade (including assessments and supporting work connected to the major work) is 91 out of 104, which is approximately 88%.

• The number of days devoted to major work (including assessments and supporting work connected to the major work) is 120 out of 144, which is approximately 83%. 

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 88% of the materials focus on the major work of the grade.

The materials reviewed for enVision Mathematics Grade 4 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 7, Lesson 7-2 connects the supporting work of 4.OA.B (Gain familiarity with factors and multiples) to the major work of 4.NBT.B (Use place value understanding and properties of operations to perform multi-digit arithmetic). In Lesson 7-2, Independent Practice, Problems 7–12, students write the factor pairs for each number. For example:  “7. 34 ___ and 34, 2 and ___”  In Problems 13–21, students write the factors of each number, using counters to help if needed. 13. 6.” In Practice Buddy: Independent Practice; Problem Solving, Problem 11, students engage with a multiple-choice question whose answer choices include the concepts of prime, factor, and multiples. “Any number that has 8 as a factor also has 2 as a factor. Why is​ this? Choose the correct answer below. A. This is because 8 and 2 are multiples of each other. If two numbers can be written as multiples of each other, then they are factors of the same number. B. This is because 2 is a prime number., making it a factor of any other number. C. If 8 is a factor, then so is 2, because 2 is a factor of 8. D. If 2 is a factor, then so is 8, because 8 is a factor of 2.”

• Topic 10, Lesson 10-4 connects the supporting work of 4.MD.A (Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit) to the major work of 4.NF.B (Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers). In Guided Practice, Do You Know How?, Problems 2 and 3, students apply their understanding of time to answer questions. The prompt states, "For 2–3, solve. Remember there are 60 minutes in 1 hour and 7 days in 1 week. 2. How many minutes are in a school day of 7 hours 25 minutes? 3. How much is 3\frac{2}{4} weeks + 2\frac{3}{4}weeks?"

• Topic 15, Lesson 15-2 connects the supporting work of 4.MD.C (Geometric measurement: understand concepts of angle and measure angles) to the major work of 4.NF.B (Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers). In Problem Solving, enVision STEM, Problem 11, students find angle measures using fractional parts of a given whole. The prompt states, “A mirror can be used to reflect a beam of light at an angle. What fraction of a circle would the angle shown turn through?” A picture is provided of a mirror and lightbulb, and the light is reflected off the mirror at a 120° angle.

The materials reviewed for enVision Mathematics Grade 4 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 2, Lesson 2-2, Enrichment, students use strategies to solve problems presented in the context of a multi-step problem. “1. The family starts their trip on Monday morning at 9 a.m. They stop at noon for lunch. How many miles have they traveled? … 4. On the second day, the Bravo family begins driving at 7 a.m. That day, they stop for a total of 2 hours. Will they get to Washington, D.C., by 5 p.m.? Explain…” The materials show travel plans for the  family vacation (e.g., They plan to drive from Miami to New York City. Mr. Bravo thinks they can average driving 60 miles an hour) and a chart that shows the distance between relevant cities (e.g., The distance from Miami to Washington, D.C., is 1,043 miles and the distance from Washington, D.C., to New York City is 237 miles). This connects the major work of 4.OA.A (Use the four operations with whole numbers to solve problems) to the major work of 4.NBT.B (Use place value understanding and operations to perform multi-digit arithmetic).

• In Topic 8, Lesson 8-3, Assessment Practice, Problems 26 and 27, students find equivalent fractions using multiplication. “26. Select the equivalent fractions.” The materials show a chart: the first column lists the fractions \frac{4}{6}, \frac{2}{8}, \frac{8}{12}, and \frac{3}{12}; the second column prompts for “Fractions Equivalent to \frac{1}{4}”; and the third column prompts for “Fractions Equivalent to \frac{2}{3}”. “27. Nia found a fraction that is equivalent to \frac{3}{8}. Is Nia’s fraction work, shown below, correct? Explain.” The materials show \frac{3\times4}{8\times3}=\frac{12}{24} . This connects the major work of 4.NF.A (Extend understanding of fraction equivalence and ordering) to the major work of 4.NBT.B (Use place value understanding and properties of operations to perform multi-digit arithmetic.). 

• In Topic 13, Lesson 13-7, Guided Practice, students perform calculations that involve fractions / whole numbers and yards/inches in a real-world context. “Jeremy uses \frac{2}{3} yard of tape for each box he packs for shipping. How many inches of tape does Jeremy need to pack 3 boxes?” This connects the major work of 4.OA.A (Use the four operations with whole numbers to solve problems) to the major work of 4.NF.B (Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers).

• In Topic 15, Lesson 15-1, Problem Solving, Problems 16–20, students identify a segment, a point, and an angle and draw angles classified by their measure. The materials show a map of Nevada “For 16–18, use the map of Nevada. Write the geometric term that best fits each description. Draw an example. 16. Be Precise The route between 2 cities 17. The cities 18. The corner formed by the north and west borders”. The materials show a map of Nevada with city landmarks and a compass rose. “19. Vocabulary Write a definition for right angle. Draw a right angle. Give 3 examples of right angles in the classroom. 20. Higher Order Thinking Nina says she can make a right angle with an acute angle and an obtuse angle that have a common ray. Is Nina correct? Draw a picture and explain.” This connects the supporting work of 4.MD.C (Geometric measurement: understand concepts of angle and measure angles) to the supporting work of 4.G.A (Draw and identify lines and angles, and classify shapes by properties of their lines and angles). 

The materials reviewed for enVision Mathematics Grade 4 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 2, Lessons 2-3 – 2-6 connect 4.NBT.4 (Fluently add and subtract multi-digit whole numbers using the standard algorithm) to the work of future grades. In Lessons 2-3 through 2-6, students use place-value concepts and the standard algorithm to add and subtract whole numbers. In Grade 5, Topic 2, students will “use models and strategies to add and subtract decimals to hundredths.”

• Topic 5, Lesson 5-9 connects 4.NBT.6 (Find whole-number quotients and remainders with up to four-digit dividends and one-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 future grades. In Lesson 5-9, students choose a strategy they previously learned to divide and solve problems. Throughout this topic, “students apply strategies for whole-number division to solve real-world problems.” In Grade 5, Topic 5, students will “use strategies to divide whole numbers by 2-digit divisors.”

• Topic 10, Lessons 10-1 – 10-3 connect 4.NF.4 (Apply and extend previous understandings of multiplication to multiply a fraction by a whole number) to the work of future grades. In Lesson 10-1, “students come to understand that they can think of a fraction as the product of a unit fraction and a whole number. This foundation is incorporated in the development of strategies for multiplying a fraction by a whole number in Lessons 10-2 and 10-3.” In Grade 5, Topic 8, students will “multiply a whole number by a fraction, a fraction by a whole number and a fraction by a fraction.”

Examples of connections to prior knowledge include:

• Topic 6, Lessons 6-1 and 6-2 connects 4.OA.1 (Interpret a multiplication equation as a comparison. Represent verbal statements of multiplicative comparisons as multiplication equations) to the work of previous grades. In Grade 3, Topics 1–5, “students solved word problems involving basic facts and the foundational understanding of multiplication and division.” In Lessons 6-1 and 6-2, students “learn how to solve problems involving multiplicative comparison and additive comparison. They also learn to distinguish between these two types of comparison.”

• Topic 9, Lesson 9-6 connects 4.NF.3a (Understand addition and subtraction of fractions as joining and separating parts referring to the same whole) to the work in previous grades. In Grade 3, Topic 12, “students learned about the meaning of fractions and used various models to represent them.” In Topic 13, “students learned how to recognize and generate simple equivalent fractions and to express whole numbers as fractions.” In this lesson, students “add and subtract fractions with like denominators” and “extend these understandings to addition and subtraction of mixed numbers.”

• Topic 14, Lessons 14-1 – 14-3 connect 4.OA.5 (Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself) to the work of previous grades. In Grade 3, “students found and explained multiplication patterns in Topic 5 and addition patterns in Topic 8.” In Lesson 14-1, “students apply an addition or subtraction rule to generate one or more numbers in a number sequence. In Lesson 14-2, students extend patterns in tables, with a focus on multiplication or division rules. For both, they analyze the patterns and look for features not given in the rule. In Lesson 14-3, they extend repeating patterns that consist of either shapes or numbers. They use the given rule to predict the pattern. For example, they find the 100th shape in a repeating pattern of four shapes.”