4th Grade - Gateway 2

The instructional materials for enVision Mathematics Common Core Grade 4 meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

Each lesson is structured to include background information for the teacher and problems and questions that develop conceptual understanding. Examples include, but are not limited to:

• Conceptual understanding for each topic is outlined in the Teacher Edition’s section Math Background: Rigor. For example, the Topic 3 Overview explains using number sense to estimate, partial products, and the distributive property to multiply multi-digit numbers by 1-digit. The Conceptual Understanding section states, “Arrays and area models are used to foster understanding of partial products. The partial products can be added in any order and the result is the same. The distributive property is the basis for breaking apart a multi-digit factor by place value and generating partial product.” 
• The Teacher Edition contains a Rigor section for each lesson explaining how conceptual understanding is developed in the lesson. For example, Lesson 8-1, the Rigor section states, “Students use an area model to demonstrate that two fractions are equivalent when they name the same part of the same whole.”
• Each lesson is introduced with a video: Visual Learning Animation Plus, to promote conceptual understanding.  For example, the Lesson 8-1 video states, “What are some ways to name the same part of a whole?” The scenario begins by saying, “James ate part of the pizza shown in the picture. He said $$\frac{5}{6}$$ of the pizza is left. Cardell said $$\frac{10}{12}$$ of the pizza is left. Who is correct?” Students should see that $$\frac{5}{6}$$ and $$\frac{10}{12}$$ are equal.
• Each lesson begins with a Visual Learning Bridge activity that provides the opportunity for a classroom conversation to build conceptual understanding for students. In Lesson 8-1, teachers ask, “How much of the pizza is left according to James? According to Cardell?  What do you need to do? What does the denominator of the fraction tell you? What does the numerator tell you? How can you tell from the picture that $$\frac{5}{6}$$ of the pizza is left? Why is the first area model labeled $$\frac{5}{6}$$? Why is the second area model labeled $$\frac{10}{12}$$? Why are $$\frac{5}{6}$$ and $$\frac{10}{12}$$ equivalent?  Does it matter what shape is used to show each of the two fractions?”
• Each lesson contains a Convince Me! section that provides opportunities for conceptual understanding. In Lesson 6-4, students use bar diagrams to model the question, “Chef Angela needs 8 cartons of eggs to make the cakes that are ordered. She has 2 cartons of eggs and 4 single eggs in the refrigerator. How many more eggs does she need to make all of the cakes?”
• Each lesson contains a Do You Understand? section which makes a connection to previous learning and provides opportunities for conceptual understanding. In Lesson 8-2,  “Students use number lines to find equivalent fractions that represent the same point on the number line.” Students also explain why two fractions are equivalent.

Practice problems provide students opportunities to independently develop conceptual understanding. Examples include, but are not limited to:

• In Lesson 5-7, Questions 5-8, students use drawings of place-value models and the concept of division as equal shares to find 2-digit quotients with and without remainders. For example, Question 6 states, “___ = 176 ÷ ____.” (4.OA.3)
• In Lesson 9-7, Questions 4-11, students use fraction strips and number lines to extend work with fractions to include mixed numbers. Question 5 shows a number line which students use to find the sum of $$1\frac{2}{3}+2\frac{2}{3}$$. (4.NF.3c)
• In Topic 8, Performance Task, students are told, “During gym class, the fourth-grade students climbed on a rope hanging from the ceiling.  The Rope Climbing table shows what part of the rope several students climbed.” The table displays students and the part of the rope that they climbed using the fractions: $$\frac{4}{6}$$, $$\frac{1}{2}$$, $$\frac{1}{3}$$, $$\frac{2}{3}$$, $$\frac{5}{6}$$, $$\frac{4}{3}$$. Problem 1, Part A, states, “Compare how high the students climbed.  Who climbed a greater part of the rope, Gia or Jim? Use benchmark fractions to compare. Explain.” Part B, states, “Who climbed a greater part of the rope, Gia or Jason? Use the number line to compare.”  Part C states, “Who climbed a greater part of the rope, Rachel or Russ? Justify your comparison using fraction strips.” (4.NF.2)

The instructional materials for enVision Mathematics Common Core Grade 4 meet expectations that they attend to those standards that set an expectation of procedural skill and fluency.

Problem sets provide opportunities to practice procedural fluency. Regular opportunities for students to attend to Standard 4.NBT.4: add and subtract fluently within 1,000,000, are provided.

The instructional materials develop procedural skill and fluency throughout the grade-level. Examples include, but are not limited to:

• Each topic contains a Math Background: Rigor page with a section entitled “Procedural Skill and Fluency.”  In Topic 2, “Students use a variety of procedures for using mental math to add and subtract multi-digit whole numbers.  They also use rounding to estimate sums and differences. Students develop fluency in adding and subtracting multi-digit whole numbers using the standard algorithms. The standard addition and subtraction algorithms are shortened versions of strategies involving partial sums and partial differences. Students begin by lining up the numbers by place value. Then they add (or subtract) the ones, then the tens, then the hundreds, and so on, regrouping as necessary."
• Each lesson contains a Visual Learning Bridge which provides instruction on  procedural skills. Students make and interpret line plots. In Lesson 11-2, the Visual Learning Bridge states, “The manager of a shoe store kept track of the lengths of the shoes sold in a day. Complete the line plot using the data from the shoe store."
• Fluency Practice Activities are found at the end of Topics 2-16 to support adding and subtracting fluently within 1,000,000. In Topic 4, the Fluency Activity provides a numbered chart and states, “Work with a partner. Point to a clue. Read the clue. Look below the clues to find a match. Write the clue letter in the box next to the match.” For example, “The sum is greater than 300 and less than 400” matches “283 + 38."
• The Performance Task for Topic 2 provides students the opportunity to demonstrate fluency when adding and subtracting multi-digit whole numbers.  Students are given a table displaying items ordered in April and May and asked to complete tasks such as, “Use an algorithm to find how many more items of fruit and yogurt were ordered to fill the vending machine in April than in May."
• Students can practice fluency skills when accessing the Game Center Online at PearsonRealize.com.

The instructional materials provide opportunities for students to independently demonstrate procedural skill and fluency throughout the grade-level. Examples include, but are not limited to: 

• In Topic 2 there are six Fluency Practice/Assessment worksheets for students to complete to practice 4.NBT.4. In Lesson 2-2, Question 11, students estimate sums and differences, “485,635 - 231,957= __."
• In Topic 6: Fluency Practice Activity: Students subtract 3-digit numbers and then find a matching clue on the page: “917-365 = _____.” (4.NBT.4) 
• In Lesson 12-5, Independent Practice, students work on adding and subtracting multi-digit numbers in world problems. Problem 5 states, "Carlos spends $14.38 on equipment.  How much change should Carlos receive if he gives the clerk $20.00?” (4.NBT.B.4, 4.MD.A.2)

The instructional materials for enVision Mathematics Common Core Grade 4 meet expectations that the materials are designed so that teachers and students spend sufficient time working with engaging applications of grade-level mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied.

Materials provide opportunities for students to solve a variety of problem types requiring the application of mathematics in context. Additionally, the materials support teachers by explaining how the students will apply concepts they have learned within each topic in the Math Background: Rigor section of the Topic Overview. 

Students are provided opportunities to work with routine problems presented in context that require application of mathematics.  Examples include, but are not limited to:

• In Lesson 6-1, the Visual Learning Bridge states, “Max said the Rangers scored 3 times as many runs as the Stars. Jody said the Rangers scored 8 more runs than the Stars. Could both Max and Jody be correct?”
• In Lesson 12-6, the Problem Solving Performance Task states, “Tomas deposits money in his savings account every month. If he continues to save $3.50 each month, how much money will he have at the end of 6 months? 12 months?”  

Students are provided opportunities to work with non-routine problems presented in context that require application of mathematics.  Examples include, but are not limited to:

• In Lesson 1-3, Solve & Share, students use a chart of ocean depths to solve,  “A robotic submarine can dive to a depth of 26,000 feet. Which oceans can the submarine explore all the way to the bottom?  Solve this problem anyway you choose.”
• In Lesson 6-5, the Performance Task states, “Rainey’s group designed the flag shown for a class project. They use 234 square inches of green fabric. After making one flag, Rainey’s group has 35 square inches of yellow fabric left. How can Rainey’s group determine the total area of the flag?” 

Students are provided opportunities to independently demonstrate the use of mathematics flexibly in a variety of contexts. Examples include, but are not limited to:

• In Lesson 2-5, Question 19, students are provided a map containing information about the areas of certain counties.  Students determine, “How much greater is the area of Hernando County than Union County?”
• In Lesson 4-2, Question 11, students use place value blocks, area models, or arrays to solve, “During a basketball game, 75 cups of fruit punch were sold. Each cup holds 20 fluid ounces. How many total fluid ounces of fruit punch were sold?” 
• In Lesson 8-7, Question 3 states, “In the after-school club, Dena, Shawn, and Amanda knit scarves that are all the same size with yellow, white and blue yarn.  Dena’s scarf is $$\frac{3}{5}$$ yellow, Shawn’s scarf is $$\frac{2}{5}$$ yellow, and Amanda’s scarf is $$\frac{3}{4}$$  yellow. The rest of each scarf has an equal amount of white and blue. Describe how Amanda could make the argument that her scarf has the most yellow.”
• In Lesson 10-4, Question 9 states, “There are 55 minutes between the time dinner ends and the campfire begins. What is the elapsed time from the beginning of dinner to the beginning of the campfire?” 

The instructional materials for enVision Mathematics Common Core Grade 4 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately.  

All three aspects of rigor are present independently throughout the program materials. Examples include, but are not limited to:

• Conceptual Understanding is needed to solve Lesson 2-2, Question 14. Students use conceptual understanding of number sense to estimate when asked, “The table shows the number of students at each school in the district. Is 2,981 reasonable for the total number of students at Wilson Elementary and Kwame Charter School? Explain."
• Fluency is practiced in Lesson 2-5, Questions 7-18. Students use the standard algorithm to subtract 3-digit numbers and use estimation to check reasonableness. Question 7 states, “289 - 145."
• Students apply mathematics to solve problems in context. In Lesson 10-3, Question 19 states, “Oscar wants to make 4 chicken pot pies. The recipe requires $$\frac{2}{3}$$ pound of potatoes for each pot pie. How many pounds of potatoes will Oscar need?”

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. Examples include, but are not limited to:

• In Lesson 3-4, students use conceptual understanding of area models to build procedural skill using the distributive property to find products. In Question 5, students use area models and partial products to solve, “3 x 185."
• In Lesson 7-5, students develop conceptual understanding of multiples and use procedural skill to list multiples of a whole number.  Question 25 states, “Latifa and John played a game of multiples. Each player picks a number card and says a multiple of that number. Latifa picked a 9. Write all the multiples of 9 from the box." 
• In Lesson 12-5, students use their understanding of bills and coins to represent money amounts to solve problems in context. Question 4 states, “Sarah bought 3 wool scarves. The price of each scarf was $23.21. How much did 3 scarves cost?”  
• In Lesson 15-6, students use their understanding of rulers and protractors to solve problems in context. Question 2 states, “Lee brought $$1\frac{3}{5}$$ pounds of apples to the picnic. Hannah brought $$\frac{4}{5}$$ pound of oranges. Less said they bought $$2\frac{2}{5}$$ pounds of fruit in all. Lee needs to justify that $$1\frac{3}{5}+\frac{4}{5}=2\frac{2}{5}$$ . How can Lee use a tool to justify the sum? Draw pictures of the tool you used to explain."

The instructional materials reviewed for enVision Mathematics Common Core Grade 4 meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade-level.

All eight Standards for Mathematical Practice (MPs) are clearly identified throughout the materials. Math Practices are identified in the Topic Planner by lesson. In addition, Math Practices and Effective Teaching Practices (ETP)  are identified for each topic, within each lesson, and for specific problems.

• In Topic 4, the Topic Planner states, “MP.4, and MP.7 are addressed in Lesson 5.”
• In Topic 4, the Math Practices and ETP addresses MP4: “Model with mathematics. Students model with math when they use arrays and equations to represent multiplication. (e.g., p. 142, Convince Me!).”
• In Lesson 3-6, Mathematical Practice states, “MP.2 Reason Abstractly and Quantitatively: Students make sense of quantities and their relationships in problem situations by representing a multiplication problem symbolically and manipulating the symbols. Also MP.3, MP.7."

The MPs are used to enrich the mathematical content and are not treated separately. A Math Practices and Problem Solving Handbook is available online at PearsonRealize.com.  This resource provides a page on each math practice for students and teachers to use throughout the year. Math Practice Animations are also available for each practice to enhance student understanding. For example:

• MP1: In Lesson 6-4, Question 8, students make sense of problems and persevere in solving them by using modeling to solve a multi-step problem, and checking reasonableness of the answer. For example, “Cody and Max both solve the problems below correctly. Explain how each solve. Emma has $79 to spend at the toy store. She wants to buy a building set, a board game, and 2 action figures from her favorite movie. What else can she buy?” A picture providing the work of both Cody and Max is provided along with a table of toys sold and their cost.
• MP2: In Lesson 3-2, Visual Learning Bridge, students reason abstractly and quantitatively when estimating products of a 1-digit number multiplied by numbers of up to 4-digits.  The student task states, “Mr. Hector’s class sold calendars and notepads for 3 weeks as a class fundraiser. About how much did Mr. Hector’s class make selling calendars? Selling notepads?” Teachers are prompted to ask, “Is the estimated earnings for the notepads greater than or less than the actual amount? How do you know?” 
• MP3: In Lesson 11-4, Solve & Share Activity, students critique the reasoning of others. For example: “A class made a line plot showing the amount of snowfall for 10 days. Nathan analyzed the line plot and said, ‘The difference between the greatest amount of snowfall recorded and the least amount of snowfall recorded is 3 because the first measurement has one dot and the last measurement has 4 dots.’ How do you respond to Nathan’s reasoning?"
• MP7: In Lesson 3-2, Look Back, students use place value structure to multiply. The student task states, “A theatre contains 14 rows of seats with 23 seats in each row. How many seats are in the theater? Solve this problem using any strategy you choose. Theater seating is an example of objects that are arranged in rows and columns, or arrays. How do the number of rows and the number of seats in each row relate to the total number of seats?”
• MP8: In Lesson 13-1, Convince Me!, students use repeated reasoning to generalize about multiplying to get a greater number of units when converting from a larger unit to a smaller unit. For example: “Maggie has a tree swing. How many inches long is each rope from the bottom of the branch to the swing? How do you know the answer is reasonable when converting a larger unit to a smaller unit?”

The instructional materials reviewed for enVision Mathematics Common Core Grade 4 meet expectations that the instructional materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

The Solve & Share activities, Visual Learning Bridge problems, Problem Sets, 3-Act Math activities, Problem Solving: Critique Reasoning problems, and Assessment items provide opportunities throughout the year for students to both construct viable arguments and analyze the arguments of others.

Student materials consistently prompt students to construct viable arguments. Examples include, but are not limited to:

• In Lesson 7-2, the Visual Learning Bridge states, “Jean wants to arrange her action figures in equal-size groups. What are all the ways Jean can arrange her action figures?” Students are presented with three different methods for arranging the action figures. The next question in the Convince Me! section states, “How do you know there are no other factors for 16 other than 1, 2, 4, 8, and 16? Explain." 
• In Lesson 8-7, the Solve & Share states, students determine how full 3 bottles are with water by comparing fractions. The problem states, “If Tia’s bottle was $$\frac{1}{3}$$ filled with water at the end of the hike, would you be able to decide who had the most water left? Construct an argument to support your answer." 
• In Lesson 16-4, Problem Solving, students answer several questions about symmetry. Question 22, labeled as “Construct Arguments,” states, “How can you tell when a line is NOT a line of symmetry?” 

Student materials consistently prompt students to analyze the arguments of others. Examples include, but are not limited to:

• In Lesson 1-2, Question 9 states, “Vin says in 4,346, one 4 is 10 times as great as the other 4. Is Vin correct? Explain.”
• In Lesson 3-6, Question 14 states, “Quinn used compensation to find the product of 4 x 307. First, she found 4 x 300 = 1,200. Then she adjusted the product by subtracting 4 groups of 7 to get her final answer of 1,172. Explain Quinn’s mistake and find the correct answer.” 
• In Lesson 8-6, Convince Me! states, “The fractions on the right refer to the same whole. Kelly said, ‘These are easy to compare. I just think about $$\frac{1}{8}$$ and $$\frac{1}{6}$$.’ Circle the greater fraction. Explain what Kelly was thinking." 
• In Lesson 15-2, Convince Me! states, “Susan thinks the measure of angle B is greater than the measure of Angle A. Do you agree? Explain."

The instructional materials reviewed for enVision Mathematics Common Core Grade 4 meet expectations that the instructional materials assist teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

Materials assist teachers in engaging students in both constructing viable arguments and analyzing the arguments of others in a variety of problems and tasks presented to students. Examples include, but are not limited to:

• In Lesson 1-5, Convince Me! states, “Construct Arguments, teachers are given an Essential Teaching Practice (ETP), Construct Arguments; providing a clear and complete explanation for conjecture involves using objects, drawings, diagrams, and actions to support the argument. When constructing a viable argument students use mathematical terms and symbols correctly. They should use definitions and previously solved problems when deciding when another students' explanation makes sense. Point out that Bella’s conjecture makes sense because it can be supported with a correct and clear explanation including diagrams, mathematical terms, and symbols.”
• In Lesson 2-3, Question 17 states, “Harmony solved this problem using the standard algorithm, but she made an error. What was her error, and how can she fix it? 437 + 175 = 5,112.” The teacher edition states, “Encourage students to estimate their sum before trying to determine Harmony’s error. Students should see because 400 + 200 = 600, Harmony’s sum is not reasonable."
• In Lesson 7-4, Convince Me! states, “Generalize: Can a number be both prime and composite? Explain.” The teacher edition states, “Students use the definitions of prime and composite numbers to generalize that all whole numbers greater than 1 are classified as either prime or composite."

The instructional materials reviewed for enVision Mathematics Common Core Grade 4 meet expectations that materials explicitly attend to the specialized language of mathematics.

The materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams, and symbols. Examples include, but are not limited to:

• Each topic contains a Vocabulary Review providing students the opportunity to show their understanding of vocabulary and use vocabulary in writing.
• The Teacher Edition provides teacher prompts to support oral language. In Topic 3, the Oral Language prompt states, “Before students complete the page, you might reinforce oral language through a class discussion involving one or more of the following activities. Have students say math sentences that use the words." 
• The Game Center at PearsonRealize.com contains an online vocabulary game. 
• A vocabulary column is provided in the Topic Planner that lists words addressed with each lesson in the topic. In Lesson 2-1, the Vocabulary List includes: Commutative Property of Addition, Associative Property of Addition, Identity Property of Addition, Count Up, Count Down, and Compensation. These words are also listed in the Lesson Overview. 
• Online materials contain an “Academic Vocabulary” and an “Academic Vocabulary Teacher’s Guide” section. The guide supports vocabulary instruction by providing information on how teachers can develop word meaning and build word power. The Academic Vocabulary section provides a variety of academic words with definitions and activities to help students learn the words. For instance, when clicking on the word, conjecture, the definition is provided: “a statement that is thought to be true but has not been proven." Next, the word is used in context: “Make a conjecture about which expression has a lesser sum. 205 + 627 or 354 + 428." Lastly, students are provided a task to help build word power: “Use the word in a sentence."
• A glossary exists in both the Student Edition and the Teacher’s Edition Program Overview. In the glossary, breaking apart is defined as: “Mental math method used to rewrite a number as the sum of the numbers to form an easier problem."
• Visual Learning Bridge activities provide explicit instruction in the use of mathematical language. The words are highlighted in yellow and a definition is provided. 
• A bilingual animated glossary is available online which uses motion and sound to build understanding of math vocabulary.

The materials use precise and accurate terminology and definitions when describing mathematics, and support students in using them. Examples include, but are not limited to:

• In Lesson 6-1, Visual Learning Bridge, students learn how to display data using a line plot. The materials define line plots as: “A line plot shows data along a number line. Each dot above a point on the line represents one number in the data set.” To further explain, a line plot is displayed and the materials state, “Here is how the data look on a line plot."
• In Lesson 12-1, the Visual Learning Bridge introduces factor pairs by highlighting in yellow and giving the definition. It states, “Pairs of whole numbers multiplied together to find a product are called factor pairs. Think about multiplication to decompose a number into its factors.” 
• In Lesson 15-1, Questions 16-18 directions state, “For 16-18, use the map of Nevada. Write the geometric term that best fits each description. Draw an example."