3rd Grade - Gateway 2

The instructional materials for enVision Mathematics Common Core Grade 3 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. The Topic 2 Overview explains Multiples of 2, 5, 9, and 10, as well as the Identify Property of Multiplication and the Zero Property of Multiplication. The Conceptual Understanding section states, “Deep understanding of these properties is important as students will use them throughout their work in mathematics, including equation solving and fraction equivalence.”
• The Teacher Edition contains a Rigor section for each lesson explaining how conceptual understanding is developed in the lesson. In Lesson 1-1, the Rigor section states, “Students explore the relationship between addition and multiplication. Multiplication is used in various applications throughout this lesson.”
• 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 1-4, students use their knowledge of multiplication to help understand and solve division problems. Teachers are provided the following prompts to facilitate discussion and conceptual understanding: “Why do you need 3 equal groups? What is the total number of toys? What part of the diagram shows the total? Why is the bar divided into 3 equal parts? How is division different from multiplication?”
• Each lesson is introduced with a video: Visual Learning Animation Plus, to promote conceptual understanding. The Lesson 12-5 video states, “How can you use a number line to represent fractions greater than 1?” The scenario begins by saying, “A marsh rabbit hopped $$\frac{7}{4}$$ the length of a rabbit trail. How can you show this on a number line?”  It then explains how number lines can represent fractions greater than one whole. Students are asked to divide each whole into fourths and explore the value of fraction units using an interactive number line.  
• Each lesson contains a Convince Me! section that provides opportunities for conceptual understanding. In Lesson 2-3, students are asked to use appropriate tools to solve the following, “How would you use counters to show 7x1? How many counters would you have in all?”  Students use counters to show their understanding of the Identity Property of Multiplication. 
• Each lesson contains a Do You Understand? section that makes a connection to previous learning that provides opportunities for conceptual understanding. In Lesson 1-2, “Students explore the relationship between equal groups on a number line and multiplication.” Teachers facilitate student understanding by asking, “On the previous page, why do you skip count by 3s on the number line?  On the previous page, why do you make five jumps on the number line? How would the jumps on the number line look different if there were 4 pens in each gift bag?”

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

• In Lesson 4-4, students use multiplication facts to find related division facts with a multiplication table. (3.OA.2)
• In Lesson 2-3, Visual Learning Bridge, students use their understanding of properties to apply procedures accurately for multiplying by 1 and 0, for example: “How would you use counters to show 7x1? How many counters would you have in all? (3.OA.1)
• In Lesson 8-2, Solve & Share, students “Explain how you can test to see if the relationship among the three sums that are next to each other is a pattern.”  Students use an addition table provided on the page. (3.OA.9)

The instructional materials for enVision Mathematics Common Core Grade 3 meet expectations that they attend to those standards that set an expectation of procedural skill and fluency.

Within the Teacher’s Edition of the lessons, if the lesson focuses on procedural skills and fluency, it is stated in the Lesson Overview page.  For example, Lesson 9-6 focuses on procedural skill and fluency. In regards to procedural skill, “Students add and subtract 3-digit numbers from other 3-digit numbers with one or more zeros using a variety of strategies.”  In terms of fluency, “Students’ fluency in place value, addition and subtraction, and regrouping benefits them as they continue to become more adept at using other strategies. The 3-digit subtraction problems in this lesson provide a challenge as students learn to regroup with zeros” (Teacher’s Edition, page 357 A).  

The Game Center Online at PearsonRealize.com provides opportunities for practicing fluency skills.

The Visual Learning Bridge integrates conceptual understanding with procedural skills.  Addition Fluency and Practice pages are in the Teacher Edition and Ancillary Books as well as online with the Practice Buddy Additional Practice.  Each topic ends with Fluency Practice/Assessment Worksheets.  

Problem sets provide opportunities to practice procedural fluency. Regular opportunities for students to attend to Standard 3.NBT.2: adding and subtracting within 1,000, and to Standard 3.OA.7: multiplying and dividing within 100, 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 5 this section states, “Fluency with multiplication and division within 100 is an expectation in this topic. There are many relevant patterns in a multiplication table that are important in building fluency. For example, noticing that 4 x 8 = 8 x 4 reinforces the Commutative Property of Multiplication. Students can also notice that if one factor is doubled, the product is also doubled. These patterns can become a powerful strategy for learning more difficult facts. Students continue to use the Distributive Property extensively in Topic 5. Students also use the relationship between multiplication and division to find quotients.”
• Each lesson contains a Visual Learning Bridge which provides instruction on  procedural skills. In Lesson 7-3, the Visual Learning Bridge states, “Greg made a table to show the amount of money he saved each month from tutoring. Use the data in the table to make a bar graph." Students practice graphing coordinates from a table. 
• Fluency Practice Activities are found at the end of Topics 5, 6, 7, 8, 11, 13, and 15 to support multiplying and dividing within 100. In Topic 5, the Fluency Activity provides a numbered chart and states, “Partner 1 and Partner 2 each point to a black number at the same time. Both partners multiply those numbers.”
• The Performance Task for Topic 9 provides students the opportunity to fluently add and subtract within 1,000.  Students are presented with a table that contains names of children and the amount of green and silver tokens they earned while playing a board game. Students are asked questions about the table, including: “How many tokens did each friend win in all? The board game comes with a total of 500 green tokens. After each friend earns his or her tokens, how many green tokens are left?”
• 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 6, Fluency Practice Activity, students practice fluently dividing to find the quotients that are odd numbers during a partner activity, this also includes an online game and an interactive practice buddy. (3.0A.C)
• In Topic 7, Fluency Practice Activity, students practice fluently multiplying numbers within 100 during a partner activity, this also includes an online game and an interactive practice buddy. (3.0A.C)
• In Lesson 5-2, Question 6-7, students fill in missing factors and products in multiplication charts provided.
• In Lesson 5-3, Question 11, students use strategies to find the product: “_____ = 5 x 9." 
• In Lesson 9-3, students estimate and then find the sum of the following three numbers stacked vertically using the standard algorithm: “164 + 68 + 35."
• The Performance Task for Topic 9 provides students the opportunity to fluently add and subtract within 1,000.  Students are presented with a table that contains names of children and the amount of green and silver tokens they earned while playing a board game.  Students are asked questions about the table, including: “How many tokens did each friend win in all? The board game comes with a total of 500 green tokens.  After each friend earns his or her tokens, how many green tokens are left?”

The instructional materials for enVision Mathematics Common Core Grade 3 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 1-5, Visual Learning Bridge, teachers guide students to solve, “June has 10 strawberries to serve her guests.  If each guest eats 2 strawberries, how many guests can June serve?” Then, Guided Practice Question 1 states, “There are 3 boxes with 2 toys in each box.  The total number of toys can be expressed as 3 x 2 = 6. What is meant by 6 ÷ 3 = 2? What is meant by 6 ÷ 2 = 3?” (3.OA.3)
• In Lesson 11-3, Solve & Share, students solve, “An aquarium had 75 clownfish in a large water tank. The clownfish represented in the graph were added to this tank.  How many clownfish are in the tank now? Write and explain how you found the answer.” (3.OA.8)

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 6-7, Visual Learning Bridge, students solve, “Janet is painting a door. She needs to paint the entire door except for the window. What is the area of the part of the door that needs paint?” Students are provided a picture of a 4 by 9 foot door with a 2 by 2 foot window. (3.OA.8)
• In Topic 7, Performance Task, Question 3, students are given a picture graph showing how many different color balloons Miles used to make different animals at a birthday party.  Students are also given a bar graph showing how many of each balloon color Miles has used. Students use the Balloons Bought picture graph and the Balloons Used bar graph to determine, “How many balloons does Miles have left?” (3.OA.3)

Students are provided opportunities to independently demonstrate the use of mathematics flexibly in a variety of contexts, especially where called for by 3.OA.3. Examples include, but are not limited to:

• In Topic 2, Performance Task, students solve, “Ms. Harris awards points to her students for reading books in the book club.  There are three levels of books. The table shows the points for each level. Students who earn 50 points get a prize.” Part A, “Luke has earned 15 points.  All of the books he read were at the same level. What level were the books he read? Explain.” 
• In Lesson 3-1, Question 11, students solve, “Paige bakes 5 cupcakes. She puts 7 jelly beans on each cupcake. How many jelly beans does Paige need? Use the bar diagram to help write an equation.” 
• In Lesson 4-2, Additional Practice, students solve, “You have 18 erasers and use 3 erasers each month. How many months will your erasers last? Identify the quotient, dividend, and divisor." 
• In Lesson 5-4, Question 7, students solve, “Bonnie buys 6 paperback books every month.  She buys 2 hardcover books every month. How many books does she buy in 4 months?”

The instructional materials for enVision Mathematics Common Core Grade 3 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 3-1, Solve & Share. Students use conceptual understanding of the distributive property to solve, “Find two ways to break the array below into two smaller arrays. What multiplication equation can you write for each array? What is the total? Tell how you decide."
• Fluency is practiced in Lesson 9-2, Questions 5-12. Students use regrouping to add 3-digit numbers. Question 8 states, “118 + 335."
• Students apply mathematics to solve problems in context.  In Lesson 9-7, the Visual Learning Animation states, “Nancy has $457 in her savings account and wants to have $500 by the end of the year. Christopher has $557 in his savings account and wants to have $600 by the end of the year. Who needs to save more money by the end of the year?”

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 6-4, students develop conceptual understanding of area by counting unit squares and then apply their understanding of multiplication and division to determine the areas of squares and rectangles in context.  Question 8 states, “Jen’s garden is 4 feet wide and has an area of 28 square feet. What is the length of Jen’s garden? How do you know?”
• In Lesson 9-3, Visual Learning Bridge, students are shown how to use partial sums or column addition to add.  Students then apply this mathematical learning in context when solving, “Different kinds of birds are for sale at a pet store.  How many birds are for sale. Find 137 + 155 + 18.”  
• In Lesson 12-6, students use their understanding of fractions to fluently measure the length of objects to the nearest half inch. Then students use their knowledge of number lines, to create a line plot.  Question 5 states, “Measure the lengths of the pieces of yarn at the right to the nearest half inch. Write the length for each piece." Question 7 states, “Make a line plot to show the measurements of the yarn."
• In Lesson 16-3, students develop conceptual understanding of perimeter and polygons by writing equations with variables that represent unknown side lengths.  Students apply the definitions and attributes of common shapes in the writing of equations. Independent Practice, Question 11 states, “These plane figures each have equal sides that are whole numbers. One figure has a perimeter of 25 inches. Which could it be? Explain.”  Students must choose from a triangle, pentagon, hexagon.

The instructional materials reviewed for enVision Mathematics Common Core Grade 3 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. Math Practices and Effective Teaching Practices (ETP) are also provided for each topic, within each lesson, and for specific problems.

• In Topic 3, the Topic Planner states, “MP.3, MP.6, and MP.7 are addressed in Lesson 1.”
• In Topic 3, the Math Practices and ETP addresses MP1: “Students make sense of problems and identify information to use to explain their solution. (e.g., p. 84, Item 14).”
• In Lesson 3-4, Mathematical Practice states, “MP.7 Look For and Make Use of Structure: Students will use known multiplication facts and skip counting to solve problems. Also MP.1."

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 3-2, Question 14, students make sense of problems and persevere in solving them. For example: “James needs to buy supplies for his trail walk.  What is the total number of cereal bars James needs to buy? Explain how you used the table to find the answer?”
• MP2: In Lesson 5-6, Question 9, students reason abstractly and quantitatively. For example: “Mrs. Kendel is making a model house. The footprint for the house is shown at the right. What is the total area? Explain your reasoning."
• MP3: In Lesson 14-5, Convince Me!, students critique the reasoning of others. For example:  “Jason says, ‘I think it is better to find the measurement of the fishbowl by filling the fishbowl with liters of water instead of emptying the fishbowl into liter beakers.’ Is Jason correct?”
• MP4: In Lesson 1-1, Convince Me!, students connect repeated addition with multiplication and use strategies to solve a real-world problem. For example:  “Suppose Jessie won 5 bags of 8 goldfish. Use math you know to represent the problem and find the number of goldfish Jessie won.”  
• MP6: In Lesson 8-5, Solve & Share, students attend to precision. For example:  “Think about ways to find numbers that tell about how much or about how many. Derek has 277 stickers. What number can you use to describe about many stickers Derek has? Explain how you decided."
• MP7: In Lesson 10-4, Question 6, students look for and make use of structure. For example: “How can you find the total amount for each student? Think about properties or patterns you know."
• MP8: In Lesson 5-5, Solve & Share, students look for both general methods and shortcuts as they apply math they know when writing and solving multiplication and division problems. For example: “Write a real-world division story for 28 ÷ 4. Then write another real-world story that shows a different way to think about 28 ÷ 4."

The instructional materials reviewed for enVision Mathematics Common Core Grade 3 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 6-1, the Solve & Share states, “Look at Shapes A-C on Area of Shapes Teaching Tool. How many square tiles do you need to cover each shape? Show your answer below. Explain how you decided. Can you be sure you have an accurate answer if there are gaps between the tiles you used? Explain.”
• In Lesson 7-3, Question 7 states, “Construct Arguments: Which two kinds of movies received about the same number of votes? Explain how to use your bar graph to find the answer."
• In Lesson 8-4, the Visual Learning Bridge states, “A store is having a sale on jackets. A jacket is on sale for $197 less than the original price. What is the sale price?” Students are shown two ways to solve the problem, one with counting back on the number line and another with counting up on the number. The next question in the Convince Me! section states, “Which of the two ways above would you use to solve 762 - 252? Explain.” 

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

• In Lesson 3-1, Question 12 states, “Critique Reasoning: Fred wants to separate the rows of the array below into a 2 x 4 array and a 3 x 4 array. Can Fred do this? Explain?”
• In Lesson 8-4, Convince Me! states, “Which of the two ways above would you use to solve 762-252?  Explain.” 
• In Lesson 9.6, Problem Solving, Problem 18 states, “The students at Cleveland School are collecting soda can tabs. The goal of each class is to collect 500 tabs. So far, the second graders have collected 315 tabs. The third graders have collected 190 more tabs than the second graders. Have the third graders reached their goal? Construct an argument to explain.” (3.NBT.1)
• In Lesson 12-4, Convince Me! states, “Jenna and Benito each marked 1⁄4 on a number line. The length of the part from 0 to $$\frac{1}{4}$$ on Jenna’s number line is shorter than on Benito’s. Did someone make a mistake? Explain your thinking.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 3 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 1-4, the Solve & Share states, “Six friends picked 48 grapefruits. They want to share them equally. How many grapefruits should each friend get?” In the teacher edition, during small group, assists teachers to engage students in constructing an argument “Could the answer by 5 grapefruits? Explain.” 
• In Lesson 2-2, the Solve & Share states, “Maria bought 4 packages of bottled water. There are 9 bottles in each package. How many bottles did Maria buy? Explain how you solved this problem." Teachers are prompted to ask, “How did Nikki use counters to represent the number of bottles of water Maria bought? What strategy is represented in her work? How did Ethan use a table to find the number of bottles of water Maria bought?”
• In Lesson 5-2, Question 10 states, “Bill used a multiplication table to find the value of $$\frac{12}{6}$$. His answer was 3. Do you agree? Why or why not?” The teacher edition states, “Ensure that students understand the mistake in reasoning Bill made by showing how to correctly use a multiplication table to find the value of $$\frac{12}{6}$$." 
• In Lesson 12-3, Problem Solving, students construct an argument regarding the following problem: “Jenna and Jamal are making rugs.  They have finished the parts shown. Draw pictures to show each whole rug. Who’s rug will be longer when it is finished? Explain.” Teachers are guided to “Remind students that the size of part of the whole determines the size of the whole.  Because $$\frac{1}{3}$$ of Jamal’s rug is longer than $$\frac{1}{3}$$ of Jenna’s rug, his whole rug will be longer.” 
• In Lesson 14-9, Performance Task, Question 7 states, “Sachi says that the 5th grade singers should begin at 7:40 p.m. Phil says that 5th grade singers should begin at 8:00 p.m. Who is correct?” Teachers are prompted to ask, “What is the total length of time for each of the acts after the break? What is the total amount of time needed to introduce each act? What mistake did Phil make?” 

The instructional materials reviewed for enVision Mathematics Common Core Grade 3 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 1, 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 define terms in their own 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 8-5, the Vocabulary List includes: Round and Place Value. 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, distribute, the definition is provided: “to give out shares of something." Next, the word is used in context: “Lisa will distribute the pieces of this pizza equally to 4 friends. How many pieces will each friend get?” 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, multiple is defined as, “The product of a given whole number and any non-zero whole number. Example: 4, 8,12, and 16 are multiples of 4.”
• 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 1-3, Visual Learning Bridge, students learn about arrays. A display shows medals in an array and a character on the page defines an array as, “The medals are in an array. An array shows objects in equal rows and columns.” 
• In Lesson 1-4, the Visual Learning Bridge states, “Three friends have 12 toys to share equally. How many toys will each friend get? Think of arranging 12 toys into 3 equal groups.” A character on the page defines division: “Division is an operation that is used to find how many equal groups there are or how many are in each group."
• In Lesson 6-1, the Visual Learning Bridge states, “Area is the number of unit squares needed to cover a region without gaps or overlaps.” Next, the materials differentiate between a unit square and square units: “A unit square is a square with sides that are each 1 unit long. It has an area of 1 square unit.”