3rd Grade - Gateway 2

The instructional materials reviewed for Everyday Mathematics 4 Grade 3 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. 

All units begin with a Unit Organizer which includes Planning for Rich Math Instruction. This component indicates where conceptual understanding is emphasized within each lesson of the Unit. Lessons include Focus, “Introduction of New Content”, designed to help teachers build their students’ conceptual understanding. The instructional materials include problems that develop conceptual understanding throughout the grade level, especially where called for in the standards (3.OA.1 and 3.OA.2). Examples include:

• In Teacher’s Lesson Guide, Lesson 1-8, Multiplication Strategies, Focus, students make sense of representations for equal-groups and array number stories. The teacher is directed to address conceptual understanding, “Bring the class together and introduce the Fact Strategy Wall as a place to record strategies for solving multiplication and division problems. Record children’s suggestions on the Fact Strategy Wall under a heading for today’s focus, such as "Strategies for Equal-Groups Problems". For example, draw pictures of equal groups, draw pictures of arrays, use addition, and use skip counting. You may also wish to post children's work directly on the Fact Strategy Wall. Encourage children to refer to the Fact Strategy Wall to help them choose efficient strategies as they complete the problems in the next activity.” (3.OA.1)
• In Teacher’s Lesson Guide, Lesson 2-6, Equal Groups, Focus-Math Message, students solve problems of equal groups. “Use your slate. Solve: You have 4 packages of pencils. There are 6 pencils in each package. How many pencils in all? Show your thinking with drawings, words, or number models.” (3.OA.1)
• In Student Reference Book, Lesson 2-12, Exploring Fraction Circles, Liquid Volume, and Area, students play Division Arrays to practice division by grouping counters equally. “Players take turns. When it is your turn, draw a card and take the number of counters shown on the card. You will use the counters to make an array. Roll the die. The number on the die is the number of equal rows you must have in your array.” (3.OA.2)
• In Student Math Journal, Lesson 5-11, Multiplication Facts Strategies: Break-Apart Strategy, Problem 1, students decompose factors to solve multiplication problems. “You have a rectangular garden that is 7 feet wide and 8 feet long. You decide to plant flowers in one section and vegetables in another. Sketch at least two different ways you can partition, or divide your garden into two rectangular sections. Label the side lengths of each of your new rectangles. Write a number model using easier helper facts for one of your ways. 7 x 8 = ___ x ____ + ____ x ___.” (3.OA.1)
• In Teacher’s Lesson Guide, Lesson 7-12, Fraction of Collection, Focus - Naming Fraction of Collections students name fractions of collections using counters. “Direct children to make collections and name fractions of those collections. For example: There are 4 pennies in $$\frac{1}{2}$$ of the pile. Show me the whole pile. There are 8 crayons in 1 box. How many crayons are in 2 boxes? In 1 $$\frac{1}{2}$$ boxes?” (3.OA.2)

The materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. These include problems from Math Boxes, Home Link, and Practice. Examples include:

• In Math Masters, Lesson 1-9, Introducing Division, Home Link, Problems 3 and 4, “3. Think of things at home that could be shared equally by your family. Record them on the back of this page. 4. Write a number story about equally sharing one of the things you wrote for Problem 3. Use the back of this paper. Then solve your number story.” (3.OA.2)
• In Math Masters, Lesson 2-6, Equal Groups, Home Link, students develop conceptual understanding when they solve problems involving multiples of equal groups by using strategies like repeated addition and skip counting. “Solve. Show your thinking using drawings, words, or number models. A pack of Brilliant Color Markers contains 5 markers. Each pack costs $2. 1. If you buy 6 packs, how many markers will you have?” (3.OA.1)
• In Math Masters, Lesson 5-3, Equivalent Fractions, Home Link, students develop conceptual understanding of fraction equivalence (3.NF.3). The directions indicate, “The pictures show three kinds of fruit pie. Use a straight edge to do the following: 1. Divide the peach pie into 4 equal pieces. Shade 2 of the pieces. 2. Divide the blueberry pie into 6 equal pieces. Shade 3 of the pieces. 3. Divide the strawberry pie into 8 equal pieces. Shade 4 of the pieces.” Later, students “Explain to someone at home how you know that all of the fractions on this page are equivalent.” 
• In Math Masters, Lesson 7-4, Fraction Strips, Home Link, Problem 1, students shade fraction strips to represent given fractions. “Shade each rectangle to match the fraction below it. ‘$$\frac{2}{3}$$’” (3.NF.1)
• In Math Masters, Lesson 8-4, Setting Up Chairs, Home Link, Problem 1, students make conjectures and arguments to explain why an arrangement of marching band members is best. “There are 24 members in the school band. The band director wants them to march in rows with the same number of band members in each row. Find two different ways that the band members can be arranged. Draw a sketch that shows each arrangement.” (3.OA.2)

The instructional materials reviewed for Everyday Mathematics 4 Grade 3 meet expectations for attending to those standards that set an expectation of procedural skill and fluency.

The instructional materials develop procedural skill and fluency throughout the grade level. Each Unit begins with Planning for Rich Math Instruction where procedural skills and development activities are identified throughout the unit. Each lesson includes Warm-Up problem(s) called Mental Math Fluency. These provide students with a variety of leveled problems to practice procedural skills. The Practice portion of each lesson provides students with a variety of spiral review problems to practice procedural skills from earlier lessons. Additional procedural skill and fluency practice is found in the Math Journal, Home Links, Math Boxes, and various games. Examples include:

• In Teacher's Lesson Guide, Lesson 2-11, Framers and Arrows, Mental Math and Fluency focuses on basic fact families. “Pose each basic fact without an answer. Have children write out the rest of the fact family, including the answers, on their slates: 6 + 4, 2 x 8, 8 + 5, 5 x 4, 9 + 7, 5 x 9.” This activity provides an opportunity for students to develop fluency of 3.OA.7, “Fluently multiply and divide within 100,” and 3.NBT.2, “Fluently add and subtract within 1,000.”
• In Teacher’s Lesson Guide, Lesson 3-3, Partial-Sums Addition, Focus, students develop procedural skill with addition by expanding addends. “Display 145 + 322 in the vertical form. Ask: What is the expanded form of each addend?” This activity provides an opportunity for students to develop fluency of 3.NBT.2, “Fluently add and subtract within 1,000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.”
• In Teacher’s Lesson Guide, Lesson 3-9, Exploring Multiplication Squares, Focus, students solve multiplication squares and record products. “Have children practice multiplication squares as they complete the Rolling and Recording Squares activity. For example, if you roll a 4, think aloud: 4 x 4 = what number? I can count by 4’s: 4, 8, 12, 16. 4 x 4 =16.” This activity provides an opportunity for students to develop fluency of 3.OA.7, “Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division.”
• In Teacher’s Lesson Guide, Lesson 5-9, Multiplication Facts Strategies: Near Squares, Math Message, students solve problems to build fluency with multiplying and dividing within 100. “Kali knows 7 x 7 = 49. How could she use 7 x 7 as a helper fact to figure out 8 x 7?” This activity provides an opportunity for students to develop fluency of 3.OA.7, “Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division.”
• In Teacher’s Lesson Guide, Lesson 6-2, Playing Baseball Multiplication, is devoted to building multiplication fact fluency as students learn how to play the game Baseball Multiplication. “Tell children that they will practice multiplication facts while playing Baseball Multiplication. Players solve multiplication facts to move counters around the bases and score runs.” This activity provides an opportunity for students to develop fluency of 3.OA.7, “Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division.”

The instructional materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade level as identified in 3.OA.7, “Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division,” and 3.NBT.2, “Fluently add and subtract within 1,000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.” Examples include:

• In Student Math Journal, Lesson 2-4, Multistep Number Stories, Part 2, Student Math Journal, students solve multi-step number stories using multiplication. Problem 2, “A package of rice cakes contains 6 rice cakes. You buy 5 packages of rice cakes. You give 15 rice cakes away. How many rice cakes do you have now?” (3.OA.7,8)
• In Student Math Journal, Lesson 2-8, Picturing Division, students add fluently using strategies or the standard algorithm. Problem 1, “Scientists counted 91 eggs in 2 clutches of python eggs. If 1 python clutch has 52 eggs, how many are in the other clutch? You may draw a diagram or picture.” (3.NBT.2)
• In Teacher’s Lesson Guide, Lesson 7-6, Fractions on a Number Line, Part 2, Practice, students practice multiplication facts by playing Baseball Multiplication. The directions in the Student Reference book include, “Pitching and Batting: Members of the team not at bat take turns ‘pitching’. They roll the dice to get two factors. Players on the ‘batting’ team take turns multiplying the two factors and saying the product.” (3.OA.7)
• An online game, Facts Workshop, focuses on building fluency with addition and subtraction (3.NBT.2). For example, students are shown a domino that has 2 dots on one side and 3 dots on the other side. Students are asked to select facts that are part of that fact family (i.e. 5 - 3 = 2, 5 - 2 = 3, 3 + 2 = 5). 
• An online game, Division Arrays, builds fluency with multiplication and division within 100 (3.OA.7) and interpreting whole-number quotients of whole numbers (3.OA.2). Students can play with a partner or against the computer, “Players take turns making arrays. During each turn, a player is given a total number of counters and numbers of rows, then uses them to build an array. The player earns points equal to the number of counters in one row of the array.”

The instructional materials reviewed for Everyday Mathematics 4 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 in the instructional materials. Student practice includes all three aspects of rigor, though there are fewer questions for conceptual understanding. 

There are instances where conceptual understanding, procedural skill and fluency, and application are addressed independently throughout the instructional materials. Examples include:

• In Teacher’s Lesson Guide, Lesson 2-1, Extended Facts: Addition and Subtraction, Math Message, students use basic addition and subtraction facts to solve multi-digit problems, emphasizing procedural skills. “Solve. Record your answers on your slate. Think about the patterns that help you solve each set. 9 - 7 = ?; 90 - 70 = ?; 900 - 700 = ?; ? = 7 + 9; ? = 70 + 90; ? = 700 + 900.” (3.NBT.2)
• In Math Masters, Lesson 3-4, Column Addition, students apply what they learned about column addition to solve mileage problems. Problem 2, “Tony drives from Washington, D.C., to Cleveland on Friday. He drives back on Sunday. How many miles did Tony drive all together?” (3.NBT.2)
• In Teacher’s Lesson Guide, Lesson 3-10, The Commutative Property of Multiplication, Math Message, students develop conceptual understanding of the Commutative Property of Multiplication as students sketch arrays. “You have 8 apples for sale and want to display them in an array. How many different ways can you arrange them? Make sketches on paper to show your thinking.” (3.OA.1, 3.OA.3)

Multiple aspects of rigor are engaged in simultaneously to develop students’ mathematical understanding of a single lesson throughout the materials. Examples include:

• In Math Masters, Lesson 4-7, Area and Perimeter, Home Link, students apply their knowledge of finding the perimeter of their bedroom while demonstrating conceptual understanding. “Your pace is the length of one of your steps. 2. Find the perimeter, in paces, of your bedroom. Walk along each side and count the number of paces. The perimeter of my bedroom is about ___ paces. 3. Which room in your home has the largest perimeter? Use your estimating skills to help you decide. The ___ has the largest perimeter. Its perimeter is about ___ paces.” (3.MD.5a,7a,8)
• In Student Math Journal, Lesson 5-4, Recognizing Helper Facts, students use procedural skills and fluency to solve an application problem. Problem 1, “Savannah earns $5 selling lemonade. Jessica earns double the amount of money that Savannah earns. How much money do they have together?” (3.OA.8)
• In Teacher’s Lesson Guide, Lesson 6-6, Multiplication and Division, Focus, Introducing Multiplication/Division Diagrams, “Students will use a multiplication/division diagram to organize the number of groups, the number in each group, and the total in each number story.” Students use conceptual understanding and procedural skill and fluency as they solve real-world application problems. “Anna has 8 bags of rubber bands. Each bag has the same number of rubber bands. Anna has 56 rubber bands in all. How many rubber bands are in each bag? Record an equation to match the story and then solve. What do you understand from reading the story? What do we know? What do we need to find out?” (3.OA.2,3,4,6,7)

The instructional materials reviewed for Everyday Mathematics 4 Grade 3 meet expectations for identifying the Standards for Mathematical Practice and using them to enrich mathematics content within and throughout the grade level.

The Implementation Guide states, “The SMPs (Standards for Mathematical Practice) are a great fit with Everyday Mathematics. The SMPs and Everyday Mathematics both emphasize reasoning, problem-solving, use of multiple representations, mathematical modeling, tool use, communication, and other ways of making sense of mathematics. To help teachers build the SMPs into their everyday instruction and recognize the practices when they emerge in Everyday Mathematics lessons, the authors have developed Goals for Mathematical Practice (GMP). These goals unpack each SMP, operationalizing each standard in ways that are appropriate for elementary students.”  

All MPs are clearly identified throughout the materials, with few or no exceptions. Examples include:

• In the Teacher’s Lesson Guide, Unit Organizer, Mathematical Background: Process and Practice, provides descriptions on how the Standards for Mathematical Practices are addressed in the unit and what mathematically proficient students should do. 
• The Unit 6 Organizer identifies MP1, “Make sense of problems and persevere in solving them.” “In order to make sense of a problem, children must learn to decipher the information that is given in the problem. They must determine what is known and what the problem is asking them to find out. Situation diagrams provide a framework to guide children as they make sense of the roles different numbers play in the number stories.” 
• Lessons identify the Math Practices within the Warm Up, Focus, and Practice sections.

The MPs are used to enrich the mathematical content. Examples include:

• MP1 is connected to mathematical content in Lesson 6-6, More Operations, as students write a number story that matches a number model with a letter. In the Student Math Journal, Problem 2, “There are 48 third graders. The gym teacher groups them into teams of 6. How many teams are there? When most children have finished, bring them together to discuss how they used diagrams to organize the information in each problem and write a number model to represent the story.”
• MP2 is connected to mathematical content in Lesson 3-5, Counting-Up Subtraction, as students reason about numbers represented on number lines and in number sentences. In the Teacher’s Lesson Guide, Unit 3 Organizer, “In Lesson 3-5 children represent counting-up subtraction as a series of ‘jumps’ on open number lines and with a string of number sentences. In Lessons, 3-9 through 3-12 children represent multiplication facts as arrays. When working to derive multiplication facts, children connect number models with the array representation. Providing opportunities to make connections among different representations can enable children to make sense of an unfamiliar representation by explicitly relating it to familiar ones.”
• MP4 is connected to mathematical content in Lesson 9-7, The Length-Of-Day Project, as students use information from a scaled bar graph. In the Teacher’s Lesson Guide, Unit 9 Organizer, “In Lesson 9-7 children revisit the Length-of-Day Graph showing data they have collected throughout the year. Children use the information on the scaled bar graph to compare the lengths of days throughout the school year. Children also interpret length-of-day graphs with data from cities in different parts of the world. They connect the data with the location of the cities on maps to draw conclusions about the lengths of day in regions in the Northern and Southern Hemisphere and closer to the Equator.”
• MP7 is connected to mathematical content in Lesson 3-10, The Commutative Property of Multiplication, as students develop a rule, “turn-around rule”, for multiplication. In the Teacher’s Lesson Guide, Unit 3 Organizer, “In Lesson 3-10 children explore their array representations of multiplication facts to discover the Commutative Property of Multiplication (the turn-around rule). Later in this lesson, children look for patterns in the multiplication facts table, and connect these patterns with square numbers and facts related by the turn-around rule.”
• MP8 is connected to mathematical content in Lesson 8-3, Multiplication and Division, as students look for and discuss generalizations about factors and multiples. In the Student Math Journal, Problem 6, “The Kim family is serving dinner for 24 people. Mrs. Kim could have 1 table with 24 people or 2 tables with 12 people at each. What are some other ways Mrs. Kim could seat 24 people in equal groups at different numbers of tables? Is 1 in a factor pair for every counting number?”

The instructional materials reviewed for Everyday Mathematics 4 Grade 3 meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

Student materials consistently prompt students to construct viable arguments. Examples include:

• In Lesson 3-2, Estimating Costs, Math Message, Student Math Journal, students analyze the thinking of others. In Strategies for Estimation, “Rosa makes an estimate for the addition problem below. She uses numbers that are close to the numbers in the problem but are easier to use. ‘322 + 487 = ?’ Another problem is provided in the thought bubble: ‘320 + 490 = 810’ 1. Explain Rosa’s thinking to a partner. 2. Make a different estimate. What close-but-easier numbers could you use? Write a number sentence in the thought bubble to show your thinking.” 
• In Lesson 6-10, Order of Operations, Student Math Journal, students construct a viable argument when they explain to their partner why they picked their answer. Problem 5, “Circle the answer that makes the number sentence true. 2 x (4 + 3 x 2) = ?. a. 28, b. 20, c. 14. Explain to a partner why you picked your answer.” 
• In Lesson 7-2, Exploring Arrays, Volume, and Equal Shares, Math Masters, students construct a viable argument when they compare two different ways of dividing a room equally. The same size shape is divided in two different ways. The square is divided in half with a vertical line down the middle. The other same size square is divided into half with a diagonal line. “Mariana says that they both get the same amount of space. Use words or drawings to explain how Mariana can prove that both drawings divided the room into the same amount of space. You may cut out the drawings of the two rooms on page 227 and fold or cut apart the pieces to compare the parts of the room.” 
• In Lesson 8-4, Setting Up Chairs, Math Masters, Home Link, Problem 1, students solve “There are 24 members in the school band. The band director wants them to march in rows with the same number of band members in each row. Find two different ways that the band members can be arranged. Draw a sketch that shows each arrangement. Which way do you think is better? Explain your reasoning.” 

Student materials consistently prompt students to analyze the arguments of others. Examples include:

• In Lesson 3-14, Progress Check, Assessment Handbook, “Mia wants to solve this problem: 552 - 153 = ? She begins by making an estimate. 550 - 150 = 440. Then she uses expand-and-trade subtraction to find an exact answer, but her answer is not close to her estimate. ‘Oops,’ said Mia, ‘I didn’t cross out 500 and write 400.’ Explain why not changing 500 to 400 is a mistake.” Students must find and explain the mistake someone made in a subtraction problem. 
• In Lesson 7-2, Exploring Arrays, Volume, and Equal Shares, Home Link, Math Masters, students explain if two fraction cards are equivalent. Problem 1, “Nash chose these two cards in a round of Fraction Memory.” One card shows $$\frac{5}{6}$$ shaded and the other card shows $$\frac{6}{8}$$ shaded. “Nash says that these cards show equivalent fractions. Do you agree or disagree? Explain.”
• In Lesson 7-7, Comparing Fractions, Practice, Student Math Journal, “Gail says the liquid volume of container C is less than the liquid volume of Container A. She says that container C holds less water because it is wide and short, and container A holds more because it is taller. Kerod says all three of their containers have the same amount of volume. Do you agree with Gail or Keron? Explain your thinking.”

The instructional materials reviewed for Everyday Mathematics 4 Grade 3 meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

The Teacher’s Lesson Guide assists teachers in engaging students in constructing viable arguments and/or analyzing the arguments of others throughout the program. Many of the activities are designed for students to work with partners or small groups where they collaborate and explain their reasoning to each other. Examples include:

• In Lesson 1-10, Foundational Multiplication Facts, Teacher’s Lesson Guide, Focus: Math Message, teachers facilitate a discussion between students regarding how many dots there are without counting. “Focus on helping children make sense of each other’s thinking and strategies with prompts such as: Did everyone understand Rebecca’s strategy? Explain it in your own words. How could you try Rebecca's strategy on the image?”
• In Lesson 2-11, Frames and Arrows, Teacher’s Lesson Guide, teachers are provided with guidance to help students analyze the arguments of their peers. “Select children to share both effective and ineffective strategies during the follow-up discussion. Encourage the class to repeat the strategies in their own words and to ask questions to help them make sense of others’ strategies. You may wish to provide children with sentence stems such as: I notice… I wonder… How did you… Why did you…”.  
• In Lesson 2-12, Exploring Fraction Circles, Liquid Volume, and Area, Teacher’s Lesson Guide, students explore fraction circles. “Encourage children to make and confirm predictions about part-whole relationships between different fraction circle pieces.  Ask: What fraction of a ______ piece is a ______ piece?  How do you know?” 
• In Unit 3 Open Response Assessment, Teachers Lesson Guide,  teachers are provided the following guidance, “This open response problem requires children to apply skills and concepts from Unit 3 to find a mistake in a subtraction problem. The focus of this task is GMP 3.2: Make sense of others’ mathematical thinking. Before starting the problem, tell children that they will make sense of another child’s work on a subtraction problem and find and explain a mistake.”
• In Unit 7, Open Response Assessment, Teacher’s Lesson Guide, teachers support students to reflect on their arguments and the arguments of others. “After children complete their work, discuss their arguments. You may wish to use this as an opportunity to review and discuss conjectures and arguments. The statements or claims that Demitrius and Emma made comparing the amount of pizza they ate are conjectures. The drawings and words that children used to tell how each statement could be correct are arguments. Ask: What was Demitrius’s claim or conjecture? What was your argument that Demitrius was correct? What was Emma’s claim or conjecture? What was your argument to support her conjecture?” 
• In Lesson 7-10, Justifying Fraction Comparisons, Teacher’s Lesson Guide, teachers allow students to work with a partner and use fraction tools to help name equivalent fractions and justify answers with an explanation. “Have children use their fraction strips and the Fraction Number-Line Poster on journal page 229 (or the Class Fraction Number-Line Poster) to show that $$\frac{1}{6}$$ > $$\frac{1}{8}$$ with other tools. Invite volunteers to justify the comparison with each representation. For example: $$\frac{1}{6}$$ is a larger part of the whole strip than $$\frac{1}{8}$$. $$\frac{1}{6}$$ is farther from 0 (or closer to 1) on the number line than $$\frac{1}{8}$$.”  “As they work, ask them to justify their answer by explaining how each fraction tool models the equivalence.”   
• In Lesson 8-4, Setting Up Chairs, Teacher’s Lesson Guide, teachers facilitate a discussion between students regarding which child’s conjecture is correct in regards to their conjectures on how many total chairs there are in the room. The teacher displays a student's work and then begins the discussion. “What is this child’s conjecture? What is this child’s argument? Explain in your own words. Why did this child use only one clue in his or her argument? Do you agree with this argument? What mathematical reasoning did this child use in the argument? How can this argument be improved?”