2nd Grade - Gateway 2

The instructional materials reviewed for Everyday Mathematics 4 Grade 2 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

The materials include problems and questions that develop conceptual understanding throughout the grade level. The Focus portion of the lesson provides opportunities for students to explore, engage in, and discuss conceptual understanding of mathematical content. Examples include:

• In Lesson 1-4, Focus: Exploring Patterns on the Number Grid, students describe and explore patterns on a number grid. Students are expected to observe such things as every other number on the grid is odd and all of the numbers in the far right column end in a 0. Next, the teacher leads the students in using the chart to skip count by 2s, 5s, and 10s. This activity supports conceptual understanding of 2.NBT.2, “Count within 1000, skip-count by 5s, 10s, and 100s.”
• In Lesson 2-5, Focus: Discussing the Near-Doubles Strategy, the teacher leads a discussion about how using a near doubles fact can help find the answer to a more difficult problem. “Display problems 5 + 7 = ? and 6 + 4 = ? and tell partners to use their knowledge of doubles facts to solve them. After a few minutes, invite volunteers to share their strategies. These may include adding or subtracting 2 from a double (for example, 5 + 7 = 5 + 5 + 2 = 12) or ‘sharing’ (for example, taking 1 from 7 and giving it to 5 to make the double 6 + 6).” This activity supports the conceptual understanding of 2.OA.2, “Fluently add and subtract within 20 using mental strategies.”
• In Lesson 4-4, Focus: Reviewing Values of Digits, “Display a cube, a long, and a flat. Remind children that these are called base-10 blocks. Hold up a cube.  Say: This is a base-10 cube. It represents 1. Then hold up a long and say: This is a long. It represents 10. Ask children to explain why a long represents 10. Hold up a flat and say: This is a flat. It represents 100. Ask children to explain why a flat represents 100.” This class discussion supports students in developing conceptual understanding of 2.NBT.1, “Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones.”
• In Lesson 5-5, Focus: Introducing Arrays, students make arrays and write number models to represent them. Students should determine that organizing dots in rows makes it easier to skip count to find the total. This activity supports conceptual understanding of 2.OA.4, “Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns, write an equation to express the total as a sum of equal addends.”
• In Lesson 9-5, Focus: Comparing Multidigit Numbers, the teacher leads a discussion about expanding three-digit numbers and asks, “How can we use the expanded form of each number to help us compare them? Sample answer: First we can look at the hundreds. Both numbers have 2 hundreds, or 200, so next we look at the tens. We see that 292 has 9 tens, or 90, but 289 has 8 tens, or 80. So we know 292 is larger.” Then the teacher asks, “Do we need to look at the ones?” Students may answer, “No. The tens told us that 292 is larger.” The teacher repeats this activity with other pairs of 3-digit numbers as needed. This activity supports conceptual understanding of 2.NBT.4, “Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record results of comparisons.”

Games, Daily Routines, and Math Journals provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Examples include:

• In Routine 2: Attendance Routine, students count the number of total students and the number of students present. This information is recorded on the attendance chart and students determine how many children were absent. This activity provides continuous conceptual understanding practice of 2.NBT.5, “Fluently add and subtract within 100 using strategies based on place value, properties of operations and/or the relationship between addition and subtraction.”
• In Lesson 4-5, Game: Introducing Number Top-It, students and a partner are given a game mat with rectangles labeled with place value levels to thousands. Using a deck of digit cards, students take turns drawing cards until each player has three cards. Students use their cards to make a 3-digit number, read the number, and compare the number to their partner’s. The player with the larger number for the round scores 1 point, and the player with the smaller number scores 2 points. Students play five rounds, and the player with the fewest points at the end of five rounds wins the game. This activity provides practice of conceptual understanding of 2.NBT.4, “Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, or < symbols to record the results of comparisons.” 
• In Lesson 1-3, Math Journal, students find the total value of coin combinations. For example, Question 3 shows 4 dimes and 1 nickel. This practice activity supports conceptual understanding of 2.MD.8, “Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies using $ and ¢ symbols appropriately.”
• In Lesson 2-8, Math Journal, students use number lines provided to find sums. For example, Question 1, “Show 62 + 10 on the number line below.” Then students record their answers. This practice activity supports the conceptual understanding of 2.NBT.1, “Add and subtract within 1000 using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method.”
• In Lesson 4-5, Math Journal, students write numbers in expanded form to compare them. For example, Question 2 asks students to compare 42 and 48. Students write 40 + 2 and 40 + 8 so they can use the ones to make a comparison. This practice activity supports conceptual understanding of 2.NBT.3, “Read and write numbers to 1000 using base-10 numerals, number names, and expanded form.”

The instructional materials reviewed for Everyday Mathematics 4 Grade 2 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. The Section Organizer provides information on which part of each lesson develops procedural skill and fluency. Opportunities are found in the Daily Routines and Focus portions of the lesson. Examples include:

• In Lesson 2-3, Focus: Using Double Ten Frames, “Flash each Quick Look Card for 2 to 3 seconds before removing it from view or covering it. Always allow a second look and follow up by asking children both what they saw and how they saw it. Asking such questions will allow a variety of strategies to emerge. Encourage children to share multiple strategies but focus attention to those that involve doubles and combinations of 10.” This activity provides an opportunity for students to develop fluency of 2.OA.2, “Fluently add and subtract within 20 using mental strategies.”
• In Lesson 5-7, Focus: Introducing Open Number Lines, students use mental strategies to solve addition number stories and record their thinking on open number lines. For example, “Peter has 64 blocks in his toy box and 30 blocks on the table. How many blocks does he have in all?” This activity provides an opportunity for students to develop fluency of 2.NBT.5, “Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.”
• In Lesson 7-3, Focus: Introducing Basketball Addition, students play in teams of 3 to 5 players. Teams roll a 20-sided polyhedral die and a 6-sided die and find their sum. Teams add the sum to their team’s total score. The team with the most points wins. This activity provides an opportunity for students to develop fluency of 2.NBT.5, “Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.”
• In Routine 3: Attendance Routine, students use data from the attendance chart to tell number stories such as, “Today, 23 children are here. If 2 more children arrive, how many will be present? There are 24 children here today. If 6 go to the library, how many are left?” This activity provides continuous opportunities for students to develop fluency of 2.NBT.5, “Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.”

The instructional materials provide opportunities for students to independently demonstrate procedural skill and fluency throughout the grade level. The Section Organizer provides information on which part of each lesson develops procedural skill and fluency. Opportunities are found in the Practice portions of the lesson, Math Journal, and Math Masters. Examples include:

• In Lesson 2-4, Math Journal 1, students use double ten frames to make 10, find the sum, and write a combination of 10 that helped them solve. In Question 1, students are shown 9 and 5. The suggested answer is, “The combination of 10 that helped: 9 + 1 = 10 and fact 9 + 5 = 10”. This activity provides an opportunity for students to independently demonstrate the procedural skill of 2.OA.2, “Fluently add and subtract within 20 using mental strategies.”
• In Lesson 4-11, Math Journal 1, students match subtraction facts to strategies they could use to solve them in their journal. Students then discuss their reasoning for their pairings in small groups, focusing on how they matched facts and strategies. This activity provides an opportunity for students to independently demonstrate the procedural skill of 2.NBT.5, “Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.”
• In Lesson 5-7, Math Journal 2, students use Open Number Lines to record their thinking when adding two 2-digit numbers. In Question 2, “You build the second tower with 37 red blocks and 32 blue blocks. How many blocks did you use?” This activity provides an opportunity for students to independently demonstrate the procedural skill of 2.NBT.5, “Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.”
• In Lesson 9-7, Math Journal, students are given a subtraction problem and find a ballpark estimate, write each number in expanded form, and solve. In Question 3, “72 - 49 = ?” This activity provides an opportunity for students to independently demonstrate the procedural skill of 2.NBT.5, “Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.”

The instructional materials reviewed for Everyday Mathematics 4 Grade 2 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. 

The materials attend to conceptual understanding. Examples include:

• In Lesson 4-4, Focus: Matching Numbers to Base-10 Block Representations, students represent 3-digit numbers with number cards, base-10 blocks, and expanded form, “Display a Place-Value Mat and place 3 flats, 5 longs, and 2 cubes. Tell children that the collection of blocks represents a number. Ask: How many hundreds are in this number? What is the value of the 3? How many tens are in this number? What is the value of the 5? How many ones are in this number? What is the value of the 2?” This activity develops conceptual understanding of 2.NBT.1, “Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones.”
• In Lesson 9-6, Focus: Representing Subtraction with Trades, students use base-10 blocks to subtract with trades, “Tell children they will now use their base-10 blocks to solve 53 - 37. Ask children to represent 53 with base-10 blocks. When they have finished, record a sketch of 5 longs and 3 cubes. Are there enough longs and cubes for me to remove 3 longs and 7 cubes? How can I get more cubes so I can remove 7 cubes?” The children make the trade with their base-10 blocks. The teacher represents the trade on the sketch by crossing out 1 long and adding 10 cubes. The class repeats the activity with other subtraction problems. This activity develops conceptual understanding of 2.NBT.7, “Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.”

The materials attend to procedural skill and fluency. Examples include:

• In Lesson 1-8, Warm Up, Mental Math and Fluency, students solve addition facts using Quick Looks, “Always allow children a second look and follow up by asking both what they saw and how they saw it.” The teacher shows the Quick Look Card 79 which shows a double ten frame with four dots at the bottom of each ten frame, “Sample answer: I saw 4 and 4. I know that 4 + 4 = 8.” This activity develops the procedural skill of 2.OA.2, “Fluently add and subtract within 20 using mental strategies.”
• In Lesson 3-2, Focus: Generating Related Addition and Subtraction Facts, students generate related addition and subtraction facts based on dominoes, “Display a domino with 5 dots on one side and 4 dots on the other, Help children discover the addition facts and subtraction facts it shows. Ask: Which addition facts describe the domino? If needed, remind children about using the turn-around rule to generate related addition facts. Which subtraction facts describe this domino?” Students repeat the activity with other dominos. This activity develops the procedural skill of 2.NBT.5, “Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.”

The materials attend to application. Examples include: 

• In Lesson 2-2, Focus: Creating and Solving Addition Number Stories, students make up and solve number stories, “1. Display the story or draw a picture that illustrates the story but doesn't suggest a solution strategy. 2. Draw an empty unit box below the story. 3. Have children write a label in the unit box and share how they would answer the question in the story. 4. Ask a volunteer to write a number model for the story. 5. Ask another volunteer to explain how the numbers in the number model connect to the story.” This activity provides the opportunity to apply understanding of 2.OA.1, “Use addition and subtraction within 100 to solve one-and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing with unknowns in all positions.”
• In Lesson 6-2, Focus: Math Message, students solve comparison number stories, “Fish A is 14 inches long. Fish B is 6 inches long. How many inches longer is Fish A than Fish B?” Students solve several other comparison number stories. This activity provides the opportunity to apply understanding of 2.MD.5, “Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units.”

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

• In Lesson 4-7, Focus: Playing Target to 50, students add and subtract using base-10 blocks, “Players take turns. When it is your turn, do the following: Turn over 2 cards. You may either use one card to make a 1-digit number or both cards to make a 2-digit number. Model your number with base-10 blocks. Put these blocks just below your Target Game Mat but not on the mat. You now have two choices: Choice 1: Add all of the base-10 blocks below the mat to the blocks already on your Target Game Mat. Choice 2: Subtract all of the blocks below the mat from the blocks already on your Target Game Mat. If you decide to subtract, you may first have to make exchanges on the mat.” The first player to have a mat with a value of exactly 50 wins. Students develop conceptual understanding of 2.NBT.1, “Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones,” and procedural skill with 2.NBT.7, “Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.”
• In Lesson 4-11, Focus: Exploration A: Matching Facts with Strategies, students match subtraction facts to possible solution strategies, “Children independently match subtraction facts from Math Masters, page 110 to strategies they could use to solve them on journal page 89. In small groups children discuss their reasoning for their pairings, focusing especially on differences in how they matched facts and strategies.” Students develop procedural skill with 2.OA.2, “Fluently add and subtract within 20 using mental strategies,” and conceptual understanding of 2.NBT.5, “Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.”
• In Lesson 6-4, Focus: Solving Silly Animal Stories and Writing Silly Animal Stories, students solve number stories comparing the heights and lengths of various animals, “Have children share the names and lengths of the longest animal and the shortest animal.” Then students write and solve their own number stories, “Children write two number stories. In each story they compare or add the lengths in feet of two animals from journal page 146.” Students develop procedural skill with 2.NBT.5, “Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction,” and application of 2.OA.1, “Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing with unknowns in all positions.”

The instructional materials reviewed for Everyday Mathematics 4 Grade 2 meet expectations for identifying the Standards for Mathematical Practice and using them to enrich mathematics content within and throughout the grade level.

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

• The mathematical practices are listed on pages xxxvi-xxxix in the Grade 2, Volume I Teacher’s Lesson Guide as “Correlation to the Mathematical Process and Practices” which states, “Everyday Mathematics is a standards-based curriculum engineered to focus on specific mathematical content, processes, and practices in every lesson and activity. The chart below shows complete coverage of each mathematical process and practice in the core program throughout the grade level.” 
• Each Unit Organizer contains a Mathematical Background: Processes and Practices component identifying the MPs addressed in the section and in individual lessons. Additionally, “The authors created Goals for Mathematical Practice (GMP) that unpack the practice standards, operationalizing them in ways that are appropriate for elementary students.” 
• Within each lesson description, GMPs appear in bold print and teacher side notes identify the MPs that are addressed in the lesson.

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

• In Lesson 2-4, Focus: Exploring the Making-10 Strategy, students use the making-10 strategy to determine the number of dots in ten frames, “Remind children that Quick Look activities help us think about addition strategies. Flash Quick Look cards 95 and 98 in sequence. Have children write words or number sentences on their slates to record how they figured out the total number of dots on each card.” The mathematical content in this activity is enriched by MP2.
• In Lesson 3-10, Focus: Summarizing Subtraction Strategies, students discuss subtraction strategies used to solve problems, “Have partners discuss the following questions: If you want to solve a subtraction fact that you don’t know, what strategies could you use? Pose a few subtraction facts to provide a context, such as 12 - 6, 15 - 8, and 16 - 7. Encourage children to refer to their My Subtraction Fact Strategies table on journal page 48. As volunteers share their ideas, record their strategies on the Class Data Pad and lead a class discussion about the similarities and the differences between the strategies.” The mathematical content in this activity is enriched by MP1.
• In Lesson 6-5, Focus: Solving Two-Step Number Stories, students solve two-step number stories, “Display and fill in a change diagram to model the first step. Write 9 in the Start box, + 12 on the Change line, and ? in the End box (see margin). Tell children that the ? represents the number of shells Anabelle has after adding 12 more to her collection. Have children suggest a number model with ? to represent the first step. Write the number model below the change diagram (see margin).” The mathematical content in this activity is enriched by MP4.
• In Lesson 8-3, Focus: Comparing Triangles, students compare triangles and discuss polygons, “Compare some of the different triangles, focusing on the sides and the angles. Be sure to display one triangle with a right angle and one with all equal-length sides and discuss these attributes.” The mathematical content in this activity is enriched by MP7.
• In Lesson 9-10, Focus: Connecting Doubles and Equal Groups, students use doubles facts to solve equal-groups stories, “After all 10 possible arrays have been recorded, have children examine the list. Ask: What patterns do you notice? Referring to the two lists of possible number models, discuss the idea that when children need to find the total number of objects in 2 equal groups (or multiply by 2), they can use addition doubles. Ask: How can we use doubles facts to help us solve number stories about 2 equal groups?” The mathematical content in this activity is enriched by MP8.

The instructional materials reviewed for Everyday Mathematics 4 Grade 2 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 and analyze the arguments of others. Lessons offer Differentiated Options where students work in small groups or with partners as teachers facilitate discussions so students have opportunities to construct viable arguments and critique the reasoning of others. Open Response and Reengagement Lessons provide opportunities for students to critique the open response answers of other students. 

Students construct arguments. Examples include:

• In Lesson 5-2, Focus: Math Message and Finding the Total, students find the total value of specified toolkit coins and share their strategies for counting their coins, “Take 10 pennies, 6 nickels, 6 dimes and 4 quarters from your toolkit. How much money is that? Ask children to share their strategies and justify their answer for finding the total value of the coins.”
• In Lesson 7-3, Focus: Introducing Basketball Addition, students learn a game for adding three or more numbers, “At halftime, invite children to examine the scoreboard. Ask: Is it possible for the team that is behind to win? Yes. Why or why not? Sample answer: In the second half, the team that is behind could roll a lot of large numbers, and the team that is ahead could roll a lot of small numbers.”
• In Lesson 8-4, Focus: Setting Expectations, students discuss solutions to the Open Response Problem and revise their solutions, “Review the open response problem from Day 1. Ask: What do you think a complete answer to this problem needs to include? Sample answer: It needs drawings of gardens for Juan and Linda and an explanation for why the circled shape has the attributes for Linda’s plan.”

Students critique the reasoning of others. Examples include:

• In Lesson 2-3, Focus: Using Double Ten-Frames, students explore strategies for finding the total number of dots shown on double ten frames, “Have children suggest number sentences that match their strategies and display them for the class. Whenever children’s strategies involve either doubles or filling a frame to make 10, follow up with questions such as the following: Can someone explain Tommy’s strategy or Tommy’s way of thinking for us again? Can you use thinking like Tommy’s on this next example?”
• In Unit 3 Open Response Assessment, A Subtraction Strategy, “Grace solved 12 - 7 this way: ‘I started at 12 and took away 2 to get to 10. Then I took away 5 more. I ended up at 5. So 12 - 7 = 5.’ Grace solved 13 - 4 this way: ‘I started at 13 and took away 3 to get to 10. Then I took away 1 more. I ended up at 9. So 13 - 4 = 9.’ Show and explain how to use Grace’s subtraction strategy to solve 14 - 8.”
• In Lesson 4-7, Practice: Practicing Place-Value Concepts, Math Journal Page 79, Item 8, “Marta wrote 24 to describe the number shown by these base-10 blocks: Do you agree with Marta? Explain your answer.”

The instructional materials reviewed for Everyday Mathematics 4 Grade 2 meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

Examples of assisting teachers in engaging students to construct viable arguments include:

• In Lesson 3-6, Focus: Discussing the -0 and -1 Strategies, students develop rules for solving -0 and -1 facts, “Display this problem: 87 - 0 = ___. After a volunteer has given the answer, ask another child to explain why this is correct. Sample answer: When we subtract 0, we are not taking anything away. So we still have 87. Ask: Do you think this will always be true? Even with very large numbers? Why? Yes. Sample explanation: It is always true because when we subtract 0 from any number, the number remains the same.”
• In Lesson 6-7, Focus: Math Message and Sharing Strategies, students solve a 2-digit addition problem and share their strategies for solving, “Solve 37 + 52. Explain your strategy to a partner. Circulate and observe as children solve the Math Message Problem and explain their strategies to their partners. Encourage partners to try using each other’s strategies to solve the problem.”
• In Unit 7 Organizer, Mathematical Background: Process and Practice, Standard for Mathematical Process and Practice 3, “According to SMP3, mathematically proficient students ‘make conjectures and build a logical progression of statements to explore the truth of their conjectures.’ In Lesson 7-1 children enter a starting number into their calculators and work to change it to a given multiple of 10. For example, they enter 17 and are asked to change it to 40. When children state whether they need to add or subtract and what number they need to add or subtract, they are making conjectures, or educated guesses, based on some given information. When they explain how they determined what number to add or subtract, they are making arguments, or logical progressions of statements, that support their conjectures.”

Materials assist teachers in engaging students to analyze the arguments of others frequently throughout the program. Examples include:

• In Lesson 1-5, Number-Grid Puzzles, Getting Ready for Day 2, students’ share and discuss their solutions to the Open Response Problem, “Display a response showing a solution to the puzzle that uses counting or operations with patterns on the number grid. Child B shows ‘two moves’ from 78 to 98, explaining that moving down on the number grid is the same as adding 10. A related example could show ‘one move’, in which 88 is left out and the explanation uses the number model 78 + 20 = 98. Encourage children to connect these explanations in the discussion. Ask questions such as: How are these strategies different? Do they both get the correct answer? Why? What’s different about how these children explained their thinking?”
• In Lesson 5-11, Adding Multidigit Numbers, Getting Ready for Day 2, students share and discuss their solutions to the Open Response Problem, “Display responses that use the same tool but different strategies to find the total. See sample work for Child A and Child B. Have children interpret and compare the strategies. Ask: What tool and strategy do you think Child A used to find the total cost? What could this child do to help us understand the strategy better? How are Child A’s and Child B’s work alike? Different? What could the second child do to help us understand the strategy better?”
• In Lesson 8-4, Drawing and Reasoning About Quadrilaterals, Reengaging in the Problems, students discuss their drawings and evaluate their written arguments that their drawings have the given attributes, “Children reengage in the problem by analyzing and critiquing other children's work in pairs and in a whole-group discussion. Have children discuss with partners before sharing with the whole group. Guide this discussion based on the decisions you made in Getting Ready for Day 2.”