2nd Grade - Gateway 2

The instructional materials for Eureka 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. For example:

• In Module 3, Lesson 1, students develop conceptual understanding of place value. Students work with teacher guidance to count 1000 straws. The teacher leads students in creating bundles using place value (ten ones to make a ten, ten tens to make a hundred) and discusses with students how each place value contains the collection of the preceding place, i.e., the number of tens contained in 100. (2.NBT.A)
• In Module 6, Lesson 6, students develop conceptual understanding of the foundation for multiplication. Students practice using addition to find the total number of objects arranged in an array. Problem Set Question 1a states, “Complete each missing part describing each array. Circle rows. 5 rows of ___ = ___ + ___ + ___ + ___ + ___ = ___.” (2.OA.4)

The materials provide opportunities for students to demonstrate conceptual understanding independentlythroughout the grade level. For example:

• In Module 3, Lesson 20, students independently demonstrate conceptual understanding of place value. Students model 10 more and 10 less when solving word problems involving changing the hundreds place. Problem Set Question 4 states, “Jenny loves jumping rope. Each time she jumps, she skip-counts by 10s. She starts her first jump at 77, her favorite number. How many times does Jenny have to jump to get to 147? Explain your thinking below.” (2.OA)
• In Module 5, Lesson 19, students independently demonstrate conceptual understanding of place value. Students choose which strategies to apply to a variety of addition and subtraction problems and explain their choices/listen to the reasoning of their peers. In the Concept Development part of the lesson, the teacher is prompted to ask the following questions, “Problem 1: 180 + 440. Give students three minutes to solve the problem using the strategy of their choice. T: Turn and talk: Explain your strategy and why you chose it to your small group. S1: I used a chip model to represent the hundreds and tens for each number because there were no ones. Then, I added the tens together and the hundreds together. Since there were 12 tens, I renamed 10 tens as 1 hundred, and that leaves 2 tens. 5 hundreds and 1 hundred more makes 6 hundreds. So, my answer is 620. S2: I used the arrow way. I started with 180, added 400 to get 580, added 20 to make 600, and added 20 more is 620. S3: I used a number bond to take apart 440. I took 20 from the 440 and added it to 180 to make 200. 200 plus 420 is 620.” (2.NBT.B)

The instructional materials for Eureka Grade 2 meet expectations that they attend to those standards that set an expectation of procedural skill and fluency.

The instructional materials develop procedural skill and fluency throughout the grade level. For example:

• In Module 4, Lesson 14, students develop procedural skill and fluency of adding and subtracting within 100 using strategies based on place value. The teacher is prompted to ask the following questions, “T: (Write 184.) Say the number in standard form. S: 184. T: What digit is in the tens place? S: 8. T: (Underline 8.) What’s the value of the 8? S: 80. T: State the value of the digit 1. S: 100. T: 4? S: 4. Repeat using the following possible sequence: 173, 256, and 398.”
• In Module 6, Lesson 12, students develop procedural skill and fluency of adding and subtracting within 100 using strategies based on place value. Students complete the activity, Compensation. This activity reviews the mental math strategy of compensation, which is, by making a multiple of 10, students solve a much simpler addition problem. “Using number bonds for visualization: T: (Write 42 + 19 = ____.) Let’s use a mental math strategy to add. How much more does 19 need to make the next ten? S: 1 more. T: Where can 19 get 1 more from? S: From the 42. T: Take 1 from 42, and give it to 19. Say the new simplified number sentence with the answer. S: 41 + 20 = 61. T: So, 42 + 19 is…? S: 61. T: 37 + 19? S: 36 + 20 = 56.”

The instructional materials provide opportunities to demonstrate procedural skill and fluency independently throughout the grade level. For example:

• In Module 4, Lesson 20, students independently demonstrate procedural skill and fluency of adding within 100 by using a place-value chart to solve a problem. Problem Set Question 1 states, “Solve vertically. Draw chips on the place-value chart and bundle, when needed. a. 23 + 57 = ____.”
• In Module 4, Lesson 27, students independently demonstrate procedural skill and fluency of subtracting within 100 by using a place-value chart to solve a problem. Problem Set Question 2a states, “Solve vertically. Draw chips on the place-value chart. Unbundle when needed. A. 100 - 61 = ____.”

Students build fluency for adding and subtracting to 20 using mental strategies in 5-10 minute fluency practice activities before lessons. These fluency practices are provided in all eight modules. For example:

• In Module 3, Lesson 12, students complete timed “sprints” to practice a variety of addition facts within 20.
• In Module 3, Lesson 14, students complete timed “sprints” to practice a variety of subtraction facts within 20.
• In Module 5, students practice subtracting from a number which has tens (primarily teens numbers) by a single-digit number. Students begin using pennies and dimes to count and add different totals.
• In Module 6, students utilize “sprints,” coins, flashcards and differentiated problems selected by the teacher to build fluency for addition and subtraction up to 20, primarily working with addition and subtraction of teen numbers.

Students build fluency for adding and subtracting within a 100 in 5-10 minute fluency practice activities before lessons. These fluency practices are provided in all eight modules. For example:

• In Module 1, students subtract a single-digit number from a multiple of ten.
• In Module 3, students use meter sticks to both add and subtract different numbers from multiples of ten within 100.
• In Module 4, students practice addition and subtraction to 100, primarily by adding and subtracting by tens (making a ten to add or to subtract, counting numbers of tens).
• In Module 5, students subtract tens within 100, use compensation to subtract, practice subtraction that crosses multiples of tens, and use linking cubes to model subtraction facts.
• In Module 8, students practice using addition to solve subtraction facts quickly.

The instructional materials for Eureka Grade 2 meet expectations that the materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied.

The instructional materials include multiple opportunities for students to engage in routine and non-routine application of mathematical skills and knowledge of the grade level. For example:

• In Module 4, Lesson 5, students engage in grade-level mathematics when solving two-step story problems using tape diagrams and number bonds. The Concept Development Problem 3 states, “Solve a two-step problem by drawing a tape diagram and using a number bond to solve. There are 31 students on the red bus. There are 29 more students on the yellow bus than the red bus. How many student are on the yellow bus? How many students are on both buses combined?” (2.OA.1)
• In Module 5, Lesson 14, students engage in grade-level mathematics when solving one and two-step word problems within 100. The Application Problem states, “Brienne has 23 fewer pennies than Alonzo. Alonzo has 45 pennies. How many pennies does Brienne have? How many pennies do Alonzo and Brienne have altogether?” (2.OA.1)
• In Module 6, Lesson 9, students engage in grade-level mathematics when solving word problems involving addition of equal groups. Concept Development Problem 2 states, “Miss Tam arranges desks into 4 rows of 5. How many desks are in her classroom? Draw a picture to solve, and write a repeated addition equation. Then, write a statement of your answer.” (2.OA.4)

The instructional materials provide opportunities for students to demonstrate independently the use of mathematics flexibly in a variety of contexts. For example:

• In Module 5, Lesson 13, students independently demonstrate the use of mathematics by solving two-step story problems using tape diagrams and number bonds. The Application Problem states, “A fruit seller buys a carton of 90 apples. Finding that 18 of them are rotten, he throws them away. He sells 22 of the ones that are left on Monday. Now, how many apples does he have left to sell? Note: This problem is designed for independent practice. Possibly encourage students to use the RDW process without dictating what to draw. Two-step problems challenge students to think through the first step before moving on to the second. The number sentences can help them to see and articulate the steps as well.” (2.OA.1)
• In Module 4, Lesson 16, students independently demonstrate the use of mathematics by using place-value strategies to solve one and two-step word problems within 100. Problem Set Question 5 states, “Thirty-six books are in the blue bin. The blue bin has 18 more books than the red bin. The yellow bin has 7 more books than the red bin. How many books are in the red bin? How many books are in the yellow bin?” (2.OA.1)
• In Module 7, Lesson 20, students independently demonstrate the use of mathematics by solving two-digit addition and subtraction problems involving length. Problem Set Question 5 states, “Solve using tape diagrams. Use a symbol for the unknown. The total length of all three sides of a triangle is 96 feet. The triangle has two sides that are the same length. One of the equal sides measures 40 feet. What is the length of the side that is not equal?” (2.MD.5)

The instructional materials for Eureka Grade 2 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. The lessons include components such as: Fluency Practice, Concept Development, and Application Problems. Conceptual understanding is addressed in Concept Development. During this time, the teacher guides students through a new concept or an extension of the previous day’s learning. Students engage in practicing procedures and fact fluency while modeling and solving these concepts. Fluency is also addressed as an independent component within most lessons. Lessons may contain an Application Problem which connects previous learning to what students are learning for the day. The program balances all three aspects of rigor in every lesson.

All three aspects of rigor are present independently throughout the program materials. For example:

• In Module 5, Lesson 18, students engage in the application of mathematics by solving a real-world problem involving subtraction and addition within 100. The Application Problem states, “Joseph collected 49 golf balls from the course. He still had 38 fewer than his friend Ethan. How many golf balls did Ethan have? If Ethan gave Joseph 24 golf balls, who had more golf balls? How many more?” (2.OA.1)
• In Module 7, Lesson 2, students develop conceptual understanding of representing a data set with up to four categories by creating a picture graph. Problem Set Question 1 states, “Use grid paper to create a picture graph below using data provided in the table. Then, answer the questions.” (2.MD.10)
• In Module 2, Lesson 8, students practice addition fluency within 100 by filling in the missing number of the given addition equation. Sprint Question 11 states, “23 cm + ___ = 100 cm” (2.NBT.5)

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

• In Module 3, Lesson 8, students develop conceptual understanding of place value and practice fluency of addition within 1000 when when writing the amount of money in expanded form. Problem Set Question 1 states, “Show each amount of money using 10 bills: $100, $10, and $1 bills. Whisper and write each amount of money in expanded form. Write the total value of each set of bills as a number bond. $136” (2.NBT.1)
• In Module 4, Lesson 25, students develop conceptual understanding of place value and practice fluency of subtraction within 100 when using place-value chips and place-value disks to solve problems. Problem Set Question 1a states, “Solve the following problem(s) using the vertical form, your place-value chart, and place-value disks. Unbundle a ten or hundred when necessary. Show your work for each problem. 72 - 49” (2.NBT.5)

The instructional materials reviewed for Eureka 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. For example:

• In Module 2, Lesson 7, the materials prompt students to analyze a measurement solution and explain why the solution was incorrect. Problem Set Question 4 states, “Christina measured Line F with quarters and Line G with pennies. Line F is about 6 quarters long. Line G is about 8 pennies long. Christina said Line G is longer because 8 is a bigger number than 6. Explain why Christina is incorrect.”
• In Module 4, Lesson 10, the materials prompt students to analyze a place-value model of an addition problem and fill in the missing addends of the equation. Homework Question 4 states, “Jamie started to solve this problem when she accidentally dropped paint on her sheet. Can you figure out what problem she was given and her answer by looking at her work?”
• In Module 5, Lesson 7, the materials prompt students to analyze a subtraction equation and the strategy used to solve it. Problem Set Question 2 states, “Circle the student work that correctly shows a strategy to solve 721 - 490. Fix the work that is incorrect by making a new drawing in the space below with a matching number sentence.”
• In Module 5, Lesson 18, the materials prompt students to analyze two subtraction equations and explain why the strategy used provided a correct solution. Problem Set Question 4 states, “Prove the students' strategy by solving both problems to check that their solutions are the same. Explain to your partner why this way works. 800 - 543 = ___, 799 - 542 = ___.”

The instructional materials reviewed for Eureka 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.

Teacher materials assist teachers in engaging students in both constructing viable arguments and analyzing the arguments of others, frequently throughout the program. The teacher materials consistently provide teachers with question prompts for student discussion and possible student responses to support that discussion. For example:

• In Module 2, Lesson 7, teachers are prompted to engage students in constructing an argument by having students measure the lengths of different objects in paperclips and discussing with a partner the reason that some of the lengths may be different. “T: Measure your straw with your paper clips. T: How long is the straw? T: Why do you think the measurements are different? Turn and talk.”
• In Module 4, Lesson 10, teachers are prompted to engage students in constructing an argument and analyze the arguments of others by having students discuss the differences and similarities between two addition models. “When students have finished, invite two volunteers to the board. One draws a model of 35 + 106 before bundling a ten. The other draws the model after bundling the ten. Encourage the remaining students to be active observers and to notice the similarities and differences between the models. T: Talk with your partner. Describe how the models are similar and different before and after bundling a ten.”
• In Module 5, Lesson 7, teachers are prompted to engage students in constructing an argument and analyzing the arguments of others by having students discuss the different strategies used to solve addition problems. “T: (Write 697 + 223) The problem is 697 + 223. Turn and talk to your partner about how you would solve this problem. T: How did Student A solve this problem? Explain to your partner what this student was thinking. What strategy did Student A use? T: Let’s look at a different way to solve this. T: What did Student B choose to do? Turn and talk. T: Which way would you do it? Discuss with your partner.”
• In Module 8, Lesson 3, teachers are prompted to engage students in constructing an argument and analyzing the arguments of others by asking students a combination of questions to facilitate a discussion about the attributes of two-dimensional shapes. “Any combination of the questions below may be used to lead the discussion. T: Look at Problems 1(b) and 2(b). How are these problems similar? How are they different? T: Look at Problems 1(d) and 2(d). Do all of your six-sided polygons look alike? What can we call a six-sided polygon? Can hexagons have five sides? Why not? T: Look closely at our polygon chart. Do you agree with the way that we sorted and named all of the polygons? If not, which do you disagree with and why?”

The instructional materials reviewed for Eureka Grade 2 meet expectations for explicitly attending to the specialized language of mathematics.

In each module, the instructional materials provide new or recently-introduced mathematical terms that will be used throughout the module. A compiled list of the terms along with their definitions is found in the Terminology tab at the beginning of each module. Each mathematical term that is introduced has an explanation, and some terms are supported with an example.

The mathematical terms that are the focus of the module are highlighted for students throughout the lessons and are reiterated at the end of most lessons. The terminology that is used in the modules is consistent with the terms in the standards.

The materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams and symbols.

• In Module 2, Lesson 6, the Notes on Multiple Means of Action and Expression states, “Couple comparative vocabulary with illustrative gestures and questions such as the following: Who is taller? Shorter? (Ask with students standing back to back.) How wide is this shoe? How long? Which shoe is longer? Which shoe is shorter? Point to visuals while speaking to highlight the corresponding vocabulary.”
• In Module 3, Lesson 20, the Notes on Multiple Means of Action and Expression states, “The complexity of moving 10 less and changing the hundreds place together can be a big jump for some students. Therefore, use the language of tens for the following problem: What is 10 less than 508? T: How many tens are in 508? S: 50 tens.”
• In Module 6, Lesson 20, the Notes on Multiple Means of Representation states, “At other times in the school day, consider relating the mathematical term even to the everyday term even by asking questions such as the following: What does it mean for kickball teams to be even? When you are playing cards with two people, why do we deal an even number? When we share our grapes with a friend, do we try to make our shares even? What does even mean then?”

The materials use precise and accurate terminology and definitions when describing mathematics and support students in using them.

• In Module 2, Lesson 4, the mathematical term meter is in bold writing within a question listed in the Student Debrief section. These questions guide teachers in leading a class discussion. “What new (or significant) math vocabulary did we learn today? (Chart student responses. Prompt students to list vocabulary from the lesson such as measure, measurement, length, height, length unit, measuring tool, meter tape, meter, and meter stick.)”
• In Module 3, Lesson 5, the materials use precise terminology of unit form and word form while using the terms when showing an example of each. The Concept Development states, “T: (Write on the board ___ hundreds ___ tens ___ ones.) Tell me the number of each unit. (Point to the number modeled in the place-value box.) S: 2 hundreds 3 tens 4 ones. T: That is called unit form. T: We can also write this number as (write on board) two hundred thirty-four. This is the word form. T. Work with your partner with your Hide Zero cards showing 234. Pull the cards apart and push them together. Read the number in unit form and in word form.”
• In Module 6, Lesson 5, the materials use accurate terminology when students create equal rows. Problem Set Question 1 states, “Circle groups of four. Then, draw the triangles into 2 equal rows.”