2nd Grade - Gateway 2

The instructional materials for Math Expressions Grade 2 meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

Materials include problems and questions that develop conceptual understanding and provide opportunities for students to independently demonstrate conceptual understanding throughout the grade where called for in the standards. The Teacher Edition page vii states, “Through using objects, drawings, conceptual language, and real-world situations, students build mathematical ideas that make sense to them.”

Materials utilize MathBoards (laminated boards) for students to make their own drawings to communicate their conceptual understanding. Students have access to iTools to model conceptual understanding. Math Talk activities provide “frequent opportunities for students to explain their mathematical thinking and to ask questions of other explainers deepens their understanding of concepts.” Additionally, "What’s The Error?" activities with the Puzzled Penguin provide students the opportunity to identify errors, discuss why it is incorrect, and how to correct it. Finally, Learning Paths, found in each unit, explain how students will build understanding of concepts throughout the unit.

Examples include but are not limited to:

• In Unit 2 Lessons 2, 3, and 4, students use math drawings (Quick Hundreds, Quick Tens, and circles), expanded form, Secret Code place value cards, and numerals to represent numbers to 200. In Unit 6, Lessons 1-4, this activity is extended to build numbers to 1,000 using the same representation models (2.NBT.1).
• In Unit 2, Lesson 9, “What’s The Error?,” the Puzzled Penguin gives students double-digit addition problems with incorrect answers. Students make proof drawings to show the mistakes Puzzled Penguin made. Students also explain the method the Puzzled Penguin used to solve incorrectly in exercise 4 (2.NBT.6).
• The Unit 2, Learning Path, explains how the materials will build understanding of place value. “In this unit, children work with place value, representing numbers in different ways, and comparing numbers. They add two, three, or four 2-digit numbers, sometimes resulting in new tens or new hundreds with sums to 200” (2.NBT.4).

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

The instructional materials develop and provide independent opportunities for procedural skill and fluency throughout the grade-level. Math Expressions includes a Path to Fluency for each grade level.  “This plan provides targeted practice in the Student Activity Books, Teacher Editions, Teacher Resource Books, Math Activity Centers, as well as Fluency Checks in the Student Activity Books” (TE I12). In Grade 1 the fluency plan contains practice problems in the Student Activity Book identified by a Path to Fluency icon, Fluency Checks, Quick Practices, Daily Routines, Math Mountain Cards, Strategy Cards, Games, Homework and Remembering pages, and online resources. Examples include but are not limited to:

• Practice for fluency within 20 is included in many Student Activity Book pages even in units where the emphasis is not number and operations, such as Unit 3, fluency appears in the Remembering spiral review pages (2.OA.2).
• Unit 4, Lesson 14, Student Activity Book, 231, students solve addition and subtraction problems within 100 (2.OA.1).
• Practice for fluency within 100 appears in many Homework and Remembering pages. For example, Unit 4, Lesson 14, Homework Practice provides problems for students to add and subtract within 100 (2.NBT.5).
• Unit 7, in Lessons 7-15, Daily Quick Practice Routines, “Unscramble the Hundreds, Tens, and Ones.”
• The Remembering Workbook pages spiral back to prior work and help build fluency. For example, in Unit 7, Lesson 10, Remembering 7-10, students add within ten (2.OA.2).
• Periodic Fluency Checks assessing addition and subtraction fluency within 10, 20, and 100 are administered throughout the units. For example, Volume 1, Fluency Check 2, page 58, students add and subtract a one-digit and two-digit number (2.OA.2).
• Online Resource, Poggles MX, interactive game providing addition and subtraction fluency practice.

The instructional materials for Math Expressions 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 and to independently demonstrate the use of mathematics flexibly in a variety of contexts. Opportunities for contextual problem solving and non-routine problems are found in Math Talks. Students are provided real-world problem scenarios throughout each lesson. Performance Tasks at the end of each unit, provide students the opportunity to solve real world situations. Also, Math Readers embed math learning in a context appropriate story. Finally, online games provide problem solving practice.

Examples include, but are not limited to:

• In Unit 2, Lesson 6, students solve word problems with a new ten. “Mr. Green puts 56 red peppers in the vegetable bin. Mrs. Green puts 28 yellow peppers in the bin. How many peppers do they put in the bin altogether?” Students are prompted to explain a method for finding the total number of peppers (2.OA.1).
• In Unit 4, Lesson 23, students read a two-paragraph description of a Stegosaurus and find that the rear foot of a Stegosaurus was about 35 centimeters long. Students then make a paper strip that is 35 cm long and measure their own foot and compare the two measurements to determine the difference. They continue by measuring strides of the Stegosaurus and comparing to their own strides (2.MD.5).
• Performance Tasks at the end of each unit provide students with the opportunity to solve a real world task. For example, Unit 4 Performance Task, students read, “Jacob has 2 quarters and 5 nickels. Cassie has 5 dimes and 3 nickels. Who has enough money to buy an apple. Explain.” (2.MD.8)
• Math Talks provide opportunities for students to engage in routine problems. In Unit 4, Lesson 4, students are asked to write a word problem to match the given subtraction equation “63 - 35 = ___ and 92 - 54 = ____” and to use any method to solve the problem. Students share their solutions and discussion follows.
• Students create their own story problems. In Unit 5, Lesson 8, students write 2-step word problems using data they previously collected on students’ favorite colors.
• Math Readers provide opportunities for students to solve word problems in a different context. Unit 6’s Reader, “The If Game” includes word problems with unknown addends based on the pictures in the story.
• Online activities provide additional opportunities for students to apply mathematical knowledge and skills to real-world contexts. For example, in the In-Depth Inquiry Based Task for Unit 6, Ticket Totals, students plan how six students will each use their 999 tickets at the carnival. Students use the information provided about the cost of food, games and activities to create six different plans.

The instructional materials for Math Expressions Grade 2 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately.

The instructional materials attend to conceptual understanding, procedural skill and fluency, and application independently to develop students’ mathematical understanding of a single topic/unit of study throughout the grade level materials. The three aspects of rigor can be found in the Daily Routines, Quick Practice, Math Talks, Fluency Checks, Puzzled Penguin, Lessons, and Homework. For example,

• Fluency can be found in any of the 20 Fluency Checks. For example, Fluency Check 11 has students adding and subtracting within 20 (2.OA.2).
• Conceptual Understanding can be found in any of the Puzzled Penguins “What’s the Error?” questions. For example, in Unit 5, Lesson 2, students are shown a clock and the Puzzled Penguin’s time. Puzzled Penguin asks if he has the right time and students must help him tell the correct time.
• Application can be found in Math Talk in Action activities. In Unit 6, Lesson 5, students work in groups of 3 to 5 to solve Problem 4, “A camping club buys some raisins. They buy 3 cartons that have 100 bags each. They also have 24 bags left from their last trip. How many bags of raisins does the club have?” Students work together using Secret Code Cards or proof drawings to find a solution.

Examples where the three aspects of rigor are treated together include, but are not limited to:

• In Unit 2, Lesson 1, students engage with fluency and conceptual understanding as they write the numbers 1-110 on their math boards, discuss and create numbers greater than 100 using Secret Code cards to develop place value understanding, and write the numbers 101-200 in the Student Activity Book.
• In Unit 4, Lesson 3, students engage with all three aspects of rigor. In the Quick Practice Teen Subtraction Flash, students practice subtraction fluency. Students develop conceptual understanding as they discuss various methods of finding unknown addends and relate to equations, for example, “73 = 34 +___” to break apart a sum. They relate addends with a sum of 100 to addends with a sum of 10, and build the concept 10 is made up of seven ones and three ones, but 100 is made up of seven tens and three tens. Students begin to explore other addends with a sum of 100. Students use their conceptual understanding to solve problems involving real-world contexts for subtracting a 2-digit number from 100, and solve a series of real-word problems involving 100. In the Student Activity Book, students engage in additional procedural skill practice subtracting from 100.
• In Unit 4, Lesson 4, students solve two word problems using any method, “There were 100 puddles on the playground. 27 of them dried up. How many puddles are left on the playground.” Students apply their understanding of subtraction to write word problems to illustrate, “76 - 48 = _____.”

The instructional materials reviewed for Math Expressions Grade 2 meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade-level.

Materials clearly identify Mathematical Practices being used in each lesson and are embedded in the content to enrich the mathematics. Instructions are provided for teachers on how to implement Mathematical Practices within the lesson. No Mathematical Practice is under or over used in the materials. While Mathematical Practices are not identified in the student materials, the Teacher Edition does provide highlighted narratives for Mathematical Practice activities found in the Student Activity Book.

Examples include, but are not limited to:

• The Teacher Edition provides guidance on how to implement the Mathematical Practices in the Student Activity Book. For example, Unit 3, Lesson 6, Student Activity Book page 163, students estimate and measure objects. In the lesson narrative, MP5 is identified and the teacher narrative states, “make sure each small group has a centimeter ruler and a meter stick to use in exercises 4-7. In each exercise, children will select the tool to use, then measure the classroom object.”
• All Mathematical Practices identified in the materials provide notes for the teacher. For example, in Unit 3, Lesson 7, Activity 2, the notes state, “MP7 Look for Structure. Hold up a yardstick and ask children to identify what it is. If a child calls it a meter stick, show the class that a yardstick is a bit shorter than a meter stick and that it has inch units instead of centimeter units.”
• The Overview of every unit contains “Common Core State Standards for Mathematical Practices in this Unit”.  A table is provided that lists every Mathematical Practice along with corresponding lessons where that practice is embedded.
• The “Using the Common Core Standards for Mathematical Practice” section contains a description of the Mathematical Practice along with examples of where to find it within the unit.  For example, in Unit 6, the MP1, “Encourage children to suggest as many ways to subtract as possible. See the next page for sample methods.”
• Focus on the Mathematical Practices lessons are the last lesson in each unit. The lessons engage students in all eight practices, however, the practices are often over-identified. For example, in Unit 2 Lesson, 15, the activities involve recycling word problems, comparing methods and discussing solutions, money story problems, and patterns.

The instructional materials reviewed for Math Expressions Grade 2 partially meets expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

There are lessons throughout the series that prompt students to make comparisons, or determine whether statements are true or false. Some of these lessons require students to construct arguments, and some of these lessons require students to provide an explanation. Examples include:

• In Unit 2, Lesson 15, Activity 3, Justify Conclusions about Money, students are presented with the following statement, “A bag holds 4 nickels, 4 pennies, and 4 dimes. If 3 coins are taken out of the bag, the value of the 3 coins can never be more than 16 cents.” Students must decide if the statement is true or false, and then volunteers share their positions and explanations with the class. 
• Unit 6, Lesson 6, Math Talk, students solve and explain word problems. Students explain why they solved word problems in a particular way, and engage in Math Talk about good types of questions to ask for justification. For example, “How did you know you were supposed to add? Why did you put the 7 under the 8? 7 +8 is 15. What happened to the 10 in 15? I see only 5.” (2.OA.1). Students need to construct arguments to justify why.
• Stretch Your Thinking problems found in the Student Activity Book and Homework and Remembering pages often require students to take a position and explain their thinking. For example, in Unit 4, Lesson 14, Stretch Your Thinking, the problem states, “Allison solved this problem. ‘46 + 17 = 53’ Is she correct? If not, explain and solve.” 

Puzzled Penguin problems are found throughout the materials and provide students an opportunity to correct errors in the penguin’s work. These tasks focus on error analysis, and many of the errors presented are procedural. Examples of Puzzled Penguin problems include:

• Unit 2, Lesson 5, Puzzled Penguin, What’s the Error, “I know that 9 is greater than 6. Did I make a mistake?” The Puzzled Penguin wrote: “149 > 176” Problem 17: Make a proof drawing to help Puzzled Penguin. Write <, >, or =.” 
• Unit 6, Lesson 3, Puzzled Penguin, What’s the Error, Compare Numbers, “I know that 3 is greater than 2. Did I make a mistake? 35 > 245.” Problem 11: “Draw boxes, sticks and circles to help Puzzled Penguin.” Students do not need to explain the error, simply show the correct comparison.

The instructional materials reviewed for Math Expressions Grade 2 meet expectations that materials use accurate mathematical terminology.

There are instances where materials use “A classroom research-based term developed for Math Expressions.” These terms are used in the Student Activity Book. Examples include but are not limited to: 

• Math Mountain: a visual representation of the partners and total of a number. The total appears at the top, and the two partners that are added to produce the total are below to the right and left.
• Secret Code Cards: student cards that display the numerals 1-9, decade numbers 10-90, and 100.
• New Group Above method: a strategy for multi-digit addition. The new groups are placed above the existing groups.
• New Group Below method: a strategy for multi-digit addition. The new groups are placed below the existing groups.
• Show All Totals method: a strategy for multi-digit addition. Add the tens column together and place the total under the problem. Next, add the ones column together and place the total under the tens total. Add up the tens and ones totals to find the answer.
• Adding Up Method for Subtraction: a method of finding an unknown addend in which children add up from the known addend until they reach the total.
• Ungroup First Method: a method of subtraction in which children check each place to see if they need to ungroup in order to subtract. Children then complete all necessary ungrouping before they subtract.

In addition to a Glossary in the Student Activity Book, there are Teaching Notes on vocabulary and language. These are found in Unit 1 as students are introduced to mathematical language. For example:

• Unit 1, Lesson 2, Teaching Notes, Vocabulary: “In the Student Activity Book, at the beginning of each unit, there are Vocabulary Cards that can be cut out. The cards can be used in the classroom or sent home for children to review the math vocabulary in Math Expressions. These cards can also be used with the Vocabulary Activities found in the back of this Teacher Edition.”