1st Grade - Gateway 2

The instructional materials for Math Expressions Grade 1 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’s 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 Units 1, 2, and 3, Daily Routines activities use the 120 Poster, Count by Tens and Ones Up to a Number to help students understand place value through the use of a concrete models. Students use columns with 10 blue circles in each to show a group of 10 which is later connected to a representation using “Secret Code Cards” a version of place value cards where a two-digit number is created by placing the ones digit card on top of the “0” in the tens place card. For example, the tens card is 90, and the 5 card gets placed on top of the 0 in 90 to make 95. Using place value cards in this manner reinforces the concept of digit 9 in 90 having a value of 90 (1.NBT.2).
• In Unit 2, the Learning Path, explains how students 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.” (1.NBT.2)
• In Unit 4, Lesson 9, students are introduced to adding a 1-digit number to a 2-digit number by counting on. A Number Path model, as well as 10-sticks and circles pictorial model are used as tools to build conceptual understanding. For example, the teacher says, “Make the number 38 (on the number path). Draw sticks through the 10-groups until you reach 30. How many sticks do you draw? Then make 8 dots until you reach 38. Write 38. Let’s say you want to add 5 to 38. Everyone make 5 more dots. What is the total? Did you make a new ten? Show it by drawing another 10-stick.” The Number Path model builds understanding of when we add, sometimes a new ten is created.  Students continue this practice independently in the Student Activity Book on page 170 (1.NBT.4).
• In Unit 4 Lesson 12, “What’s The Error?,” the Puzzled Penguin illustrates an incorrect comparison of two 2-digit numbers and asks students, “Am I correct?” This student discussion builds and assesses students’ conceptual understanding of comparison (1.NBT.3).
• Unit 4, Lesson 17, Math Talk, students play One Hundred Ants in small groups. Students come to an agreement on the number of ants playing the game. Students then draw a card and state how many more ants came. Students add more dots to their visual model and check to see if they are all in agreement (1.OA.8).

The instructional materials for Math Expressions Grade 1 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, Count-On Cards, Games, Homework and Remembering pages, and online resources. Examples include but are not limited to:

• Unit 1, Student Activity Book, students practice fluency within 10. Students practice equations involving addition and subtraction fluency within 5 and problems of plus and minus 1. As various strategies and pattern types are introduced, the set of equations is added to the practice pages to give students ongoing exposure and practice in order to develop fluency (1.OA.6).
• In the Student Activity Book, during the Daily Routine - Remembering, students engage in rote counting practice throughout the units. For example, students practice numeral writing to 20 (1.NBT.1).
• Periodic Fluency Checks assess addition and subtraction fluency within 10 are administered throughout the units beginning in Unit 2 after Lesson 4. For example, Volume 1, Fluency Check 6, page 122, students subtract within 10 (1.OA.6).
• Unit 4, Lessons 1-8, Daily Quick Practice Routines, Add a Ten, students use the strategy of adding a ten to quickly find sums (1.NBT.2).  
• Unit 4, Lesson 8, Student Activity Book, 167, students practice writing numbers and number words.  For example, students are given word form and asked to write the numeral and vice-versa (1.NBT.1).
• Online Resource, Poggles MX, interactive game providing addition and subtraction fluency practice.

The instructional materials for Math Expressions Grade 1 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. For example:

• Math Talks provide opportunities for students to engage in routine problems. In Unit 1, Lesson 8, students share stories with partners about each set of partners of 10. For example, “Guy: There are 9 girl cousins and 1 boy cousin in my family. Zaraya: I picked 8 red roses and 2 pink roses.”
• Math Readers provide opportunities for students to solve word problems in a different context. Unit 4’s Reader, “Comic Books for Sale” includes subtraction story problems based on the pictures in the story.
• Performance Tasks at the end of each unit, provide students with the opportunity to solve a real world task. For example, the Unit 6 Performance Task, students are directed, “To play a game, Celia and Anthony must first put their marbles into bags of 10. How many bags of 10 marbles can Celia fill? How many bags of 10 marbles can Anthony fill? If they put their marbles together first, can they fill the same total number of bags? Explain.” Non-routine practice occurs when students create their own story problems. In Unit 3, Lesson 12 students write their own story problems about sports. Students use information from a picture to write an addition or subtraction story problem. Students then share their story problems and solutions with the class.
• In Unit 6, Lesson 9, Activity 2, students are shown the following statement, “Chen has 2 marbles. His friend gives him 4 marbles. Then another friend gives him more marbles.” They are then asked to explain why Chen now has more than 6 marbles (1.OA.2).
• 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 8, Fruit Pop Party, students plan a fruit pop party. Students use the information provided on charts to determine how many ice pops they will need for the party.

The instructional materials for Math Expressions Grade 1 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 18 Fluency Checks. For example, Fluency Check 12 has students adding and subtracting within 20 (1.OA.6).
• Conceptual Understanding can be found in any of the Puzzled Penguins “What’s the Error?” questions. For example, in Unit 6, Lesson 7, the teacher writes the following story problem on the board with an incorrect solution, “There are 11 ants and 7 beetles on the log. How many more ants than beetles are on the log? 11 + 7 = 18.” Students must determine if Puzzled Penguin is correct or incorrect and identify his mistake.

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

• In Unit 1, Lesson 2, students count 1 - 10 together and hold up the appropriate number of fingers to match. Students play Five Crows in a Row by holding up five fingers and some more to represent the number the teacher says, building a conceptual understanding of five and number compositions. Students also work on procedural skill when they sequence number cards 1-10 correctly, and correctly show the number card for the number the teacher calls out. Students use the number cards which have a dot model imprinted on each, showing the number as a group of five with some more, students begin to visualize a 5-group and extra ones for each number, and the teacher writes a connecting expression “9 is 5 + 4”. Then students draw their own fives and ones representations for a given number. Lastly, students apply their understanding when they create their own story for a number as a group of five and some more. For example, “There are 8 apples. 5 of them belong to Antonio, and 3 of them belong to Megan.”
• In Unit 2, Lesson 13, students build conceptual understanding when they write an equation to match a story, and draw a pictorial model to show, “There are 7 monkeys swinging in a tree. Then 4 of them leave. Now there are 3 monkeys swinging.” Students apply this understanding to write an equation, draw a model, solve the story problem, and discuss the reasonableness of their answers. Students are invited to share their own story problems and solutions with the class.
• In Unit 6, Lesson 4, in the Student Activity Book, students count and record the data in each category to demonstrate procedural skill and fluency. They show conceptual understanding and engage in application when answering questions about the data, “How many more fish are there than ducks?” Students also create their own questions about the data.

The instructional materials reviewed for Math Expressions Grade 1 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, in Unit 6, Lesson 6, students solve 4 story problems using comparison bars. In the lesson narrative, MP8 is identified, and the teacher guidance states, “Children will notice that each problem on Student Activity Book page 261 is the same for all three quantities. In Problems 1 and 2, the difference is unknown. For Problem 3, the smaller quantity is unknown, and for Problem 4, the bigger quantity is unknown.”
• All Mathematical Practices identified in the materials provide notes for the teacher. For example, in Unit 5, Lesson 10, teacher notes include, “MP2 Reason Abstractly and Quantitatively, ask children to look at this set of equations.  Invite a volunteer to explain how to solve them. Then solve the set as a class.”
• 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 8, the MP7 suggestion directs to teachers to, “Invite children to relate the stick-and-circle drawing to the written methods. Guide them to see that they add ones and ones, tens and tens, and in this example, make a new ten.”
• 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 3, Lesson 12, the activities have students solve story problems about sports, write equation relationships, and explain their conclusions.

The instructional materials reviewed for Math Expressions Grade 1 meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

Math Talk is a critical component of the instructional materials and presents opportunities for students to use a consistent structure: Solve, Explain, Question, and Justify. Math Talk activities are identified in the Teacher Edition, and the structure is a familiar routine for students. In addition, students are presented with opportunities to use pictures to create math stories. Students need to explain how their story represents the picture. Examples include but are not limited to:

• In Unit 2, Lesson 4, Write and Solve Equations activity, students write one true and one false equation. Partners then exchange their equations and solve them. Students are asked to rewrite the equation that is not true in a way that makes it true.
• Unit 3, Lesson 12, students write an explanation supporting their analysis as to whether the statement “I can solve 7 - 4 = ___ with the equation 4 + ___ = 7” is true or false. 
• In Unit 4, Lesson 18, students share reasoning as they determine the statement “19+5 < 20 True or false.” 

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: 

• In Unit 7, Lesson 4, students discuss the Puzzled Penguin problem. “Draw the hour hand between the 12 and 1. Is Puzzled Penguin correct? No? What did Puzzled Penguin do wrong?” Puzzled Penguin showed the time 12:30, not 1:30. Students are expected to cross out the incorrect hour hand and discuss how they can help Puzzled Penguin. Responses should include that 1:30 means it is 30 minutes after 1:00. Children draw the correct hour hand halfway between the 1 and 2 to match the time on the digital clock.
• Unit 2, Lesson 6, Puzzled Penguin, the teacher writes 5 + 3 = ___ on the board. The Puzzled Penguin says “I have 5. 5, 6, 7.” Students analyze the Puzzled Penguin’s thinking to identify a counting error when he counted 5 twice. 
• In Unit 4, Lesson 12, the Puzzled Penguin writes 29 > 36. Students use their knowledge of place value and counting on to describe the error.
• In Unit 5, Lesson 11, Activity 3 - Is This Statement True?, students discuss two equations: 70 - 10 = 60 and 60 + 10 = 70. Students then determine: “These questions have the same total. True or False?” 

The instructional materials over-identify MP3. For example, in Unit 7, Lessons 8, 9, 10, and 11 all identify MP3, but there are no activities that engage students in constructing arguments or analyzing the arguments of others.

The instructional materials reviewed for Math Expressions Grade 1 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: 

• “Tiny Tumblers” are used with Math Mountains. Tiny Tumblers represent an imaginative way for children to visualize the partners of a number. If the total represented on a Math Mountain is 10, 7 Tiny Tumblers might play on the left side of the mountain and 3 play on the right side to show 7 and 3 as partners of 10.
• “Math Mountain” is 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. 
• “Break Apart Stick” is a simple stick, such as a coffee stirrer, children can use to help break apart numbers. Children lay out a certain number of counters and then use the Break-Apart Stick to separate the counters into two groups.
• “Partner House” is a pictorial representation of all the sets of partners for a total.
• “Secret Code Cards” are student cards that display the numerals 1-9, decade numbers 10-90, and 100.
• “Step Stairs” are strips of rectangles that have dots on one side and on the other side a small number that displays the total number of rectangles.
• “Sticks and Circles” are a visual representation of groups of tens as sticks and individual ones as circles.
• “Switch the Partners/Switched Partners” refers to changing the order of the partners in an addition equation.
• “New Group Above method/New Group Below method” is a strategy for multi-digit addition. The new groups are placed above the existing groups.
• “Show All Totals Method” is 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.

In addition to a Glossary in the Student Activity Book, there are Teaching Notes on vocabulary and language and Vocabulary Activities in the back of the book. For example:

• “Math Web: Make a word web for vocabulary words in a unit. Ask children to fill in the web with words, phrases, or examples that are related to the vocabulary word.”