2nd Grade - Gateway 1

The instructional materials reviewed for Math Expressions Grade 2 meet expectations that they assess grade-level content.

The materials provide an Assessment Guide for summative assessments. Within the Assessment guide there is a Prerequisite Skills Inventory Test and Beginning of Year Test. Each of the assessments contain a correlations document between standard and question number. Within each Unit, there is a Test Form A and Test Form B with a correlations document, and a Performance Assessment with a grading rubric and sample student responses. A Middle of Year Test is located at the end of Unit 4 and an End of Year Test at the end of Unit 8, contain correlation documents. Additionally, the Student Activity Book contains Unit Quick Quizzes and Unit Reviews.

Assessment items containing grade level content include, but are not limited to:

• Unit 3, Form A, Question 7 states, “Estimate the length of the key in inches. Write your estimate. Measure the key to the nearest inch. Circle the actual measure.” (2.MD.1 and 2.MD.3)
• Unit 4, Form A, Question 9 states, “Jasmine, Marcus, and Ian bring cans of food to school. Jasmine brings 22 cans. Marcus brings 13 more cans than Jasmine. Ian brings 15 fewer cans than Marcus. How many cans does Ian bring?” (2.OA.1)
• Unit 7, Form A, Question 6 states, “Draw lines in each shape to make equal shares.” Students are given squares and circles and partition into halves, thirds, and fourths. (2.G.3)

Some above grade-level assessment item are included, but could be modified or omitted without a significant impact on the underlying structure of the instructional materials. Assessment items containing above grade-level content include:

• Beginning of Year Test, Question 25 states, “Write the total amount of money. Jaden has 1 dollar, 1 quarter, 2 dimes, and 3 pennies.” The answer is in decimal format (4.MD.2).

The instructional materials reviewed for Math Expressions Grade 2 meet expectations for spending a majority of instructional time on major work of the grade.

• The approximate number of units devoted to major work of the grade (including assessments and supporting work connected to the major work) is 7 out of 7, 100%.
• The number of lessons devoted to major work of the grade (including assessments and supporting work connected to the major work) is 96 out of 99, approximately 97%.
• The number of days devoted to major work (including assessments and supporting work connected to the major work) is 148 out of 153, approximately 97%.

A number of days level analysis is most representative of the instructional materials because some lessons require two days of instruction. As a result, approximately 97% of the instructional materials focus on major work of the grade.

The instructional materials reviewed for Math Expressions Grade 2 meet expectations that supporting work enhances focus and coherence simultaneously by engaging students in the major work of the grade.

Supporting standards/clusters are connected to the major work of the grade. In the Teacher Edition, the Day at a Glance for each lesson contains a bubble identifying the mathematical content standards and mathematical practices covered in the lesson, and can include connections between supporting and major work.

Examples include, but are not limited to:

• In Unit 3, Lesson 6, students estimate the heights of students, measure actual heights of students, and use the data to create a line plot. This connects supporting standard represent and interpret data (2.MD.9) to the major work of measure and estimate length (2.MD.1 and 2.MD.3).
• In Unit 4, Lesson 10, students view five items of different prices and choose the correctly priced item to complete the problem, “I have 162 cents in my pocket. I buy the _________. I have ________ cents left.” This connects supporting standard work with time and money (2.MD.8) to the major work of add and subtract within 1000 (2.NBT.7).
• In Unit 5, Lesson 3, students work in a group and are given 8 pennies (Maya), 5 pennies (Anthony), and 2 pennies (Sona). As a group, students create a graph on their MathBoards. Students answer the following questions, “Who has the most pennies? Who has the fewest pennies? How many more pennies does Maya have than Anthony? How many fewer pennies does Anthony have than Maya?” This connects the supporting standard draw a picture and bar graph (2.MD.10) to the major work of use addition and subtraction within 100 to solve problems (2.OA.1).

Instructional materials for Math Expressions Grade 2 meet expectations that the amount of content designated for one grade-level is viable for one year.

As designed, the instructional materials can be completed in 142 days. The Pacing Guide can be found on page I18 in the Teacher Edition. The suggested amount of time and expectations for teachers and students of the materials are viable for one school year as written and would not require significant modifications.

• There are 99 lessons, and most lessons require one day of instruction.
• The Pacing Guide notes lessons that may take two days, but this is not noted in the Day at a Glance for each lesson.
• All seven units designate two days for Unit Assessments.
• Unit 1 designates one day for the Prerequisite Skills Inventory Test.
• Units designate 2-4 days for Quick Quizzes/Fluency Checks in the Students Activity Book for each Big Idea.

Materials indicate a teacher will need to spend more than 60 minutes providing math instruction daily. Teaching the lesson is comprised of several activities with an estimated time equating to 60 minutes.

The instructional materials for Math Expressions Grade 2 meet expectations for the materials being consistent with the progressions in the Standards.

The instructional materials clearly identify content from prior and future grade-levels and use it to support the progressions of the grade-level standards. The Teacher Edition (page I19) states, “Standards are designed around coherent progressions from grade to grade. Standards are not stand alone events; they form cohesive links to major work within and across grade levels.” In the Teacher’s Edition (page I12) there is a chart, A Path to Fluency: Kindergarten through Grade 6 to “help students achieve fluency as described in the Common Core State Standards.” Each unit contains a Math Background section referencing the progressions in text boxes, and how they relate to work of the grade. Each Unit Overview section lays out how the progression of the grade moves towards standards in future grades. Examples of grade level progressions include, but are not limited to:

• Teacher Edition, Unit 4, Unit Overview, the text box states, “In Grade 2, children will compute sums and differences within 1000 using place value. In Grade 3, children will build or maintain fluency for addition and subtraction within 1000.”
• Path to Fluency, TE page I12, the chart notes in Grade 2 students, “Fluently add and subtract within 20 using mental strategies. By end of Grade 2 know from memory all sums of two one-digit numbers (2.OA.2).” In Grade 3 students, “Fluently multiply and divide within 100. By the end of Grade 3, know from memory all products of two one-digit numbers.” (3.OA.7)
• Unit 2, Lesson 7, Addition-Show All Totals Method; and Lesson 8, Addition-New Groups Below Method, students progress from drawing models to add 2-digit numbers to connecting the models to two new methods including, “show all totals” and the “new groups below” methods. Both methods are consistent with the Grade 2 Progressions Document. These methods are continued and applied to 3-digit numbers in Unit 6.

The instructional materials attend to the full intent of the grade-level standards by giving all students extensive work with grade-level problems. The structure of the Math Expressions program provides a variety of learning opportunities for students to master the grade-level curriculum and provide extensive work with grade-level problems. Knead Knowledge through Practice ensures students gain understanding and fluency with desired methods. Daily Routines, Quick Practice, Student Practice pages, Homework practice, Remembering pages, and Math Activity Centers support the classroom learning in every lesson. Some instances are evident where the full intent of the standard is not met, and there are some lessons that would need to be omitted as they attend to future grade standards.

Examples of materials meeting full intent of the standards include, but are not limited to:

• Unit 1, Lesson 20 includes a Math Activity Center for on-level, challenge, and intervention work. On-Level Resources include Activity Card 1-20 which has students solve a word problem containing extra information. Challenge Resources include Activity Card 1-20 which has students solve a two-step word problem. Students needing intervention complete Activity Card 1-20 which has students solve a word problem (2.OA.1).
• Unit 4, Daily Routines, the student leader points to a projected group of pennies and asks the students, “Each group of pennies shows the number of pennies in one dollar. How many pennies are in each dollar? How do you know? We are going to count these hundreds to find out how many pennies in all.” (2.NBT.2)
• Unit 6, Lessons 3 through 8, Quick Practice, students use place value to add 100 to or subtract 100 from any 3-digit number. For example, “329 + 100 = and 957 - 100 =.” (2.NBT.8)

Instances where full intent of the standard is not met:

• Two lessons in Unit 7 (Lessons 3 and 5) address standard 2.MD.6, relate addition and subtraction to length. No additional work to this standard is found in Daily Routines or Quick Practice.

Some lessons would need to be omitted as they attend to future grade standards. For example:

• In Unit 2, Lesson 11, students are introduced to dollar notation with a decimal point and are expected to write money amounts using decimals. According to the progressions, decimals should not be introduced until Grade 4 (4.NF.6) and this is not explicitly stated as future content. Subtraction problems with decimal notation are also found in the Student Activity Book.
• In Unit 6, Lesson 6, students solve 3-digit addition computations with a variety of strategies. The “New Groups Above Method” is the standard algorithm (4.NBT.4) and is not clearly identified as future grade level content.

The instructional materials relate grade-level concepts explicitly to prior knowledge from earlier grades. Learning paths are identified at the beginning of each unit that relate grade-level concepts to prior knowledge from earlier lessons. Examples where content is connected to prior grades includes, but is not limited to:

• Teacher Edition, Unit 2, Learning Path in the Common Core Standards, the author states, “Relate Addition and Subtraction: Diagrams used in Grade 1 to show how quantities in the situation are related continue to be useful in Grade 2, and students continue to relate the diagrams to situation equations. Such relating helps students rewrite a situation equation like, __ - 38 = 49 as 49 + 38 = ___.”
• Teacher Edition, Unit 3, Learning Path for Measurement, the author states, “Grade 2 length measurement with rulers begins the formal use of measurement tools that use a length scale in which numbers indicate the total number of unit lengths from the zero of the scale. Such length scales are number-line diagrams that are also used as bar graph scales and as scales on a line plot (dot plot).” The author cites two National Research Council reports which “recommend that work with such scales not begin before Grade 2 because of the conceptual difficulties described above in using such scales: Many students count the numerals on the scale rather than seeing and counting the length units. To overcome this difficulty, Math Expressions emphasizes marking and counting the length units. Initially, children draw marks to show the length units on things they are measuring, and then they count these length units. Through such activities, they mathematize length as consisting of small units of the same size that compose the total length....[D]rawing and then counting of the length units also helps children extend the work they did in earlier grades with seeing different kinds of measurement units for different kinds of measures. This is the basis for future work with other units: square units compose and are counted to make an area (initially of rectangles), and cubes compose and are counted to make volumes (initially of rectangular prisms).”
• Teacher Edition, Unit 4, Relating Addends to Subtraction, Lessons 3 and 4, the author states, “children use their mathematical understanding of the relationship between finding an unknown addend and subtracting. They may use drawings or methods that they learned in Grade 1. These lessons prepare the children for the more formal introduction of methods of subtracting in Lesson 5.”

The instructional materials for Math Expressions Grade 2 meet expectations that materials foster coherence through connections at a single grade, where appropriate and required by the Standards.

The instructional materials foster coherence through connections at a single grade, where appropriate and required by the Standards. Overall, the materials include learning objectives visibly shaped by CCSSM cluster headings, and provide problems and activities connecting two or more clusters in a domain or two or more domains when the connections are natural and important. ​

Examples where materials include learning objectives visibly shaped by CCSSM cluster headings include, but are not limited to:

• Each unit is broken into several “Big Ideas” shaped by the CCSSM cluster headings. For example, Unit 6, 3-digit Addition and Subtraction, contains four Big Ideas:
• Big Idea 1: Understanding Numbers to 1,000 is shaped by 2.NBT.A: Understand place value.
• Big Idea 2: Adding to 1,000 is shaped by 2.NBT.B: Use place value understanding and properties of operations to add and subtract.
• Big Idea 3: 3-Digit Subtraction is shaped by 2.NBT.B: Use place value understanding and properties of operations to add and subtract.
• Big Idea 4: 3-Digit Addition and Subtraction is shaped by 2.NBT.B: Use place value understanding and properties of operations to add and subtract.
• The Teacher Edition Planning Chart for each unit identifies the Big Ideas, CCSSM Domains, Clusters, and Standards for each lesson.
• Unit 3, Big Idea 2, is designed around Grade 2 cluster 2.MD.A: Measure and estimate lengths in standard units. For example, in Lesson 7, students estimate and measure to the nearest inch. Students also estimate to the nearest foot (2.MD.1 and 2.MD.3).

Examples of connections between clusters and/or domains include, but are not limited to:

• Unit 7, Lesson 7, connects Operations and Algebraic Thinking Clusters 2.OA.B: Add and Subtract within 20, and 2.OA.C: Work With Equal Groups of Objects to Gain foundations for multiplication. Students practice finding “unknown partners” as they solve, add to situations with with the change unknown, and put together/take apart problems with one addend unknown.
• Unit 5, Lessons 4 through 10, connects 2.MD.B: Relate Addition and Subtraction to Length, to 2.OA.A: Represent and Solve Problems Involving Addition and Subtraction. Students have opportunities throughout the unit to use the data from graphs to solve word problems. For example, Homework and Remembering page 146, students use the picture graph of the number of crayons to answer the problem, “Five of Yoko’s crayons are new. The rest of her crayons are old. How many are old?”
• Unit 7, Lesson 3, connects 2.NBT.B: Use Place Value Understanding and Properties of Operations to Add and Subtract, to 2.MD.B: Relate Addition and Subtraction to Length. Students solve word problems involving length by adding two- and three-digit numbers together.