2nd Grade - Gateway 1

The instructional materials reviewed for Achievement First Mathematics Grade 2 partially meet expectations for being consistent with the progressions in the Standards. Overall, the materials do not provide all students with extensive work on grade-level problems. The instructional materials develop according to the grade-by-grade progressions in the Standards. Content from future grades are not clearly identified and does not relate to the grade-level work. The instructional materials relate grade-level concepts explicitly to prior knowledge from earlier units. Within the overview for each unit, there is “Identify the Narrative,” which provides a description of connections to concepts in prior and future grade levels.

The lessons follow a workshop model, including a math lesson, math stories, and calendar/fluency. Most lessons do not provide enough opportunity or resources for students to independently demonstrate mastery. The lessons include teacher-directed problems that the class solves together. Math stories are intended to occur every day there is a lesson; however there are insufficient math stories for each lesson day. In addition, many practice workbook pages are repeated across multiple units.

The materials develop according to the grade-by-grade progressions in the Standards. The Unit Overview documents contain an Identify the Narrative component that looks back at previous content or grade level standards and looks ahead to content taught in future grades. In addition, the Linking section includes connections taught in future grades, units, or lessons. Evidence of prior and future grade-level work supporting the progressions in the standards is identified. Examples include: 

• In Unit 2, Addition and Subtraction to 100 Unit Overview, Identify The Narrative, Linking states, “Looking ahead to 3rd and 4th grade, 2nd grade mastery of this unit is vital considering 3rd math has a small amount of time dedicated to addition and subtraction where they are meant to build stronger fluency. This is also means 2nd grade success here is very important for the 4th grade math. Expanded notation sets up scholars to use the standard algorithm with ease.” 
• In Unit 3, Story Problem Unit Overview, Identify the Narrative, Linking states, “Scholars need to master the addition and subtraction story problem types with 2 steps to be ready for third grade where they represent and solve story problems with multiplication, division, elapsed time, and rounding. Additionally, students need to be proficient in independently going through the story problem protocol so that they are able to make sense of all story problem types. The work students do in this unit will carry over into math stories in second grade as they continue to practice all addition and subtraction problem types, including 2-step, and begin to explore problems with equal groups. In third grade, scholars begin multiplication and division, and they continue to solve equal groups/array story problems within 100. Scholars also do 2-step story problems with all four operations. By the end of fourth grade, scholars have mastered all addition, subtraction, multiplication, and division story problem types (including multiplicative compare) with all whole numbers for addition and subtraction and two-digit multipliers and one-digit divisors for multiplication and division. Furthermore, they master multi-step problems with mixed operations, including measurement contexts.” 
• In Unit 4, Data Unit Overview, Identify the Narrative, Linking states, “In 3rd grade, students draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. In 2nd grade, each picture on a pictograph or square on a bar graph stood for one object/item. In 3rd grade, students will draw pictographs and bar graphs where the picture/square stands for more than one object.”
• In Unit 7, Addition and Subtraction within 1000 Unit Overview, Identify the Narrative, Linking states, “In third grade, students are no longer using flats, sticks, and dots or place value blocks as a strategy to solve. Students are exclusively using the more abstract place value strategies to solve 3-digit addition and subtraction. Students are fluently using expanded notation, number line and other strategies to solve. This is all done in preparation for the standard algorithm which is taught in fourth grade.” 
• In Unit 9, Geometry--Fractions Unit Overview, Identify the Narrative, Linking states, “In 3rd grade, students work with fractions that have numerators that are more than one. They also work with simple equivalent fractions and comparing fractions with the same numerator or same denominator. When the fractions have the same numerator we can think about the size of the unit fraction. For example, $$\frac{2}{3}$$ is bigger than $$\frac{2}{6}$$ because as the denominator gets bigger, the size of the part/fraction gets smaller. 2 pieces of an object partitioned into thirds are larger than 2 pieces of a same-sized object partitioned into sixths because the whole is divided into fewer pieces.”

Overall, the materials do not provide all students with extensive work on grade-level problems. The majority of the lessons implement 45 minutes of math workshop with a whole group introduction, workshop in pairs or small groups, mid-workshop interruption, whole group discussion, and closing with an exit slip. As it is unclear if students are working together or individually, workshop lessons may not provide enough opportunity for students to independently demonstrate mastery. The Guide to Implementing AF, Grade 2, describes the workshop component as, “Collaborative processing time to continue to develop understanding of prioritized concept and strategy.” The lessons include a teacher-directed introduction to the workshop “game” and follows up with students tasked to participate in the “game.” Most lessons include an exit ticket with one or two questions for the students to complete individually and some lessons include Independent Practice problems. 

Beyond the lesson component of the math time, the Guide to Implementing AF Math, Grade 2 suggests 15 minutes of daily calendar and practice. Each unit indicates the Grade 2 Practice Workbook pages to be implemented during this time. However, the practice workbook pages contain a limited number of practice items and are recommended to be used repeatedly in different units. On Fridays, students have 25 minutes of Cumulative Review problems “to facilitate the making of connections and build fluency or solidify understandings of the skills and concepts students have acquired throughout the week and to strategically review concepts.” Exit Tickets, Independent Practice, and Cumulative Review provide limited independent practice on some grade level standards. As a result of the limited number of opportunities to practice grade-level standards, the materials do not give students extensive work with grade-level problems. 

Examples where the full intent of a standard is not met and/or extensive work is not provided include:

• In Unit 1, Lesson 13, Exit Ticket, students engage in 2.MD.2 as they measure the length of an object twice, using length units of different lengths for two measurements. This standard is only addressed in this lesson and extensive practice is not given. Problem 1 states, “Measure the length of the book to the nearest inch and to the nearest centimeter.” 
• In Unit 1, Lesson 14, Workshop Worksheet, students engage with 2.MD.6 as they represent whole numbers on a number line and solve addition and subtraction equations within 100 on a number line diagram.  The full intent of this standard is not met as students measure the lengths of objects using a ruler and are not provided the opportunity to create their own ruler. In addition, this standard is only addressed in this lesson. Problem 2  states, “Peter was measuring his shoe using the centimeter ruler below. About how long is Peter’s shoe?” Students are given a picture of a shoe and a centimeter ruler. 
• In Units 4 and 5, Practice Workbook B, students engage in 2.NBT.6 where they add up to four two-digit numbers. However, there are no lessons or instruction addressing 2.NBT.6, and, therefore, the full intent of the standard is not met as they are not provided instruction on using strategies based on place value and properties of operations as the standard requires. Students are not provided with extensive work as the only materials provided for 2.NBT.6 are 24 Practice Workbook problems and six Cumulative Review problems. Problem 19 states, “What are two ways that you can make a total of 50 using 3 addends?”  
• In Unit 8, Lesson 6, Workshop and Exit Ticket, students engage with the standard 2.MD.3 by the use of mental benchmarks of a meter and a centimeter to estimate objects in the classroom. As this is the only lesson addressing 2.MD.3 where they are expected to estimate using the units of inches, feet, centimeters, and meters, the full intent of the standard is not met since students do not estimate using the units of inches and feet, nor do students have the opportunity to engage in the extensive work with the standard. During Workshop, they are given a worksheet with a variety of estimation problems to solve. To close the lesson, the students are given an Exit Ticket with two problems. Exit Ticket Problem 1 states, “Circle the most reasonable estimate for each object. a. Length of an eraser - 5 cm or 1 m.”
• In Unit 10, Lesson 2, Workshop and Independent Practice Worksheet, students engage with 2.G.1 as they recognize and draw shapes having specified attributes such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, hexagons, and cubes. The full intent of this standard is not met as none of the lessons addressing 2.G.1 include a cube, as required by the standard. During Workshop, students are given a description of shape attributes and asked to draw and name the shape. Independent Practice Problem 1 states, “A flat closed shape with 6 sides and 6 angles. What shape did you build?”

The instructional materials do not clearly identify content from prior and future grade-levels, and, as a result, do not give students extensive work with grade level standards. Examples include:

• In Unit 5, Lesson 2, Independent Practice Worksheet aligns with the intent to engage students with 2.MD.8 as they solve word problems involving dollar bills, quarters, nickels, and pennies, using $ and $$\cancel{C}$$ symbols appropriately. However, as this lesson requires subtraction of decimals, it is more accurately aligned to 4.MD.2 where students use the four operations to solve word problems involving distances, intervals of time, liquid volume, masses of objects, and money, including problems involving simple fractions or decimals. Problem 9 states, “Jordan emptied his pocket and found this: (Images of bills and coins adding to $22.87 are shown) a. How much money does Jordan have? b. Jordan wants to buy a toy that costs $23.50. Can he afford it? Why or why not?”
• In Unit 6, Lesson 7, Independent Practice Worksheet aligns with the intent to engage students with 2.NBT.1 as they are to understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones, 2.NBT.2 as they count within 1000; skip-count by 5s, 10s, and 100s, and 2.NBT.3 as they read and write numbers to 1000 using base-ten numerals, number names, and expanded form. However, this is more accurately aligned to 4.NBT.1 where students are to recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. Problem 3 states, “Julisa says that 4 hundreds, 13 tens and 15 ones is the same as 5 hundreds, 4 tens and 5 ones. Is she correct? Why or why not? Use the space in the box to prove your thinking.”
• In Unit 8, Lesson 9, Independent Practice aligns with the intent to engage students with 2.OA.3 where students determine whether a group of objects (up to 20) has an odd or even number of members (e.g., by pairing objects or counting them by 2s) and write an equation to express an even number as a sum of two equal addends. Of the 26 Independent Practice problems provided in Unit 8, 23 are above 20. In addition, the Practice Workbook contains 33 problems, 18 of which extend beyond 20. None of the problems within the Independent Practice or the Practice Workbook require that students write an equation to express an even number as a sum of two equal addends. As it extends beyond 20 and requires addition beyond two equal addends, it is more accurately aligned to 3.OA.9 where students are to identify arithmetic patterns (including patterns in the addition table or multiplication table) and explain them using properties of operations. Problem 1 states, “Jezdon has 17 cards. Angel has 12 pokemon cards. Jezdon says if they put them together they will have an even number of pokemon cards. Is he right? How do you know? Draw a picture and use words to explain your thinking.” 
• In Unit 9, Lesson 2, Workshop Worksheet aligns with the intent to engage students with 2.G.3 as they are to partition circles and rectangles into two, three, or four equal shares. On the Workshop Worksheet, students are provided a table with the headings: “Shape to Make,” “Equal Parts,” and “Build & Record.” While the standard, 2.G.3, states that students are to partition circles and rectangles into two, three, or four equal shares, the table includes a hexagon, trapezoid, and rhombus to partition into halves and thirds. 
• In Unit 9, Lesson 8, Independent Practice Worksheet aligns with the intent to engage students with 2.G.3 as they are to partition circles and rectangles into two, three, or four equal shares; describe the shares using the words halves, thirds, half of, a third of, etc.; describe the whole as two halves, three thirds, or four fourths; and recognize that equal shares of identical wholes need not have the same shape. The lesson content is more accurately aligned to 3.NF.3d where students are to compare two fractions with the same numerator or the same denominator by reasoning about their size. Problem 2 states, “Enrique cut his pie in half and ate one piece. Melvin cut his pie into fourths and ate one piece. A. Show each pie. Shade the fraction each boy ate. B. Enrique says he ate more pie than Melvin. Is he correct? Why or why not?” 

The Unit Overview supports the progression of Second Grade standards by explicitly stating connections between prior grades and current grade level work. Each Unit Overview contains an Identify the Narrative component that identifies connections to what students learned before this Second Grade Unit and/or concepts previously learned in previous grade levels. Each Unit Overview also contains an Identify Desired Results: Identify the Standards section that makes connections to supporting standards learned prior to the unit. In addition, some lessons make connections to previous grade-level learning in the Narrative section. Examples include: 

• In Unit 1, Lesson 2, Identify the Narrative, How does the learning connect to previous lessons? What do students have to get better at today? states, “In the previous lesson, scholars reviewed measurement strategies they learned in first grade. They measured the length of items using nonstandard units (linking cubes) and focused on precision going endpoint to endpoint without gaps or overlaps. In this lesson, students will move to using centimeter cubes. They will focus on precision in measuring and also accuracy in labeling their measurement using units.”
• In Unit 2, Addition and Subtraction to 100 Unit Overview, Identify the Narrative states, “Next, students move into addition of two-digit numbers. Scholars were exposed to two-digit addition with regrouping at the end of first grade.”
• In Unit 4, Data Unit Overview, Identify the Narrative states, “Unit 4 opens with students representing and interpreting categorical data. In Grade 1, students learned to organize and represent data with up to three categories. Now, in Grade 2, students build upon this understanding by drawing both picture and bar graphs with up to four categories.” 
• In Unit 6, Place Value - Three Digit Numbers Unit Overview, Identify the Standards, the materials identify 2.NBT.1 (Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones), 2.NBT.2 (Count within 1000; skip-count by 5s, 10s, and 100s), 2.NBT.3 (Read and write numbers to 1000 using base-ten numerals, number names, and expanded form), and 2.NBT.4 (Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.) as standards being addressed in the unit. Identify the Standards also shows 1.NBT.2 (Understand that the two digits of a two-digit number represent amounts of tens and ones) and 1.NBT.3 (Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, <) as a “Previous Grade Level Standard/ Previously Taught & Related Standard.”
• In Unit 10, Geometry- Shapes Unit Overview, Identify the Narrative states, “In Unit 10, scholars continue to develop their geometric thinking from Grade 1, progressing from a descriptive to an analytic level of thinking, where they can recognize and characterize shapes by their attributes and properties.”