The instructional materials reviewed for Achievement First Mathematics Grade 2 meet expectations for assessing grade-level content. Above-grade-level assessment questions are present and could be modified or omitted without a significant impact on the underlying structure of the instructional materials.

The series is divided into ten units, and each unit contains a Unit Assessment available online in the Unit Overview document and can also be printed for students. Each unit has a Pre- and Post-Unit Assessment. However, the Pre-Assessments do not identify the standards being pre-assessed.

Examples of assessment questions aligned to grade-level standards include:

• In Unit 2, Addition & Subtraction to 100 Unit Assessment, Question 1b states, “$$89 - 52 =$$ _____.”  (2.NBT.5)
• In Unit 3, Story Problems Unit Assessment, Question 3 states, “Some cookies are on the plate. Leann ate 19 cookies. Now there are 12 cookies on the plate. How many cookies were on the plate before?” (2.OA.1)
• In Unit 6, Three Digit Numbers Unit Assessment, Question 13 states, “_____ $$- 100 = 280$$.” (2.NBT.8)
• In Unit 9, Fractions Unit Assessment, Question 3 states, “Use lines to partition the rectangles into fourths in different ways:” Below the question are three rectangles of equal size. (2.G.3)

There are off-grade-level assessment items included in the Unit Assessments that can be modified or omitted without impacting the underlying structure of the materials. For example:

• In Unit 4, Data Unit Assessment, Question 5, which is identified at 2.MD.10, states, “How many more students own pets with 4 legs than students who own pets with fewer than 4 legs?” Students are given a bar graph with the categories of rabbit, dog, cat, and goldfish and solve a multi-step word problem. Question 5 is more accurately aligned to 3.MD.3 (Draw a scaled picture graph and a scaled bar graph to represent data with several categories. Solve one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs.) 
• In Unit 5, Length, Money, Graphing and Time Unit Assessment, Question 5, which is identified as 2.MD.8, states, “A. Kelly has 1 five dollar bill, 3 quarters, 2 nickels, and 3 pennies. She wants to buy something that costs $5.95. Can she afford it? Why or why not? B. How much more does she need?” Question 5 is more accurately aligned to 4.MD.2 (Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including simple fractions or decimals.)
• In Unit 8, Arrays Unit Assessment, Question 5, which is identified as 2.OA.3, states, “Alex says that if he adds 5 to any odd number his answer will be an even number. Do you agree with him? Explain your thinking using pictures and words.” Question 5 is more accurately aligned to 3.OA.9 (Identify arithmetic patterns (including patterns in the addition table or multiplication table, and explain them using properties of operations.)
• In Unit 9, Fractions Unit Assessment, Question 8, which is identified as 2.G.3, states, “A.J., Jorge, and Jack were at a birthday party. AJ ate half of the birthday cake. Jorge ate one-fourth of the cake. Jack ate one-fourth of the cake. Show how the cake below could be divided so that each person gets the amount they ate. Clearly label each person’s piece of cake.” Question 8 is more accurately aligned to 3.NF.1 (Understand a fraction $$\frac{1}{b}$$ as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction $$\frac{a}{b}$$ as the quantity formed by a parts of size $$\frac{1}{b}$$.) 
• In Unit 10, Shapes Unit Assessment, Question 6, which is identified as 2.G.1, states, “Why is a square always a rectangle but a rectangle is not always a square?” Question 6 is more accurately aligned 3.G.1 (Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals)).

The instructional materials reviewed for Achievement First Mathematics 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 6.5 out of 10, which is approximately 65%.
• The number of lessons devoted to major work of the grade, including assessments and supporting work connected to the major work, is approximately 100 out of 141, which is approximately 71%. 
• The instructional block includes a math lesson, cumulative review, math stories, and math practice components. The non-major component minutes were deducted from the total instructional minutes resulting in 9,320 major work minutes left out of 12,690 total instructional minutes. As a result of dividing the major work minutes by the total minutes, approximately 73% of the instructional materials focus on major work of the grade. 

A minute-level analysis is most representative of the instructional materials because the minutes consider all components included during math instructional time. As a result, approximately 73% of the instructional materials focus on major work of the grade.

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

The publishers identify connections between supporting content and major work within the lesson plan in the “Standards in Lesson” section, as well as in the Guide to Implementing AF Math: Grade 2. Additional connections exist within the materials, although not always stated by the publisher. For example, in Unit 4, Lesson 6, 2.MD.D, represent and interpret data, is listed as the cluster in the unit. However, 2.OA.A, represent and solve problems involving addition and subtraction, is connected to Lesson 6 in Unit 4. Examples of the connections between supporting work and major work include the following:

• In Unit 4, Lesson 6, Independent Practice, students engage with the supporting work of 2.MD.10, compare problems using information presented in a bar graph, and the major work of 2.OA.1, use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing with unknowns in all positions. Problem 3 states, “11 of the kids who own pets are boys. The rest are girls. How many girls own pets?” In order to solve, students would need to first add all of the students represented on the table to find out how many total students own pets $$(4 + 7 + 1 + 8 + 4 = 24)$$, then subtract the 11 boys to solve for girls $$(24 - 11 = 13.)$$ 
• In Unit 5, Lesson 1, Independent Practice Worksheet, students engage with the supporting work of 2.MD.8, solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, and the major work of 2.OA.1, use addition and subtraction within 100 to solve one-and two-step word problems while students find the value of groups of coins solve related word problems. Problem 12 states, ”Jenny has 2 quarters, 1 dime, 3 nickels, and 2 pennies. a) Draw and label Jenny’s coins. b) How much money does Jenny have in all? c) Jenny finds 1 dime, 2 nickels, and 3 pennies. How much money does Jenny have now?”
• In Unit 5, Lesson 11, Workshop worksheet, students engage with the supporting work of 2.MD.9,  generate measurement data by measuring lengths of several objects to the nearest whole unit and show the measurements by making a line plot, and the major work of 2.MD.1, measure the length of an object by selecting and using appropriate tools while students measure several lines provided, create a line plot from the data, analyze the data to answer questions. Problem 4 states, “Kaylee was measuring string for an art project and she needs to find out how many pieces she has that are greater than 5 cm. Measure the strings below to the nearest centimeter. Then create a line plot to help her answer the question...How many pieces of string are longer than 5 cm?”
• In Unit 5, Lesson 13, Introduction, students engage with the supporting work of 2.MD.7, learn to tell time to the nearest five minutes, and with the major work of 2.NBT.2, count by 5s. Students are shown a clock with 1:45 displayed. The teacher asks, “What time does this clock show? How did you figure it out?” The sample student response is, “The clock shows the time 1:45. I figured it out by looking at the hands on the clock. The little hand is between the 1 and the 2, but hasn’t passed the 2 yet so it’s 1 o’clock, and the big hand is on the 9 to show the minutes. If we count them by 5s and start at the 12, which is 0 minutes, we get 45 so it’s 1:45.”
• In Unit 10, Lesson 5, Workshop worksheet, students engage with the supporting work of 2.G.1, recognize and draw shapes having specified attributes, and the major work of 2.OA.1, use addition and subtraction within 100 to solve one-and two-step word problems while students solve word problems related to the defining properties of polygons. Problem 1 states, “Ava, Chris, and Natalie are making polygons using gumdrops and pasta. They use a gumdrop for a vertex. They use pasta for a side. Ava makes three quadrilaterals using pasta and gumdrops. Chris makes three pentagons using pasta and gumdrops. Natalie makes two hexagons using pasta and gumdrops. Ava says they will each use the same number of gumdrops and pasta to make their shapes. Natalie says Chris will use more. Who is correct, Ava or Natalie?”
• Practice Workbook C, students engage with the supporting work of 2.MD.10, solve simple put-together problems using information presented in a bar graph and also addresses, although not stated, the major work of 2.NBT.5, fluently add and subtract within 100. Problem 3 states, “Use the Animal Habitats table to answer the following questions.” Tally marks are used to record data for Forest, Wetlands, and Grasslands. Problem 3d states, “How many total animal habitats were used to create this table?”

The instructional materials reviewed for Achievement First Mathematics Grade 2 meet expectations that the amount of content designated for one grade-level is viable for one school year. The Guide to Implementing AF, Grade 2 includes a scope and sequence which states, “Not every lesson is entirely focused on grade level standards, and, therefore, some lessons can be used for either remediation or enrichment. As designed, the instructional materials can be completed in 141 days. One day is provided for each lesson and one day is allotted for each unit assessment.

• Ten units with 131 lessons in total.
• The Guide identifies lessons as either R (remediation), O (on grade level), or E (enrichment).  There are 0 lessons identified as E (enrichment), 4 identified as R (remediation), and 127 identified as O (on grade level). 
• Ten days for unit assessments.
• Unit 1 has an instructional day listed as a “flex day.” However, as there are no materials identified for instruction on the flex day, the flex day was not included in the count for the review. 

When reviewing the materials for Achievement First, Grade 2, a difference in the number of total instructional days was found. Although the publisher states the curriculum will encompass 140 days, there are 141 days of lessons and unit assessments. The Unit 6 Overview allocates 18 days of instruction to the unit. However, the Guide to Implementing AF, Grade 2 and the Unit 6 Overview lesson breakdown allocates 19 days of instruction. In addition, the Grade 2 Unit Overview for Unit 8 shows 10 days for the unit while the Guide to Implementing AF, Grade 2 provides 11 days for the unit. The unit has 11 lessons including the unit assessment. The Unit 10 Overview states that five days of instruction are needed for the unit and does not include a day for assessment in the breakdown. A Unit Assessment is included as a resource in the Unit Overview document. The Guide to Implementing AF, Grade 2 provided six days for Unit 10.

The publisher recommends 90 minutes of mathematics instruction daily.

• There are three lesson types, Game Introduction Lesson, Exercise Based Lesson, or Task Based Lesson. Each lesson is designed for 55 minutes. 
• Math stories are designed for 25 minutes on Monday-Thursday.
• Cumulative review is designed for 25 minutes. 
• Practice is designed for 10 minutes.

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.”

The instructional materials reviewed for Achievement First Mathematics Grade 2 meet expectations for fostering coherence through connections at a single grade, where appropriate and required by the Standards. 

The materials include learning objectives, or Aims, that are visibly shaped by CCSSM cluster headings. Examples include: 

• In Unit 2, Lesson 5, Aim is shaped by 2.OA.B, add and subtract within 20. The materials state, “SWBAT use doubles to solve addition and subtraction problems.”
• In Unit 3, Lesson 3, Aim is shaped by 2.OA.A, represent and solve problems involving addition and subtraction. The materials state, “SWBAT represent and solve C-DU-M, C-BU-M, and C-SU-F story problems by following the story problem protocol and using a strategy that makes sense to them.” 
• In Unit 8, Lesson 3, Aim is shaped by 2.OA.C, work with equal groups to gain foundation for multiplication. The materials state, “SWBAT compose arrays to show equal group situations by representing the groups with the rows and the amount in each group with the columns.”
• In Unit 9, Lesson 3, Aim is shaped by 2.G.A, reason with shapes and their attributes. The materials state, “SWBAT interpret equal shares in composite shapes as halves, thirds, and fourths.” 
• In Unit 10, Lesson 2, Aim is shaped by 2.G.A, reason with shapes and their attributes. The materials state, “SWBAT describe, build, identify, and analyze two-dimensional shapes with specified attributes.” 

Materials include problems and activities that connect two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important. Examples of connections include:

• In Unit 3, Lesson 5, Independent Practice, students engage with 2.MD.B, relate addition and subtraction to length, and 2.OA.A, represent and solve problems involving addition and subtraction, as they solve a story problem in a five-step process provided. Problem 5 states, “Jasmine has a jump rope that is 84 inches long. Marie’s is 13 inches shorter than Jasmine’s. What is the length of Marie’s jump rope?” A visual is provided in the answer space with the following steps to complete the word problem: visualize, represent, retell, solve, and finish the story. 
• In Unit 7, Lesson 1, Exit Ticket, students engage with 2.OA.B, add and subtract within 20, and 2.NBT.B, use place value understanding and properties of operations to add and subtract by adding with expanded notation, as they solve an expression using expanded notation. Problem 1 states, “Solve using expanded notation. $$362+427$$” 
• In Unit 8, Lesson 5, Independent Practice, students engage with 2.OA.C, work with equal groups of objects to gain foundations for multiplication, and 2.G.A, reason with shapes and their attributes, as they use a grid to draw arrays of squares and use repeated addition to determine the total number of squares. Problem 7 states, “Draw a rectangular array with 4 rows of 3. Write a repeated addition sentence to match.  ______rows of _______ squares = ________in all.” 
• Practice Workbook F, students engage with 2.OA.C, work with equal groups of objects to gain foundations for multiplication, and, although not stated, 2.G.A, reason with shapes and their attributes, as they create arrays to solve an equation. Problem 4 states, “Create a rectangular array using circles to solve the equation below. $$4 + 4 + 4 + 4 + 4 =$$ _____.”

The instructional materials reviewed for Achievement First Mathematics Grade 2 meet expectations for developing 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 throughout the grade-level. Examples include:

• In Unit 2, Lesson 7, Introduction and Workshop, students engage with 2.NBT.5, fluently add and subtract within 10 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction, as they solve problems about tens and ones by using a variety of representations (stick and dots, expanded form). Students roll two number cubes, record the two-digit number, represent the number using sticks and dots, and represent the number using expanded form. The teacher asks, “How will you figure out how to represent 2-digit numbers using sticks and dots and expanded form?” The students may reply with, “I will look at the digits in each place and think about the value of each digit.” 
• In Unit 6, Lesson 2, Introduction, students engage with 2.NBT.1, understand that the three digits of a three-digit number represent the amounts of hundreds, tens, and ones, as they model three-digit numbers with place value blocks, then read and write the numbers. The materials state, “Kaleb has 3 boxes of 100 crayons, 6 boxes of 10 crayons, and 2 single crayons. How many crayons does Kaleb have?”
• In Unit 6, Lesson 14, Introduction, students engage with 2.NBT.4, compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the result of the comparison, as they compare two three-digit numbers written in different forms using <, >, and =. The teacher poses the following comparison problem to students, “562 __ 5 hundreds, 2 tens, 6 ones.” A sample student response states, “We wrote 562 > 5 hundreds, 2 tens, 6 ones. We figured it out by showing both numbers in flats, sticks, and dots. For 562 we drew 5 flats, 6 tens, 2 dots. Then for 5 hundreds, 2 tens, 6 ones, we drew 5 flats, 2 sticks, 6 dots. We looked at the hundreds place and saw that they had an equal number of hundreds, so then we looked at the tens and saw that 562 has more tens than 5 hundreds, 2 tens, 6 ones, so 562 is greater than 5 hundreds, 2 tens, 6 ones.” 
• In Unit 7, Lesson 2, Aim, students engage with 2.NBT.7, add and subtract within 1000 using concrete models or drawings, as they complete 2-digit addition problems using flats, sticks, and dots. The materials state, “SWBAT add 2-digit numbers with regrouping in one place by using flats, sticks, and dots.” Workshop Worksheet example, Problem 2A states, “$$550 + 268 =$$ ______.”
• In Unit 7, Lesson 7, Workshop Worksheet, students engage with 2.NBT.9, explain why addition and subtraction strategies work using place value and properties of operations, as they solve a three-digit subtraction problem, and write a written explanation of why their strategy worked. The materials state, “Solve. ______ $$- 246 = 568$$. Explain how you solved the problem above. What strategy did you use? What steps did you take? Why did your strategy work?”

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Examples include: 

• In Unit 2, Lesson 10, Exit Ticket, students engage with 2.NBT.5, fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction, as they add two-digit numbers by using a strategy that makes sense to them (sticks and dots, expanded notation/use known facts). Problem 1 states, “$$62 + 27 =$$ ___.” Students are directed to use sticks and dots or expanded notation to solve. 
• In Unit 3, Practice Workbook B, students engage with 2.NBT.5, fluently add and subtract within 100 using strategies based on place value, as they identify the proper model for a given problem. Problem 34 states, “Circle which set of sticks and dots will help to find the total? $$62 + 24 =$$ ______.” 
• In Unit 6, Lesson 2, Independent Practice Worksheet, students engage with 2.NBT.1, understand that the three digits of a three-digit number represent the amounts of hundreds, tens, and ones, as they independently model three-digit numbers with place value blocks, then read and write the numbers. Problem 1 states, “Draw flats, sticks, and dots to represent each number. 258. How many more ones will make a ten? How many more tens will make a hundred? How many more hundreds will make a thousand?”
• In Unit 7 Lesson 7, Exit Ticket, students engage with 2.NBT.9, explain why addition and subtraction strategies work using place value and properties of operations, as they solve a three-digit subtraction problem, and write a written explanation of how they solved the problem. Problem 2 states, “_____ $$- 567 = 293$$. Explain how you solved.”
• In Unit 7, Lesson 10, Independent Practice, students engage with 2.NBT.7, add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction and relate the strategy to a written method. Problem 2 states, “Solve using flats, sticks, and dots. $$531 - 258 =$$ ____.”

The instructional materials reviewed for Achievement First Mathematics Grade 2 meet expectations for attending to those standards that set an expectation of procedural skill and fluency. These skills are delivered throughout the materials in the use of games, workshop, practice workbook pages and independent practice, such as exit tickets. 

The instructional materials develop procedural skill and fluency throughout the grade-level. Examples include: 

• In Unit 2, Lesson 2, Introduction and Workshop, students engage with 2.OA.2, fluently add and subtract within 20 using mental strategies, as they determine the missing part to make 10 using a strategy that works for them (count up, count back, just know). Students use a dot cube to roll for a number to subtract from 10 in a number bond. Potential strategy examples state, “Count up: You can start at 6 because that’s the first part and count up until 10 because that’s the whole. Like this… 6 -- 7, 8, 9, 10. So the missing part is 4. Subtract: You can start with the whole -- 10 and subtract 6 because that’s the part we know. The answer is 4, so the missing part is 4. Count back: I started at 10 because that is the whole and then I counted back 6 because that’s the part we know. Like this 10 -- 9, 8, 7, 6, 5, 4. So the missing part is 4. Just know: I just know that 6 and 4 make 10 because they’re number pairs. So the missing part must be 4.” 
• In Unit 3, Practice Workbook B, Activity: Building Toward Fluency, students engage with 2.OA.2, fluently adding and subtracting within 20 using mental strategies, as they use various strategies to complete and discuss addition problems. The materials state, “Write the expression on the board or chart paper. Start with 4 + 10. Ask students to describe their strategy for solving the problem. Choose one or more students to explain their strategy to the class. Represent each strategy on the board using the number line or magnetic cubes. Once the student’s strategy is understood by the class, continue with the next sum.”
• In Unit 5, Practice Workbook B, Ten Plus Number Sentences, students engage with 2.NBT.5, fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or relationship between addition and subtraction, as they practice proficiency with their ten plus facts. The teacher says, “I will flash two ten-frame cards, ten and another card. Wait for the signal. Then tell me the addition sentence that combines the numbers.” The teacher flashes a 10 and 5. Students respond with, “$$10 + 5 = 15$$”
• In Unit 6, Lesson 9, Workshop Worksheet, students engage with 2.NBT.2, skip-count by 5s, 10s, and 100s, as they use skip counting by 10s and 100s to count up. Problem 1 states, “Count from 90 to 300.”

The instructional materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade-level. Examples include: 

• In Unit 2, Lesson 3, Exit Ticket, students engage with 2.OA.2, fluently add and subtract within 20 using mental strategies, as they independently complete a number bond with one unknown number and write 2 addition and two subtraction problems to match. Problem 1 states, “Finish the number bond and write number sentences to match.” Students are provided with a number bond diagram with 11 and 6 as addends and an unknown sum.
• In Unit 3, Practice Workbook B Pairs To Make Ten With Number Sentences, students engage with 2.OA.2, fluently add and subtract within 20 using mental strategies, as they solve doubles +2 facts. Problem 3 states, “$$2 + 4 =$$ ___.”
• In Unit 4, Practice Workbook B, students engage with 2.NBT.5, fluently add and subtract within 100 using strategies based on place value, properties of operations and/or the relationship between addition and subtraction, as they independently solve two-digit addition and subtraction problems. Problem 21 states, “$$53 -$$ ______ $$= 28$$.”
• In Unit 8, Practice Workbook D, students engage with 2.NBT.8, mentally add 10 or 100 to a given number 100-900, as they independently add 10 or 100 to given numbers betwembers under 1000. Problem 1a states, “Solve each problem using mental math, $$678 + 100 =$$ ____.”

The instructional materials for Achievement First Mathematics Grade 2 partially 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 series includes limited opportunities for students to independently engage in the application of routine and non-routine problems due to lack of independent work during Workshop and lack of non-routine problems. 

The instructional materials present opportunities for students to engage in application of grade-level mathematics; however, the problems are scaffolded through teacher led questions and partner work. According to the Guide to Implementing AF Grade 2, “Task Based Lesson Purpose: Students make sense of the mathematics they’ve learned by  working on a problem solving task and leveraging the knowledge they bring to math class to apply their math flexibly to non-routine, unstructured problems, both from pure math and from the real world. To shift the heavy lifting to scholars.” However, there are not any “Task Based Lessons” labeled in the Guide to Implementing AF Grade 2. 

Routine problems are found in the Independent Practice and Exit Tickets. For example:

• In Unit 3, Lesson 2, Exit Ticket, students engage with 2.OA.1, using addition and subtraction within 100 to solve one- and two-step word problems, as they calculate take apart problems with the addend unknown. Problem 2 states, “Mr. Cruz has 19 basketballs. Mr. Hogan gave him some more basketballs. Now Mr. Cruz has 63 basketballs. How many basketballs did Mr. Hogan give to Mr. Cruz?”
• In Unit 3, Lesson 7, Independent Practice, students engage with 2.OA.1, use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing with unknowns in all positions, as they solve a story problem. Problem 7 states, “Enmaries has a jump rope that is 68 inches long. Giada’s is 33 inches shorter than Enmaries’s jump rope. What is the length of Giada’s jump rope?”
• In Unit 5, Lesson 3, Independent Practice, students engage with 2.MD.8, solve word problems involving money, as they independently solve word problems with money. Problem 5 states, “King Jamonie has 3 quarters, 1 dime, 2 nickels, and 4 pennies. How much money does he have?” 
• In Unit 5, Lesson 9, Exit Ticket, students engage with 2.MD.8, solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, as they solve all types of story problems using the most efficient strategy. Problem 2 states, “Carlos bought a pack of gum using exactly $1. The cashier gave him 3 nickels, 2 pennies, and a quarter back. How much did the gum cost?”

Math Stories provide opportunities for students to engage in routine applications of grade-level mathematics. Students engage with Math Stories for 25 minutes, four days per week. The Guide to Implementing AF Grade 2 page four states the purpose of Math Stories, “Purpose: 

• To enable students to make connections, identify and practice representation and calculation strategies, and develop deep conceptual understanding through the introduction of a specific story problem type in a clear and focused fashion with deliberate questioning and independent work time.
• To reveal and develop students’ interpretations of significant mathematical ideas and how these connect to their other knowledge.
• To shift the heavy lifting to scholars.”

Examples of routine Grade 2 Math Stories:

• In Unit 2, Guide to Implementing AF Math, Math Stories, October, students engage with 2.OA.1, use addition and subtraction to solve one- and two-step word problems with unknowns in all positions, as they solve add to/change unknown word problems. Sample Problem 2 states, “Jose has 27 erasers. Kate gave him some more. Now he has 53 erasers. How many erasers did Kate give him?”
• In Unit 3, Guide to Implementing AF Math, Math Stories, December, students engage with 2.OA.1, using addition and subtraction within 100 to solve one- and two-step word problems, as they calculate take from problems with results unknown. Sample Problem 2 states, “Antonio gave 27 tomatoes to his neighbor and 15 to his brother. He had 72 tomatoes before giving some away. How many tomatoes does Antonio have remaining?”
• In Unit 4, Guide to Implementing AF Math, Math Stories, January, engages with 2.MD.5, use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, as they use a variety of strategies to solve measurement word problems. Sample Problem 6 states, There are 98 frogs on a log. 63 frogs jump into the water. 29 frogs hop back onto the log. How many frogs are on the log now?”
• In Unit 8, Guide to Implementing AF Math, Math Stories, May, students engage with 2.OA.1, use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing with unknowns in all positions, as they complete math story problems. Sample Problem 1 states, “There were 45 students on the playground. 33 Students joined them. Then 37 students went inside. How many students are now on the playground?”

Non-routine examples could only be found in Unit 3, not across the yearly curriculum. For example:

• In Unit 3, Lesson 10, Independent Practice, students engage with 2.OA.1, use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing with unknowns in all positions, as they determine if the thinking and process complete by another is correct. Problem 5 states, “Grace represented the problem below. What mistake did she make? Explain what mistake she made and what she can do to fix it. Mr. Cruz has 42 orange basketballs and 29 black basketballs. He gives away 32 basketballs. How many basketballs does Mr. Cruz have now? $$42 + 29 + 32 = N$$”
• In Unit 3, Lesson 11, Independent Practice, students engage with 2.OA.1, use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing with unknowns in all positions, as they determine if the thinking and process complete by another is correct. Problem 7 states, “Lisa represented the problem below. Is she correct? If not, explain her mistake and how she could fix it. There are 28 lions at the zoo. There are 14 fewer lions than tigers at the zoo. 18 tigers come to the zoo. Now how many tigers are at the zoo?” (Two diagrams follow the question that represent her thinking towards the problem.)

It is important to note that some of the recommended Math Stories go beyond the expectations of Grade 2 mathematics standards and some are considered extensions. The Guide to Implementing AF Grade 2 states, “The problem types covered within each unit are marked with an O (on grade level) or an E (enrichment) to reflect their alignment to grade level standards.” For example:

• In Unit 7, Math Stories, April, Extension Problem 4 states, “(AA-UF) There are 32 stickers in a sticker book page. They are arranged so that there are 8 equal rows of stickers. How many stickers are in each road?” This word problem goes beyond Grade 2 Mathematics Standards, as it is a multiplication word problem, and better aligned with 3.OA.8, solve two-step word problems using the four operations.
• In Unit 7, Math Stories, April Extension Problem 12 states, “(C-DU with AA-UP) Nithi has 3 rows of 7 brownies on her tray. Raj has 3 rows of 5 brownies on his tray. How many more brownies does Nithi have than Raj? (Note: As an extension, look for students that see that Nithi has 3 rows of 2 brownies extra instead of calculating both and subtracting!)” This word problem is better aligned with 3.OA.8, solve two-step word problems using the four operations.
• In Unit 8, Math Stories, May, Problem 9 states, “(AT-SU with AA-UP) There are some desks in a classroom. Then, Ms. Colville brings in 3 rows of 4 desks. Now there are 30 desks in the classroom. How many desks were in the classroom originally?” This word problem goes beyond Grade 2 Mathematics Standards, as it is a two-step word problem including multiplication, and is better aligned with 3.OA.8, solve two-step word problems using the four operations.

The instructional materials for Achievement First Mathematics Grade 2 meet expectations that the materials reflect the balance in the standards and help students meet the standards’ rigorous expectations by helping students develop conceptual understanding, procedural skill and fluency, and application. The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade.

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

Conceptual understanding

• In Unit 6, Lesson 2, Exit Ticket, students engage with 2.NBT.1, understand that the three digits of a three-digit number represent the amounts of hundreds, tens, and ones, as they read and write numbers within 1,000 after modeling with place value blocks (flats, sticks, and dots). Problem 2 states, “Draw models of ones, tens, and hundreds.” Students are given the number 508 and asked to answer the following questions, “How many more ones will make a ten? How many more tens will make a hundred? How many more hundreds will make a thousand?”
• In Unit 6, Lesson 3, Independent Practice, students engage with 2.NBT.1, understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones, and 2.NBT.3, read and write numbers to 1000, as they represent a three-digit numbers in a variety or forms and models. Problem 10 states, “Alexander has 529 M&Ms. Write the amount of M&Ms Alexander has in three different ways by filling in the blanks. (Unit Form, Base Ten Numeral Form, Place Value Models)”
• In Unit 7, Lesson 1, Independent Practice, students engage with 2.NBT.7, add and subtract within 1000 using concrete models or drawings and strategies based on place value, as they independently solve a three-digit subtraction. The third item states, “Solve using flats, sticks, and dots ______ $$- 348 = 650$$. Explain how you solved ______ $$- 348 = 650$$.”
• In Unit 9, Practice Workbook E, students engage with 2.NBT.7, add and subtract within 1000 using concrete models or drawings and strategies based on place value, as they independently solve a three-digit addition problem using a number line. Problem 4 states, “Use the number line to solve. Show your work. $$578 + 237 =$$ ___.” Students are provided a blank number line. 

Procedural skills (K-8) and fluency (K-6)

• In Unit 2, Lesson 4, Exit Ticket, students engage with 2.OA.2, fluently add and subtract within 20 using mental strategies, as they solve addition problems. Problem 1 states, “Solve. $$7 + 8 =$$ _____.”
• In Unit 2, Lesson 24, Exit Ticket, students engage with 2.NBT.5, fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition/subtraction, as they solve two-digit subtraction problems with missing minuends by relating addition and subtraction. Problem 1 states, “Solve. $$93 -$$ ____ $$= 62$$.” Students are provided with a blank number bond model.
• In Unit 3, Practice Workbook B, students engage with 2.NBT.5, fluently adding and subtracting within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction, as they solve put together problems with an unknown addend. Problem 5 states, “$$35 +$$ ___ $$= 50$$.”
• In Unit 5, Practice Workbook B, students engage with 2.NBT.5, fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or relationship between addition and subtraction, as they practice fluently adding and subtracting. Problem 4 states, “$$45 +$$ ___ $$= 100$$”

Application

• In Unit 2, Guide to Implementing AF Math, Math Stories, October, students engage with 2.OA.1, adding and subtracting within 100 to solve one- and two-step word problems, as they solve addition problems with the change unknown. Sample Problem 2 states, “Jose has 27 erasers. Kate gave him some more. Now he has 53 erasers. How many erasers did Kate give him?”
• In Unit 3, Lesson 4, Independent Practice, students engage with 2.OA.1, use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing with unknowns in all positions, as they solve a story problem. Problem 4 states, “Ms. Reinhardt has 42 books. Ms. Gomez has 18 fewer books than Ms. Reinhardt. How many books does Ms. Gomez have?”
• In Unit 3, Lesson 4, Exit Ticket, students engage with 2.OA.1, use addition and subtraction within 100 to solve one- and two-step word problems, as they independently solve compare/smaller unknown word problems. Problem 1 states, “There are 59 girls on the bus. There are 26 more girls than boys on the bus. How many boys are on the bus?”
• In Unit 5, Lesson 8, Exit Ticket, students engage with 2.MD.8, solve word problems involving dollar bills, quarters, dimes, nickels and pennies, using $ and $$\cancel{C}$$, as they solve one-step story problems of all types that involve bills and coins by using the most efficient strategy. Problem 2 states, “Kevin has 75 cents. He spent 3 dimes, 3 nickels, and 4 pennies on a slice of cake. How much money does he have left?”

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 Unit 2, Lesson 6, Exit Ticket, students engage with 2.NBT.9, explaining why addition and subtraction strategies work, using place value and the properties of operations, and 2.OA.1, adding and subtracting within 100 to solve one- and two-step word problems, as they solve take from problems with the result unknown (application) and show their thinking (conceptual understanding). Problem 1 states, “Represent and solve. Amya has 17 pencils. 13 are red and the rest are green. How many green pencils does Amya have? Describe how you solved.”
• In Unit 3, Lesson 2, Independent Practice, students engage with 2.NBT.5, fluently add and subtract within 100 using strategies based on place value, and 2.OA.1, use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing with unknowns in all positions, as they determine if a representation is correct (application) and how they know (conceptual understanding). Problem 10 states, “Mr. Johnson has 46 pens. 24 are blue and the rest are black. How many of Mr. Johnson’s pens are black? Charlie and Henry represented the problem below. (Charlie $$46 + 24 =$$ ? represented/ Henry $$46 - 24 =$$ ? represented)”
• In Unit 5, Lesson 8, Exit Ticket, students engage with 2.MD.8, solve word problems involving money, as they represent (conceptual understanding) and solve story problems (application) using the most efficient strategy (procedural skill). Problem 1 states, “Jacob bought a piece of gum for 26 cents and a newspaper for 61 cents. He gave the cashier $1. How much money did he get back?”
• In Unit 8, Lesson 2, Independent Practice, students engage with 2.OA.4, use addition to find the total number of object arranged in a rectangular array, as they draw a rectangular array and write addition equations (conceptual understanding) to represent and solve word problems (application) involving equal groups of objects. Problem 6 states, “Ja-yier put 5 toys into 4 different baskets. How many toys does Ja-yier have in all?”

The instructional materials reviewed for Achievement First Mathematics Grade 2 partially meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade-level. All MPs are clearly identified throughout the materials, with few or no exceptions. However, there are inconsistencies between the identified MPs in the Unit Overview and the MPs identified in the Lesson Plans. The materials provide little direction as to how the MPs enrich the content and do not make connections to enhance student learning. The MPs are not treated separately from the content.

Evidence that all eight MPs are clearly identified throughout the materials, with few or no exceptions, though they are not always accurate. For example:

• In the Unit Overviews, the bolded MPs are the Focal MPs for the unit.
• Unit 2, Unit Overview, Standards for Mathematical Practice identifies MP8, look for and express regularity in repeated reasoning, as embedded in addition and subtraction lessons within 100. 
• In Unit 5, Unit Overview, Standards for Mathematical Practice identifies MP6, attend to precision, as embedded in the measurement lessons of Unit 5. 
• The MPs are listed at the beginning of each lesson in the Standards section. For example, in Unit 4, Lesson 3, the following MPs are identified as in the lesson: MP 2  and MP 6. 
• The Mathematical Practices are not always identified accurately. For example: 
• In Unit 3, MP 6 is not bolded as a focus MP. However, it is identified in all 14 lessons. MP 7 is identified as a focus MP but is not identified in any of the 14 lessons. 
• At the unit level for Unit 7, MP 6 is not identified as a focus MP. However, at the lesson level, all 18 lessons identify it as connected. At the unit level, MP 3 is listed as a focus, but it is only connected to 3 of the 18 lessons. 
• All MPs are represented throughout the materials, though lacking balance. For example, MP 8 is the focus of none of the second grade units, while MP 5 is the focus of all ten units. 
• There are no stated connections to the MPs to the Math Stories component, Math Practice component, Cumulative Review component, or Assessments. 

There are instances where the MPs are addressed in the content. However, these connections are not clear to the teacher.

• It is left to the teacher to determine where and how to connect the emphasized mathematical practices within each lesson. 
• There are connections to the content described in the Unit Overview. However, if a teacher is not familiar with the MPs, the connection may be overlooked as there are no connections within the specific lesson content to any MPs. Examples include: 
• In Unit 2, Unit Overview, Standards for Mathematical Practice identifies MP1, make sense of problems and persevere in solving them, as embedded in the lessons of Unit 2. The materials state, “Students make math drawings and use composition and decomposition strategies to reason through the relationships in word problems. They write representations, equations, and word sentences to explain their solutions.”
• In Unit 5, Unit Overview, Standards for Mathematical Practice identifies MP5, use appropriate tools strategically, as embedded in the measurement lessons of Unit 5. The materials state, “Students apply their measurement skills and knowledge of the ruler to measure a variety of objects using appropriate measurement tools, such as inch rulers, centimeter rulers, meter sticks, and yardsticks.”

The materials reviewed for Achievement First Mathematics Grade 2 partially meet expectation for meeting the full intent of the math practice standards. The Mathematical Practices (MPs) are represented in each of the nine units in the curriculum and labeled on each lesson. Math Practices are represented throughout the year and not limited to specific units or lessons. The materials do not attend to the full meaning of MPs 1 and 5.

The materials do not attend to the full meaning of MP1 because students primarily engage with tasks that replicate problems completed during instructional time. Examples include:

• In Unit 2, Lesson 6, Narrative, What is new/or hard about the lesson? states, “This lesson is challenging because it’s pushing students to apply their understanding of part-part-whole relationships and pushing them to become fluent within 20. Students may have difficulty understanding the context of the story problem. They may also have difficulty calculating fluently.” Introduction, Pose the Problem states, “Carla baked 15 desserts. 9 of them were chocolate chip cookies and the rest were brownies. How many brownies did Carla bake?” 
• In Unit 4, Assessment, students solve two-step story problems from data presented in a graph. Problem 4 states, “19 of the scholars who like fruit are girls. How many of the scholars are boys?” Students are provided a bar graph showing survey data regarding favorite fruits.
• In Unit 7, Lesson 17, Introduction, Check for Understanding states, “I’m going to solve ___ $$- 315 - 545$$ using expanded notation. (Set up number bond correctly and label parts and wholes correctly, but set up problem to subtract 545 - 315.) (EV) Am I right? No. (TT) Why not? What mistake did I make? Your number bond is correct, but if the whole is missing and you know that both parts are 315 and 545 you need to add the 2 parts to find the whole, not subtract. (TT) What strategies will you use to solve problems today? First we need to figure out what’s missing - a part or the whole. then we can use expanded notation, flats, sticks, and dots, or a number line to solve.”

The materials do not attend to the full meaning of MP5 because students do not choose their own tools. Examples include:

• Unit 2, Lesson 1, Step 1, “says, Roll, so I’m going to roll my dot cube gently. How many? SMS: 6!” The students are given the dot cube to roll, that is the only tool they are given and use during the lesson.
• Unit 4, Lesson 4, Students create survey questions with a partner and then use tally marks to organize the results.  Students are told to use tally marks and no other tools are available or used during the lesson. 
• Unit 9, Lesson 1, Materials, “pattern blocks” Students are given pattern blocks to build larger shapes to explore fraction concepts.  Students do not have a choice in the tool they use. 

Examples of the materials attending to the full intent of specific MPs include:

• MP2: In Unit 5, Lesson 9, Independent Practice Worksheet, students solve story problems involving coins, requiring them to abstract the value of a set of coins, to find the total value. Problem 8 states, “Anna gave Sean a dime, a nickel, and some more coins. Now Sean has $1.00 in coins. Draw and label two possible pictures of Sean’s coins.”
• MP4: In Unit 3, Lesson 2, Narrative, How does the learning connect to previous lessons? What do students have to get better at today? states, “In this lesson, scholars will focus on accurately representing the problem and then using that representation to choose the correct operation to solve. They will work with Add To/Take From - Change Unknown and Put Together/Take Apart - Addend Unknown story problems.” Introduction task states, “There are some birds on a fence. 19 birds flew away. Now there are 52 birds on the fence. How many birds on the fence were there to start?"
• MP6: In Unit 4, Lesson 3, Narrative, What is new and/or hard about the lessons? states, “Students may still be struggling to categorize data, and may still be struggling to create accurate, organized representations of data.” Introduction states, “(TT) How do I know where to stop on my scale? You need to find the category with the largest number, then you need to make sure your scale goes up to that number so that all your data fits. BPQ: Would it make sense to have my scale go to 100 if my largest category had 5 things? Why?”
• MP7: In Unit 8, Lesson 7, Workshop Worksheet, students decompose numbers and apply repeated addition to the structure of arrays to both the rows and columns. Problem 1 states, “Jeremiah drew an array with 20 squares. Draw 3 different arrays that have 20 squares in all. Write a repeated addition sentence to match each array you drew.”
• MP8: In Unit 9, Lesson 7, Narrative, How does the learning connect to previous lessons? What do students have to get better at today? states, “In the previous lesson, students partitioned shapes into the same fraction in more than one way and came to the understanding that the same fraction can have a different shape. Students also named and wrote unit fractions. Today, for the first time, students will partition rectangles into fractions in more than one way and prove the fractions are the same by cutting the parts and manipulating the pieces.”

The instructional materials reviewed for Achievement First Mathematics Grade 2 meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. The student materials prompt students to both construct viable arguments and analyze the arguments of others even though mathematical dialogue is mainly between the teacher and individual students.  

Examples of constructing viable arguments include: 

• In Unit 2, Lesson 21, Exit Ticket, students explain the strategies they used to solve two problems, and why they chose each strategy. Problem C states, “How did you solve $$68 - 21$$? Explain the steps you took to solve question A. How is this different from how you solved $$68 + 21$$ in question B? Why?”
• In Unit 5, Lesson 3, Workshop Worksheet, Problem 3 states, “Megan has 4 dimes and 15 pennies. Joshua has the same amount of money as Megan but none of the same coins. What coins does Joshua have? How do you know?” 
• In Unit 7, Lesson 19, Assessment, Problem 4 states, “Find the missing numbers to make each statement true. Show your strategy to solve. a. ___ $$= 407 - 159$$. Explain how you solved this using what you know about place value.”
• In Unit 9, Lesson 4, Intro, Problem 2 states, “Chase offers to share his pie with Jariah and Luke. They want to have the largest pieces possible. Should Chase cut it into thirds or fourths? Why?” 

Examples of analyzing the arguments of others include: 

• In Unit 2, Lesson 24, Workshop Worksheet, students use knowledge of place value and expanded notation to critique the work of other students, and show how their work can be fixed. Problem 2 states, “Cat solved the problem below using expanded notation. $$95 -$$ _____ $$= 27$$, $$90 + 5$$, - (subtraction symbol)  $$20 + 7$$,  _______, $$70 + 2 = 72$$, Is she correct? If not, how can she fix her work?” (Commas separate different lines of example student work.)

• In Unit 3, Lesson 2, Introduction, Problem 2 states, “Khaleel and Mauricia represented the problem below. There were 36 kids on the playground. Some more kids came over to join them. Now there are 62 kids on the playground. How many kids came to join them? Look at Khaleel and Mauricia’s representations. Who is correct? How do you know?” 
• In Unit 4, Lesson 4.2, Cumulative Review, Problem 6 states, “Kimberly was solving the problem below using expanded notation. What mistake did she make? Fix Kimberly’s work in the box and then explain her mistake on the lines below. 67 - 39 = ___, $$60 + 7$$, - (subtraction symbol) $$30 + 9$$, ______, $$30+2=32$$.” (Commas separate different lines of example student work.)
• In Unit 5, Lesson 9, Independent Practice, Problem 7 states, “Calvin has 72 cents. Michelle says he needs 1 quarter and 5 pennies to make a dollar. Sarah says he needs 3 pennies, 2 dimes, and a nickel to make a dollar. a) How much money does Calvin need to make a dollar? b) Who is right and how do you know?” 
• In Unit 8 Assessment, students use their knowledge of arrays to analyze the mathematical thinking of a fictitious student. Item 1 states, “Angela wants to make 3 pins. Angela wants to put 5 beads on each pin. Angela has a bead box with three rows in it. Each row has five sections. Angela has one bead in each section. Angela says that she has enough beads to make three pins. Is Angela correct? Show all of your mathematical thinking.”

The instructional materials reviewed for Achievement First Mathematics 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. Examples of the materials assisting teachers in engaging students to construct viable arguments and analyze the arguments of others include: 

• In Unit 2, Lesson 3, Introduction, students are prompted to analyze the thinking of others as they determine the mistakes made by the teacher as he/she models how to find the unknown number in a number bond. The materials state, “(Model an intentional mistake--add when you should subtract.) (Show number bond with 9 in the center/whole and 5 in the part--label 9 red and blue crayons, 5 red crayons, missing # blue crayons.) Oh! I can find the missing number by adding 5 and 9. (EV) What do you think? No. (TT) What mistake did I make? How can I fix it? You added the 5 red crayons to the 9 red and blue crayons and we know that 5 of those crayons are red. We need to figure out how many of those crayons are blue. We need to subtract to find the missing part, not add.”
• In Unit 2, Lesson 21, Workshop, students solve two-digit addition and subtraction problems on a number line. While circulating during Workshop, the teacher may ask the following questions to check for understanding, “How did you solve the problem? Why did you make ___ jumps of ___? Which direction did you hop and why? (Where did you start with and where did you hop to and why?) Why did you start at XX and make jumps of XX?”
• In Unit 3, Lesson 6, Independent Practice, Check for Understanding, teachers are provided with a list of prompts designed to engage students in constructing viable arguments as they represent and solve story problems. The materials state, “How did you know that $$x$$? What strategy did you use? Why are you adding $$x$$ and x/subtracting $$x$$ and $$x$$? What in the story made you think that? Why did you have to regroup?”
• In Unit 5, Lesson 5, Independent Practice, Check for Understanding states, “Why did you trade ___ for ___? What strategy did you use to make trades?”
• In Unit 6, Lesson 9, Introduction, students count up between 90 and 1,000 by using skip counting. The materials state, “Pose the Problem: I have 286 gumballs. I want to see how many more I need to get to 500.” Students are given three minutes to represent and solve with a partner. The teacher then asks, “How did you find out how many gumballs I need to make 500 (call on someone with correct representation who solved on a number line)?” After the student shares, the teacher follows up with the following questions, “Agree/Disagree/Clarification; Why does that work? Why do you think we counted by ones first? Why do we count by tens next?” 
• In Unit 7, Lesson 7, Introduction, students work to answer three-digit addition problems. Questioning guidance is provided to assist the teacher for two different scenarios, $$>\frac{2}{3}$$ correct and $$<\frac{2}{3}$$correct. The materials state, “Agree/Disagree/Clarification (TT) Why did they have to regroup? (CC) How did they show regrouping?”

The instructional materials reviewed for Achievement First Mathematics Grade 2 meet expectations that materials explicitly attend to the specialized language of mathematics.

The materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams, and symbols. The materials also use precise and accurate terminology and definitions when describing mathematics, and support students in using them. 

Examples of explicit instruction on the use of mathematical language include:

• In Unit 4, Lesson 2, Introduction, students are provided explicit instruction in the meaning of a pictograph as they learn to use them to represent data. The materials state, “Yesterday we worked so hard to categorize data and represent it in tally charts. Today we are going to represent using a new kind of graph--a pictograph! A pictograph represents data using pictures. There are important elements that must be part of a pictograph so people can understand it. (Reveal top quality graph VA as you go.) We need a title, category labels, and a KEY!”
• In Unit 5, Overview, Major Misconceptions & Clarifications states, “Misconception: Students struggle with the terms ‘quarter after’ and ‘quarter of’ since the coin ‘quarter’ represents 25 cents and is taught during this unit. Clarification: Have students shade their paper clocks into 4 quarters. Call attention to the fact that 4 quarters equal 1 dollar - and we name it a ‘quarter’ because it’s a quarter of a dollar.”
• In Unit 8, Lesson 3, Introduction, students are provided explicit instruction in the definition of an array, horizontally, and vertically, as they organize objects of equal groups into rows and columns. The materials state, “An array has rows that go horizontally, or side to side, and it has columns that go vertically, or up and down (fill in on VA). In an array, all of the rows are equal and all of the columns are equal. We can think of the rows as our groups (teacher rows and labels groups) and the columns help us see how many we have in each group (label on VA).”
• In Unit 9, Lesson 3, Introduction states, “Yesterday you used pattern blocks to divide shapes into equal parts-halves, thirds, and fourths. (TT) Teach your partner what you know about halves, thirds, and fourths. Halves means 2, the whole is divided into 2 equal parts, thirds means the whole is divided into 3 equal parts, fourths means the whole is divided into 4 equal parts.”

Examples of  the materials using precise and accurate terminology and definitions: 

• In Unit 2, Lesson 5, Introduction, accurate terminology is used as students use doubles to solve addition and subtraction problems. The materials state, “Skeleton VA: (remember that strategies section should be added/co-created with students during the intro and should include visual representations of strategies at work) 2. Solve. Addition: - Count on -Known doubles facts; Subtraction: -Count up -Count back -Known doubles addition facts.” 
• In Unit 3, Lesson 4, $$>\frac{2}{3}$$ Correct representation, students use accurate terminology as they discuss solving a word problem. The materials state, “How did you represent the problem? I represented with a tape diagram. I put 29 in the long box because Anthony has 29 cars and he has more cars than Jason. I put 16 in one small box because Anthony has 16 more cars than Jason. Then I put J in the other small box because we don’t know how many cars Jason has.” 
• In Unit 4, Lesson 3, Introduction, accurate terminology is used as students learn to create bar graphs to represent data. The materials state, “You know how to sort objects into categories to create tally charts and pictographs. Today we are going to use those categories to make bar graphs. (VH) Before we can make a bar graph what are some things we need in our graph? (Reveal top quality graph VA as you go) We need a title, categories, category labels, scale, scale labels, and bars! The scale is the number on the side of the bar graph that tells us how many in each category. (TT) How do I know where to stop on my scale? You need to find the category with the largest number, then you need to make sure the scale goes up to that number so that all of your data fits.”
• Unit 6, Lesson 2, Introduction, accurate terminology is used on a skeleton visual aid as students learn to represent three-digit numbers using multiple forms. Teachers fill in the different forms during instruction and leave the VA posted in the classroom. The materials state, “Writing 3-Digit Numbers, Standard Form 235, Place Value Models (Flats, Sticks, and Dots), Base Ten Numeral Form, Unit Form, Word Form, Expanded Form.”
• In Unit 9, Overview, Major Misconceptions & Clarifications states, “Misconception: Students confuse numerator and denominator. Clarification: Have students label their fraction with words. The numerator as the part and the denominator as all of the parts. Have VA posted and labeled and help students reference it until it becomes second nature.”