## Achievement First Mathematics

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### Overall Summary

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for Alignment to the CCSSM. In Gateway 1, the materials meet expectations for focus and coherence, and in Gateway 2, the materials meet expectations for rigor and practice-content connections.

###### Alignment
Meets Expectations
###### Usability
Does Not Meet Expectations

### Focus & Coherence

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for focus and coherence. For focus, the materials assess grade-level content and provide all students extensive work with grade-level problems to meet the full intent of grade-level standards. For coherence, the materials are coherent and consistent with the CCSSM.

##### Gateway 1
Meets Expectations

#### Criterion 1.1: Focus

Materials assess grade-level content and give all students extensive work with grade-level problems to meet the full intent of grade-level standards.

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for focus as they assess grade-level content and provide all students extensive work with grade-level problems to meet the full intent of grade-level standards.

##### Indicator {{'1a' | indicatorName}}

Materials assess the grade-level content and, if applicable, content from earlier grades.

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

• Unit 2, Addition & Subtraction to 100 Unit Assessment, Question 1b, “$$89 - 52 =$$ _____.”  (2.NBT.5)

• Unit 3, Story Problems Unit Assessment, Question 3, “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)

• Unit 6, Three Digit Numbers Unit Assessment, Question 13, “_____ - 100 = 280.” (2.NBT.8)

• Unit 9, Fractions Unit Assessment, Question 3, “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:

• Unit 4, Data Unit Assessment, Question 5, which is identified at 2.MD.10, “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.)

• Unit 5, Length, Money, Graphing and Time Unit Assessment, Question 5, which is identified as 2.MD.8, “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.) • Unit 8, Arrays Unit Assessment, Question 5, which is identified as 2.OA.3, “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.) • Unit 9, Fractions Unit Assessment, Question 8, which is identified as 2.G.3, “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}.) • Unit 10, Shapes Unit Assessment, Question 6, which is identified as 2.G.1, “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)). ##### Indicator {{'1b' | indicatorName}} Materials give all students extensive work with grade-level problems to meet the full intent of grade-level standards. The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards. There is one exception - a major work standard in Number & Operations Base Ten is not addressed as an instructional lesson. Students do not have the opportunity to engage with the full intent or extensive work of 2.NBT.6, Add up to four two-digit numbers using strategies based on place value and properties of operations. Students do not use strategies based on place value or properties of operations, and this standard is only practiced in workbook pages. • Units 4 and 5, Practice Workbook B, students engage in 2.NBT.6 by adding 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 students are not provided instruction on using strategies based on place value and properties of operations as the standard requires. Each unit consists of lessons that are broken into four components: Introduction, Workshop/ Discussion, Independent Practice, and Exit Ticket. In addition to lessons, there are Math Stories “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,” and Math Practice (Practice Workbook) for students “to build procedural skill and fluency.” Examples include: • Unit 1, Lesson 6, Workshop and Exit Ticket, students use mental benchmarks of a meter and a centimeter to estimate objects in the classroom (2.MD.3). The full intent of the standard is met as students estimate using the units of inches, feet, centimeters, and meters. 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, “Circle the most reasonable estimate for each object. a. Length of an eraser - 5 cm or 1 m.” • Unit 1, Lesson 14, Workshop Worksheet, students represent whole numbers on a number line and solve addition and subtraction equations within 100 on a number line diagram (2.MD.6). The full intent of this standard is met as students measure the lengths of objects using a ruler and are provided the opportunity to model word problems using the number line. Problem 2, “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. • Unit 10, Lesson 2, Workshop and Independent Practice Worksheet, students engage with 2.G.1 as they draw shapes having specified attributes such as a given number of angles or a given number of equal faces. Students identify triangles, quadrilaterals, hexagons, and cubes. During Workshop, students are given a description of shape attributes and asked to draw and name the shape. Independent Practice Problem 1, “A flat closed shape with 6 sides and 6 angles. What shape did you build?” In Lesson 3, students continue to engage with 2.G.1 as they recognize shapes having specified attributes such as a given number of angles or a given number of equal faces. During Introduction and Workshop, students play a game, Guess My Rule, where they are directed to sort quadrilaterals by thinking about how they are the same and how they are different based on their attributes. “Guess the Rule: Look at the quadrilaterals to figure out how they were sorted. Ask what does this group of quadrilaterals have that the other group doesn’t? How many equal angles? How many equal sides?” The full intent of this standard is met as the lessons addressing 2.G.1 engage students in both recognizing and drawing shapes based on specified attributes. • Unit 8, Lesson 2, Workshop and Independent Practice Worksheet, students engage with 2.OA.4 as they write repeated addition equations to match equal groups. Workshop, “Put the objects into equal groups and then write an addition sentence to match. (image shows 15 stars)” Lesson 3, Assessment Criteria, “Students should create arrays by putting the objects into equal groups, then putting those groups into rows. Lesson 3, Independent Practice Problem 7, “Draw two different rectangular arrays with 20 circles. Write a repeated addition sentence to match.” The full intent of the standard is met as students engage with rectangular arrays with up to five rows and five columns and they express the total as a sum of equal addends. #### Criterion 1.2: Coherence Each grade’s materials are coherent and consistent with the Standards. The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for coherence. The materials: address the major clusters of the grade, have supporting content connected to major work, make connections between clusters and domains, and have content from prior and future grades connected to grade-level work. ##### Indicator {{'1c' | indicatorName}} When implemented as designed, the majority of the materials address the major clusters of each grade. The materials reviewed for Achievement First Mathematics Grade 2 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each 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 materials focus on major work of the grade. A minute-level analysis is most representative of the materials because the minutes consider all components included during math instructional time. As a result, approximately 73% of the materials focus on major work of the grade. ##### Indicator {{'1d' | indicatorName}} Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade. The materials reviewed for Achievement First Mathematics Grade 2 meet expectations that supporting content 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: • 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, “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) • 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, ”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?” • 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, “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?” • 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.” • 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, “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, “Use the Animal Habitats table to answer the following questions.” Tally marks are used to record data for Forest, Wetlands, and Grasslands. Problem 3d, “How many total animal habitats were used to create this table?” ##### Indicator {{'1e' | indicatorName}} Materials include problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade. The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for including problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade. Examples of connections include: • 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, “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. • 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, “Solve using expanded notation. 362+427” • 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, “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, “Create a rectangular array using circles to solve the equation below. 4 + 4 + 4 + 4 + 4 = _____.” ##### Indicator {{'1f' | indicatorName}} Content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades. The materials reviewed for Achievement First Mathematics Grade 2 meet expectations that content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades. 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: • Unit 1, Lesson 2, Identify the Narrative, How does the learning connect to previous lessons? What do students have to get better at today?, “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.” • Unit 2, Addition and Subtraction to 100 Unit Overview, Identify the Narrative, “Next, students move into addition of two-digit numbers. Scholars were exposed to two-digit addition with regrouping at the end of first grade.” • Unit 4, Data Unit Overview, Identify the Narrative, “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.” • 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.” • Unit 10, Geometry- Shapes Unit Overview, Identify the Narrative, “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 Unit Overview documents contain an Identify the Narrative component that 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: • Unit 2, Addition and Subtraction to 100 Unit Overview, Identify The Narrative, Linking, “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.” • Unit 3, Story Problem Unit Overview, Identify the Narrative, Linking, “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.” • Unit 4, Data Unit Overview, Identify the Narrative, Linking, “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.” • Unit 7, Addition and Subtraction within 1000 Unit Overview, Identify the Narrative, Linking, “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.” • Unit 9, Geometry--Fractions Unit Overview, Identify the Narrative, Linking, “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.” ##### Indicator {{'1g' | indicatorName}} In order to foster coherence between grades, materials can be completed within a regular school year with little to no modification. The instructional materials reviewed for Achievement First Mathematics Grade 2 foster coherence between grades and can be completed within a regular school year with little to no modification. The Guide to Implementing AF, Grade 2 includes a scope and sequence. “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. ###### Overview of Gateway 2 ### Rigor & the Mathematical Practices The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for rigor and balance and practice-content connections. The materials reflect the balances in the Standards and help students develop conceptual understanding, procedural skill and fluency, and application. The materials make meaningful connections between the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs). ##### Gateway 2 Meets Expectations #### Criterion 2.1: Rigor and Balance Materials reflect the balances in the Standards and help students meet the Standards’ rigorous expectations, by giving appropriate attention to: developing students’ conceptual understanding; procedural skill and fluency; and engaging applications. The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for rigor. The materials develop conceptual understanding of key mathematical concepts, give attention throughout the year to procedural skill and fluency, spend sufficient time working with engaging applications of mathematics, and do not always treat the three aspects of rigor together or separately. ##### Indicator {{'2a' | indicatorName}} Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. he 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 content standards or cluster headings. Materials include problems and questions that develop conceptual understanding throughout the grade-level. Examples include: • 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.” • 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. “Kaleb has 3 boxes of 100 crayons, 6 boxes of 10 crayons, and 2 single crayons. How many crayons does Kaleb have?” • 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, “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.” • 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. “SWBAT add 2-digit numbers with regrouping in one place by using flats, sticks, and dots.” Workshop Worksheet example, Problem 2A, “$$550 + 268 =$$ ______.” • 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. “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: • 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, “$$62 + 27 =$$ ___.” Students are directed to use sticks and dots or expanded notation to solve. • 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, “Circle which set of sticks and dots will help to find the total? 62 + 24 = ______.” • 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, “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?” • 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, “_____ - 567 = 293. Explain how you solved.” • 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, “Solve using flats, sticks, and dots. 531 - 258 = ____.” ##### Indicator {{'2b' | indicatorName}} Materials give attention throughout the year to individual standards that set an expectation for procedural skill and fluency. The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for giving attention throughout the year to individual 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 materials develop procedural skill and fluency throughout the grade-level. Examples include: • 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, “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.” • 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. “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.” • 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, “$$10 + 5 = 15$$ • 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, “Count from 90 to 300.” The materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade-level. Examples include: • 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, “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. • 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, “$$2 + 4 =$$ ___.” • 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, “$$53 -$$ ______ = 28.” 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, “Solve each problem using mental math, 678 + 100 = ____.” ##### Indicator {{'2c' | indicatorName}} Materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Students are given multiple opportunities to engage in real-world applications, especially during Math Stories, which include both guided questioning and independent work time, and Exit Tickets to independently show their understanding. Materials include multiple routine and non-routine applications of the mathematics throughout the grade level. Examples include: • Unit 1, Guide to Implementing AF Math, Math Stories, September, students engage with 2.G.1, recognize and draw shapes with specified attributes, in a non-routine problem. Sample Problem 3, “Corie is making three-dimensional shapes in art class. He has made 18 faces in total. What kind of shapes could he have made and how many of each?” • Unit 3, Guide to Implementing AF Math, Math Stories, December, students engage in a routine problem 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, “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?” • Unit 8, Guide to Implementing AF Math, Math Stories, May, students engage with 2.OA.4, using addition to find the total number of objects in a regular array, in a non-routine problem. Sample Problem 15, “A tic-tac-toe board has 3 columns with 3 rows in each. How many different ways could someone win the game (get 3 in a row)? ” Materials provide opportunities for students to independently demonstrate routine and non-routine applications of the mathematics throughout the grade level. Examples include: • Unit 3, Lesson 7, Independent Practice, students engage in a routine word problem 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, “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?” • Unit 5, Lesson 3, Independent Practice, students engage in a routine word problem with 2.MD.8, solve word problems involving money, as they independently solve word problems with money. Problem 5, “King Jamonie has 3 quarters, 1 dime, 2 nickels, and 4 pennies. How much money does he have?” • Unit 6, Lesson 17, Workshop, students engage in a non-routine word problem with 2.NBT.4, compare two three-digit numbers. “Nehemiah and Sonya are exercising to stay healthy. In the month of January, Nehemiah exercised 198 minutes for the first week and 277 the second week. Sonya exercised 309 the first week and 172 the second week. Write a comparison statement to show who exercised more. Show all of your mathematical thinking.” ##### Indicator {{'2d' | indicatorName}} 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. The materials reviewed for Achievement First Mathematics Grade 2 meet expectations in that 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. Examples include: Conceptual understanding • 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, “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?” • 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, “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)” • 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, “Solve using flats, sticks, and dots ______ - 348 = 650. Explain how you solved ______ - 348 = 650.” • 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, “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) • 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, “Solve. 7 + 8 = _____.” • 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, “Solve. 93 - ____ = 62.” Students are provided with a blank number bond model. • 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, “$$35 +$$ ___ = 50.” • 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, “$$45 +$$ ___ = 100 Application • 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, “Jose has 27 erasers. Kate gave him some more. Now he has 53 erasers. How many erasers did Kate give him?” • 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, “Ms. Reinhardt has 42 books. Ms. Gomez has 18 fewer books than Ms. Reinhardt. How many books does Ms. Gomez have?” • 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, “There are 59 girls on the bus. There are 26 more girls than boys on the bus. How many boys are on the bus?” • 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, “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. Examples include: • 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, “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.” • 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, “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)” • 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, “Jacob bought a piece of gum for 26 cents and a newspaper for 61 cents. He gave the cashier1. How much money did he get back?”

• 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 7, “Ja-yier put 5 toys into 4 different baskets. How many toys does Ja-yier have in all?”

#### Criterion 2.2: Math Practices

Materials meaningfully connect the Standards for Mathematical Content and Standards for Mathematical Practice (MPs).

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for practice-content connections. The materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs).

##### Indicator {{'2e' | indicatorName}}

Materials support the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. The Standards for Mathematical Practice are identified and incorporated within mathematics content throughout the grade level. The Mathematical Practices are listed in the Unit Overviews as well at the beginning of each lesson.

There is intentional development of MP1 to meet its full intent in connection to grade-level content. Examples include:

• Unit 2, Lesson 6, Narrative, What is new/or hard about the lesson? provides an explanation to the teacher about how the content engages with MP1. “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, “Carla baked 15 desserts. 9 of them were chocolate chip cookies and the rest were brownies. How many brownies did Carla bake?”

• Unit 4, Assessment, students make sense of data presented in a graph to solve two-step story problems. Problem 4, “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.

• Unit 8, Lesson 8, Introduction, students make sense of even/odd. “There are 6 students in Mr. Johnson’s reading group. If he wants his students to work in pairs, will everyone have a partner? Use pictures to prove your thinking.”

There is intentional development of MP2 to meet its full intent in connection to grade-level content. Examples include:

• Unit 1, Lesson 7, Narrative, “Students also engage with MP 2 by reasoning abstractly and quantitatively when they discuss and explain how much longer/shorter one line is than the other line. Students’ understanding of quantities will help then determine if their calculations are reasonable by determining if their answer makes sense. Students use reasoning of abstract length through questions like ‘how can we find out how long is Line A that Line B?’ and ‘How does this work?’”

• Unit 5, Lesson 5, Independent Practice Worksheet, students reason through story problems involving coins, requiring them to abstract the value of a set of coins, to find the total value. Problem 4, “Enrique has 3 quarters, 1 dime, 2 nickels, and 4 pennies. How much money does he have?”

• Unit 7, Lesson 15, Narrative, “Students engage with MP 2 as students reason abstractly when they discuss the relationship between subtraction and addition and the part-part-whole relationship to solve problems with unknowns in all positions.”

##### Indicator {{'2f' | indicatorName}}

Materials support the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

There is intentional development of MP3 to meet its full intent in connection to grade-level content. Examples include:

• 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. “(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. What do you think? What mistake did I make? How can I fix it?”

• Unit 3, Lesson 2, Introduction, Problem 2, “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?”

• Unit 5, Lesson 5, Independent Practice, Check for Understanding, “Why did you trade ___ for ___? What strategy did you use to make trades?”

• Unit 7, Lesson 19, Assessment, Problem 4, “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.”

• Unit 8 Assessment, students use their knowledge of arrays to analyze the mathematical thinking of a fictitious student. Item 1, “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.”

• Unit 9, Lesson 4, Intro, Problem 2, “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?”

##### Indicator {{'2g' | indicatorName}}

Materials support the intentional development of MP4: Model with mathematics; and MP5: Choose tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Choose tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

There is intentional development of MP4 to meet its full intent in connection to grade-level content. Examples include:

• Math Stories Guide, Promoting Reasoning through the Standards for Mathematical Practice, MP4, “Math Stories help elementary students develop the tools that will be essential to modeling with mathematics. In early elementary, students become familiar with how representations like equations, manipulatives, and drawings can represent real-life situations.” Within the K-4 Math Stories Representations and Solutions Agenda, students are given time to represent, retell, and solve the problem on their own.

• Unit 3, Lesson 2, Narrative, “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, “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?"

• Unit 8, Lesson 6, Narrative, “Students engage with MP4 as they model with math tiles and drawings to analyze the relationship between rows and columns of arrays to rectangular arrays. The use of the math models to help deepen their understanding of rectangular arrays and why they work.” Pose the Problem, “Use your tiles to make a rectangular array with 12 total squares and 4 columns. Then draw the array and write a repeated addition sentence to match.”

• Unit 9, Lesson 8, Exit Ticket, “Leani and Carla each baked a cake in the same rectangular pan. Leani ate one-fourth of her cake. Carla ate one-third of her cake. a. Show how Leani and Carla cut their cake. Share the fraction that they ate. Who ate more cake? How do you know?” Narrative, “Students also engage in MP4 through the story problem protocol to use mathematical models and connect it back to the story problem. Students model the story problem with an appropriate representation and use an appropriate strategy to explain their answer. Students also engage with MP4 as they determine if their answer makes sense and connect their answer back to the story problem by finishing the story. This process occurs with each story problem they solve throughout the lesson.”

There is intentional development of MP5 to meet its full intent in connection to grade-level content. Examples include:

• Unit 1, Lesson 6, Narrative, “Students engage with MP 5 as they explain and justify which math tools to use for estimating the measurements either with centimeters or a meters. Students will decide and justify which mental benchmarks of an M&M is about the same size as a centimeter and their opposite shoulder to fingertips is about the same length as a meter to estimate the best measurement as they solve each problem throughout the lesson.”

• Unit 2, Lesson 23, Narrative, “Students engage with MP5 as they solve problems with missing addends using a strategy of their choice: sticks and dots, expanded notation, or a number line to solve. The teacher and students make connections between the various strategies and help students become more efficient in solving throughout the lesson.”

• Unit 6, Lesson 12, the Exit Ticket shows a table with the columns from left to right having 100 less, 10 less, Starting Number-565, 10 more, 100 more. Students are to complete the table and are not guided to use any specific tools or strategies. “Students engage with MP5 by using a strategy to represent the problem that works best for them to add or subtract by 10 or 100.” In the lesson, the first problem of the bingo game, “How can I represent 38 + ____ = 72 with a number bond? How can we figure out the missing part? We can use a number line and count up, we could subtract using sticks and dots or expanded notation.”

At times, the materials are inconsistent. The Unit and Lesson Overview narratives describe explicit connections between the MPs and content, but lessons do not always align to the stated purpose.

The materials do not provide students with opportunities or guidance to identify and use relevant external mathematical tools and resources, such as digital content located on a website.

##### Indicator {{'2h' | indicatorName}}

Materials attend to the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

There is intentional development of MP6 to meet its full intent in connection to grade-level content. Many problems present students with the opportunity to attend to precision within the mathematics and the reasoning of the answer. Examples include:

• Unit 1, Lesson 1, Introduction, students attend to precision with measurement. “How long is it? How can I measure the crayon? Note: Teacher should model each measuring strategy as it is named. Measure the longest part: When we measure length we measure the LONGEST part of the object. Line up the endpoints.” Students might say, “You need to start at one end and go to the other end. No gaps or overlap. You need to make sure the cubes are right next to each other and there aren’t gaps or overlaps.”

• Unit 4, Lesson 2, Introduction, students attend to precision when categorizing items in a picture graph. “Why is it important to draw all your pictures the same size? It is important to draw all the pictures the same size because it makes it easier to read the graph. If all the pictures are the same size we can easily tell which group has more or less.”

• Unit 9, Lesson 3, Introduction, students attend to precision when differentiating between parts using the concept of division. “Yesterday you used pattern blocks to divide shapes into equal parts-halves, thirds, and fourths. 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.”

The instructional materials attend to the specialized language of mathematics. The materials use precise and accurate mathematical terminology. Examples include:

• Unit 4, Lesson 3, Introduction, teachers use accurate terminology to create bar graphs. “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. Before we can make a bar graph what are some things we need in our graph? 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. 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 8, Lesson 3, Introduction, teachers provide explicit instruction in the definition of an array, horizontally, and vertically, as they organize objects of equal groups into rows and columns. “An array has rows that go horizontally, or side to side, and it has columns that go vertically, or up and down. 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 and the columns help us see how many we have in each group.”

• Unit 9, Overview, Major Misconceptions & Clarifications, “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.”

##### Indicator {{'2i' | indicatorName}}

Materials support the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards.

There is intentional development of MP7 to meet its full intent in connection to grade-level content. Examples Include:

• Unit 2, Lesson 13, Narrative, “Students will learn the structure within the number line representation to explain why and how it works.”

• Unit 6, Lesson 12, Narrative, “Students also engage with MP 7 as they discuss and make connections between the place values and adding/subtracting by 10 or 100. Students are able to recognize that adding/subtracting by 10 changes the number on the tens place by 1 and adding/subtracting by 100 changes the digit in the hundreds place by 1.” Exit Ticket Problem 1 has a table with the following listed from left to right: “100 less, 10 less, starting number -565, 10 more, and 100 more.”

• Unit 8, Lesson 7, Workshop Worksheet, students decompose numbers and apply repeated addition to the structure of arrays to both the rows and columns. Question 1, “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.”

There is intentional development of MP8 to meet its full intent in connection to grade-level content. Examples Include:

• Unit 7, Lesson 4, Narrative, “Students continue to deepen their understanding of the structure of adding starting with the correct place and also recognizing the repeated reasoning and calculations involved in regrouping when there are 10 or more in any place to make a bundle of a ten or a hundred when adding.”

• Unit 8, Lesson 10, Introduction, Pose the Problem, students engage in repeated reasoning. “(Have VA (visual aid) with #s 1-20 written across the top, drawing of 2 empty 10-frames.) How many counters do we have? Is 2 an even number? How do you know? How many counters now? Is 3 even or odd? How do you know? How many counters altogether? Is 4 odd or even? How do you know? What numbers on the poster are even? How do you know? What do you notice about the numbers we circled? Do you see a pattern?”

• Unit 9, Lesson 7, Narrative, “How does the learning connect to previous lessons? What do students have to get better at today? 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.”

### Usability

The materials reviewed for Achievement First Mathematics Grade 2 do not meet expectations for Usability. The materials partially meet expectations for Criterion 1, Teacher Supports, partially meet expectations for Criterion 2, Assessment, and do not meet expectations for Criterion 3, Student Supports.

##### Gateway 3
Does Not Meet Expectations

#### Criterion 3.1: Teacher Supports

The program includes opportunities for teachers to effectively plan and utilize materials with integrity and to further develop their own understanding of the content.

The materials reviewed for Achievement First Mathematics Grade 2 partially meet expectations for Teacher Supports. The materials: provide teacher guidance with useful annotations and suggestions for enacting the materials, include standards correlation information that explains the role of the standards in the context of the overall series, and provide a comprehensive list of supplies needed to support instructional activities. The materials contain adult-level explanations and examples of the more complex grade-level concepts, but do not contain adult-level explanations and examples and concepts beyond the current grade so that teachers can improve their own knowledge of the subject. The materials provide explanations of the instructional approaches of the program but do not contain identification of the research-based strategies.

##### Indicator {{'3a' | indicatorName}}

Materials provide teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development. Teacher guidance is found throughout the materials in the Implementations Guides, Unit Overviews, and individual lessons.

Materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials. Examples include:

• The Guide to Implementing AF Math provides a Program Overview for the teacher with information on the program components and scope and sequence. This includes descriptions of the types of lessons, Math Stories, Math Practice, and Cumulative Review.

• The Math Stories Guide (K-4) provides a framework for problem solving.

• Each Unit Overview includes a section called “Key Strategies” that describes strategies that will be utilized during the unit.

• The Teacher’s Guide supports whole group/partner discussion, ask/listen fors, common misconceptions and errors.

• In the narrative information for each lesson, there is information such as “What do students have to get better at today? Where will time be focused/funneled?”

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. Each lesson includes anticipated challenges, misconceptions, key points, sample dialogue, and exemplar student responses. Examples from Unit 7, Addition and Subtraction with 1000, Lesson 6 include:

• “What is new and/or hard about the lesson? Students need to mentally decompose numbers into hundreds, tens, and ones in order to make jumps on the number line to add two 3-digit numbers together. Some scholars may also make jumps of other quantities (ex. Jump of 3 from 237 to get to 240 instead of making 3 jumps of 1). They also make use of their understanding of part-part-whole relationships. Scholars need to understand that when adding on a number line, they can start at one of the parts and make jumps forward of the other part to get the whole or the total. If students are still struggling to add on a number line, they will likely struggle to subtract 3-digit numbers on a number line, especially if the misunderstanding is rooted in part-part-whole relationships.”

• “Exemplar Student Response: I solved 436 + 249 on a number line. I started at 436. Then I made 2 jumps up of 100 b/c there are 2 hundreds in 249. Next I made 4 jumps of 10 b/c there are 4 tens in 249. Last I made a jump of 4 + 6 jump of 5 b/c 4 + 5 = 9. There are 9 ones in 249. I counted up on the number line and got 675. This proves 436 + 249 = 675.”

• “Potential Misconception: Not knowing the next number in the sequence when skip counting or counting by ones. Starting with smaller number (this works but is less efficient). Starting at one part and jumping up the same part (ex. Solving 356 + 291 by starting at 356 and making 3 jumps of 100, 5 jumps of 10, and 6 jumps of 1).”

• “Mid-Workshop Interruption: If > \frac{2}{3} of students are successfully solving, share out someone whose work is neat and organized and is clearly showing regrouping. Discuss the jumps they made and why they worked. If < \frac{2}{3} of students are successful, call students back together to clear up the misconception through a misconception protocol. Continue to circulate and check for students to apply the learning. Make note of student success in applying in your Rapid Feedback tracker to inform the path for the Discussion.”

• “Share/Discussion: Direct students to the Discussion work space in their packets as needed. Use workshop data to determine the appropriate discussion path: Facilitate a discussion around a major misconception, Share example where student skip counted incorrectly (likely over decades or centuries) or made incorrect jumps). Show non-example and related example: Which is correct? Why doesn’t ___’s work? OR, 2-3 students share their work/strategies, What is the same about these strategies? What is different? OR, ask students to apply their learning in a new way with an additional exercise.”

Each lesson includes both “What” and “How” Key Point sections that describe what students should know and be able to do and how they will do it. Examples from Unit 7, Addition and Subtraction with 1000, Lesson 6 include:

• “What Key Points: We can solve 3-digit addition problems on a number line by starting at one of the parts and jumping forward the other part to find the whole/total. When adding on a number line, we can use place value to help us tell how many jumps of hundreds, ten, and one to make. Understand that if we start at a part and jump up the other part we will get the whole/total because part + part = whole.”

• “How Key Points: When we add 3-digit numbers on a number line…. First we start at a part, Then we jump up the other part by making jumps of 100s, 10s, and 1s.”

##### Indicator {{'3b' | indicatorName}}

Materials contain adult-level explanations and examples of the more complex grade-level/course-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject.

The materials reviewed for Achievement First Mathematics Grade 2 partially meet expectations for  containing adult-level explanations and examples of the more complex grade/course-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject. There is very little reference or support for content in future courses.

Materials contain adult-level explanations and examples of the more complex grade/course-level concepts so that teachers can improve their own knowledge of the subject. Examples include:

• Unit Overviews provide thorough information about the content of the unit which often includes definitions of terminology, explanations of strategies, and the rationale about incorporating a process. Unit 4 Overview, Identify the Narrative, “Students may be challenged by reading a graph. They are used to reading from left to right. Reading a graph requires students to interpret the information both horizontally and vertically. Students may need to put a finger on the horizontal axis and another finger on the vertical axis and then move the fingers until they intersect.”

• The Unit Overview includes an Appendix titled “Teacher Background Knowledge” which includes a copy of the relevant pages from the Common Core Math Progression documents which includes on grade-level information.

Materials do not contain adult-level explanations and examples of concepts beyond the current course so that teachers can improve their own knowledge of the subject. Examples include:

• The Common Core Math Progression documents in the Appendix are truncated to the current grade level and do not go beyond the current course.

##### Indicator {{'3c' | indicatorName}}

Materials include standards correlation information that explains the role of the standards in the context of the overall series.

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.

Correlation information is present for the mathematics standards addressed throughout the grade level/series. Examples include:

• Guide to Implementing AF Grade 2, Program Overview, “Scope and Sequence Detail is designed to help teachers identify the standards on which each lesson within a unit is focused, whether on grade level or not. You will find the daily lesson aims within each unit and the content standards addressed within that lesson. A list of the focus MPs for each lesson and unit and details about how they connect to the content standards can be found in the Unit Overviews and daily lesson plans.”

• The Program Overview informs teachers “about how to ensure scholars have sufficient practice with all of the Common Core State Standards. Standards or parts thereof that are bolded are addressed within a lesson but with limited exposure. It is recommended that teachers supplement the lessons addressing these standards by using the AF Practice Workbooks to ensure mastery for all students. Recommendations for when to revisit these standards during Math Practice and Friday Cumulative Review are noted in the Practice section of each unit.”

• The Unit Overview includes a section called Identify Desired Results: Identify the Standards which lists the standards addressed within the unit and previously addressed standards that relate to the content of the unit.

• In the Unit Overview, the Identify The Narrative provides rationale about the unit connections to previous standards for each of the lessons. Future grade-level content is also identified.

• The Unit Overview provides a table listing Mathematical Practices connected to the lessons and identifies whether the MP is a major focus of the unit.

• At the beginning of each lesson, each standard is identified.

• In the lesson overview, prior knowledge is identified, so teachers know what standards are linked to prior work.

Explanations of the role of the specific grade-level/course-level mathematics are present in the context of the series.

In the Unit Overview, the Identify the Narrative section provides the teacher with information to unpack the learning progressions and make connections between key concepts. Lesson Support includes information about connections to previous lessons and identifies the important concepts within those lessons. Examples include:

• Unit 7, Lesson 8 Narrative, “How does the learning connect to previous lessons? What do students have to get better at today? In the previous lessons, students added 3-digit numbers with regrouping in both places using flats, sticks, and dots, a number line, and expanded notation. Today, for the first time, students will subtract 3-digit numbers using flats, sticks, and dots, and expanded notation.”

##### Indicator {{'3d' | indicatorName}}

Materials provide strategies for informing all stakeholders, including students, parents, or caregivers about the program and suggestions for how they can help support student progress and achievement.

The materials reviewed for Achievement First Mathematics Grade 2 provides strategies for informing all stakeholders, including students, parents, or caregivers about the program and suggestions for how they can help support student progress and achievement.

• The Unit Overview includes a parent letter in both English and Spanish for each unit that includes information around what the students are working on and example strategies students will use. The letter includes information about common mistakes that parents can watch for as well as links to websites that can provide assistance.

• There is also a suggestion related to the Unit Overview, “This guide can be printed and sent home to families so that parents/guardians can better support their scholars with homework.”

##### Indicator {{'3e' | indicatorName}}

Materials provide explanations of the instructional approaches of the program and identification of the research-based strategies.

The materials reviewed for Achievement First Mathematics Grade 2 partially meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies.

Materials explain the instructional approaches of the program.

• The Implementation Guide states, "Our program aims to see the mathematical practices come to life through the shifts (focus, coherence, rigor) called for by the standards. For students to engage at equal intensities weekly with all 3 tenets, we structured our program into three main daily components Monday-Thursday: Math Lesson, Math Stories and Math Practice. Additionally, students engage in Math Cumulative Review each Friday in order for scholars to achieve the fluencies and procedural skills required."

• The Implementation Guide includes descriptions of “Math Lesson Types.” Descriptions are included for Game Introduction Lesson, Task Based Lesson, Math Stories, and Math Practice. Each description includes a purpose and a table that includes the lesson components, purpose, and timing.

Materials do not include and reference research-based strategies.

• The materials do not explicitly name any strategies as research-based strategies.

##### Indicator {{'3f' | indicatorName}}

Materials provide a comprehensive list of supplies needed to support instructional activities.

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for providing a comprehensive list of supplies needed to support instructional activities.

Each lesson includes a list of materials specific to the lesson. Examples include:

• Unit 2, Lesson 1, the Lesson Overview includes, “Materials: dot cubes, recording sheets, What’s Missing VA, and Rapid Feedback tracker.”

• Unit 9, lesson 7, the Lesson Overview includes, “Materials: handouts, scissors, glue, posters.”

##### Indicator {{'3g' | indicatorName}}

This is not an assessed indicator in Mathematics.

##### Indicator {{'3h' | indicatorName}}

This is not an assessed indicator in Mathematics.

#### Criterion 3.2: Assessment

The program includes a system of assessments identifying how materials provide tools, guidance, and support for teachers to collect, interpret, and act on data about student progress towards the standards.

The materials reviewed for Achievement First Mathematics Grade 2 partially meet expectations for Assessment. The materials identify the standards, but do not identify the practices assessed for the formal assessments. The materials provide multiple opportunities to determine students' learning and sufficient guidance to teachers for interpreting student performance but do not provide suggestions for following-up with students. The materials include opportunities for students to demonstrate the full intent of grade-level/course-level standards and practices across the series.

##### Indicator {{'3i' | indicatorName}}

Assessment information is included in the materials to indicate which standards are assessed.

The materials reviewed for Achievement First Mathematics Grade 2 partially meet expectations for having assessment information included in the materials to indicate which standards are assessed. There are connections identified for standards, but not the mathematical practices.

Materials identify the standards assessed for the formal assessments. Examples include:

• Each Unit Overview provides a chart that identifies CCSS Math Content standards for each item on the Unit Assessment. Occasionally, an individual item on the assessment identifies the standard, but in general, student-facing assessments do not include the standards.

• Each lesson includes an Exit Ticket that aligns with the standard of the lesson.

Materials do not identify the practices assessed for the formal assessments.

##### Indicator {{'3j' | indicatorName}}

Assessment system provides multiple opportunities throughout the grade, course, and/or series to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

The materials reviewed for Achievement First Mathematics Grade 2 partially meet expectations for including an assessment system that provides multiple opportunities throughout the grade, course, and/or series to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

The assessment system provides multiple opportunities to determine students' learning and sufficient guidance to teachers for interpreting student performance but does not provide suggestions for following-up with students. Examples include:

• Assessments include an informal Exit Ticket in each lesson and a formal Unit Assessment for every unit.

• There is guidance, or “look-fors,” for teachers about what the student should be able to do on the assessments.

• All Unit Assessments include an answer key with exemplar student responses.

• Some units include a section titled “Evaluating Student Learning Outcomes” which provides a detailed description of what the student is expected to do.

• There are no strategies or suggestions if students do not demonstrate understanding of the concept, and no next steps based on the results of the assessment.

##### Indicator {{'3k' | indicatorName}}

Assessments include opportunities for students to demonstrate the full intent of grade-level/course-level standards and practices across the series.

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series. There are a variety of question types including multiple choice, short answer, and constructed response. Mathematical practices are embedded within the problems.

Assessments include opportunities for students to demonstrate the full intent of grade-level standards across the series. Examples include:

• In the Unit 3 assessment, the full-intent of standard 2.OA.1 (use addition and subtraction within 100 to solve one- and two-step word problems) is met. Item 6, “Mayshon’s string is 21 cm shorter than Bernadette’s string. Mayshon’s string is 18 cm long. Bernadette cut 12 cm off her string. Now how long is Bernadette’s string?” There are nine available items, varied addition and subtraction situations including two-step problems, and space is provided for students to use drawings and equations to solve.

• In the Unit 5 assessment, the full-intent of standard 2.NBT.3 (read and write numbers to 1000 using base-ten numerals, number names, and expanded form) is met. Item 2, “Write the value of 17 tens three different ways. Use the largest numerals possible (standard form, expanded form, unit form). Item 6, “Draw flats, sticks, and dots on the place value chart to show 348.”

• In Unit 9, the full-intent of standard 2.G.3 (partition circles and rectangles into two, three, and four equal shares) is met. Item 2, “1 whole = ___ halves; __ fourths = 1 whole; ___thirds = 1 whole.” Item 3, “Use lines to partition the rectangles into fourths in different ways. (three rectangles are provided).” Item 5, “Circle all the rectangles that are partitioned into fourths and cross out any rectangle that is not partitioned into fourths. (4 rectangles provided; 3 that are correctly partitioned into fourths and one that is not correctly partitioned into fourths). Explain why the rectangles that you crossed out do not show fourths.”

Assessments include opportunities for students to demonstrate the full intent of grade-level practices across the series. Examples include:

• Unit 3 Assessment, Item 1, supports the full development of MP1: Make sense of problems and persevere in solving them and MP2: Reason abstractly and quantitatively. “Miss Taylor had a ribbon. She cut off 23 inches of the ribbon. Now the ribbon is 57 inches long. How long was Miss Taylor’s ribbon to start?”

• Unit 5 Assessment, Item 6, students engage with MP3: Construct viable arguments and critique the reasoning of others. “Dan has 35 cents. Emily says he could have 3 dimes and a nickel. Jeremy says he could have a quarter and a dime. Who is right and how do you know?”

• Unit 7 Assessment, Item 5, supports the full development of MP3: Construct viable arguments and critique the reasoning of others, MP6: Attend to precision, and MP7: Look for and make use of structure. “Ava solved the problem below using expanded notation. What mistake did she make? How can she fix it? 603 - 246 = ____(work shown of the vertical form of expanded notation with the mistake made of regrouping the tens place of 603 to 100).”

##### Indicator {{'3l' | indicatorName}}

Assessments offer accommodations that allow students to demonstrate their knowledge and skills without changing the content of the assessment.

The materials reviewed for Achievement First Mathematics Grade 2 do not provide assessments which offer accommodations that allow students to demonstrate their knowledge and skills without changing the content of the assessment. This is true for both formal unit assessments and informal exit tickets.

#### Criterion 3.3: Student Supports

The program includes materials designed for each child’s regular and active participation in grade-level/grade-band/series content.

The materials reviewed for Achievement First Mathematics Grade 2 do not meet expectations for Student Supports. The materials do not provide strategies and supports for students in special populations or for students who read, write, and/or speak in a language other than English to support their regular and active participation in learning grade-level/series mathematics. The materials provide some extensions and/or opportunities for students to engage with grade-level/course-level mathematics at higher levels of complexity.

The materials do provide manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

##### Indicator {{'3m' | indicatorName}}

Materials provide strategies and supports for students in special populations to support their regular and active participation in learning grade-level/series mathematics.

The materials reviewed for Achievement First Mathematics Grade 2 do not meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level/series mathematics.

The materials do not provide specific strategies and supports for differentiating instruction to meet the needs of students in special populations.

##### Indicator {{'3n' | indicatorName}}

Materials provide extensions and/or opportunities for students to engage with grade-level/course-level mathematics at higher levels of complexity.

The materials reviewed for Achievement First Mathematics Grade 2 partially meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.

Some lessons include an Extension/Application Problem that allows teachers to extend the thinking of the lesson during discussion. However, extension problems are intended for all students as there is no identified differentiation for advanced students. Examples Include:

• Unit 1, Lesson 13, Extension/Application Problem, “Hudson measured a desk using inches. Logan measured the same desk using feet. Which boy will have more units? How do you know?” This is an extension of this lesson because the problem focuses on analyzing the same object using two different units and the lesson focuses on measuring and comparing using inches, feet, and yards.

• Unit 8, Lesson 5, Extension, students build on their knowledge of rectangular arrays in the Extension problem. The materials state, “Students who are showing mastery can make the connection to multiplication and also show a multiplication sentence to match their rectangular arrays.”

##### Indicator {{'3o' | indicatorName}}

Materials provide varied approaches to learning tasks over time and variety in how students are expected to demonstrate their learning with opportunities for students to monitor their learning.

The materials reviewed for Achievement First Mathematics Grade 2 provide varied approaches to learning tasks over time and variety in how students are expected to demonstrate their learning; however, there are no opportunities for students to monitor their learning.

The program uses a variety of formats and methods over time to deepen student understanding and ability to explain and apply mathematics ideas. These include: Exercise Based Lessons, Task Based Lessons, Math Stories, Math Practice, and Cumulative Review.

In the lesson introduction, the teacher states the aim and connects it to prior knowledge. In Pose the Problem, the students work with a partner to represent and solve the problem. Then the class discusses student work. The teacher highlights correct work and common misconceptions. Then students work on the Workshop problems, Independent Practice, and the Exit Ticket. Students have opportunities to share their thinking as they work with their partner and as the teacher prompts student responses during Pose the Problem and Workshop discussions. Math Stories provide opportunities for students to question, investigate, sense-make, and problem-solve using a variety of formats and methods.

##### Indicator {{'3p' | indicatorName}}

Materials provide opportunities for teachers to use a variety of grouping strategies.

The materials reviewed for Achievement First Mathematics Grade 2 provide some opportunities for teachers to use a variety of grouping strategies. Grouping strategies within lessons are not consistently present or specific to the needs of particular students. There is no specific guidance to teachers on grouping students.

The majority of lessons are whole group and independent practice; however, the structure of some lessons include grouping strategies, such as working in a pair for games, turn-and-talk, and partner practice. Examples include:

• Unit 1, Lesson 1, Narrative, “Teachers should encourage students in the turn and talk to agree or disagree with their partners and explain why.”

• Unit 7, Lesson 7, Introduction, “Give students a few minutes to work with a partner and to come up with a solution. Circulate as they work, see how they are doing.”

##### Indicator {{'3q' | indicatorName}}

Materials provide strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

The materials reviewed for Achievement First Mathematics Grade 2 do not meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

Materials do not provide any resources for students who read, write, and/or speak in a language other than English to meet or exceed grade-level standards through regular and active participation in grade-level mathematics.

##### Indicator {{'3r' | indicatorName}}

Materials provide a balance of images or information about people, representing various demographic and physical characteristics.

The materials reviewed for Achievement First Mathematics Grade 2 provide a balance of images or information about people, representing various demographic and physical characteristics. Examples include:

• Lessons portray people from many ethnicities in a positive, respectful manner.

• There is no demographic bias seen in various problems.

• Names in the problems include multi-cultural references such as Mario, Tanya, Kemoni, Jiang, Paige, and Tomi.

• The materials are text based and do not contain images of people. Therefore, there are no visual depiction of demographics or physical characteristics.

• The materials avoid language that might be offensive to particular groups.

##### Indicator {{'3s' | indicatorName}}

Materials provide guidance to encourage teachers to draw upon student home language to facilitate learning.

The materials reviewed for Achievement First Mathematics Grade 2 do not provide guidance to encourage teachers to draw upon student home language to facilitate learning.

The materials do not provide suggestions or strategies to use the home language to support students in learning mathematics. There are no suggestions for teachers to facilitate daily learning that builds on a student’s multilingualism as an asset nor are students explicitly encouraged to develop home language literacy. Teacher materials do not provide guidance on how to garner information that will aid in learning, including the family’s preferred language of communication, schooling experiences in other languages, literacy abilities in other languages, and previous exposure to academic everyday English.

##### Indicator {{'3t' | indicatorName}}

Materials provide guidance to encourage teachers to draw upon student cultural and social backgrounds to facilitate learning.

The materials reviewed for Achievement First Mathematics Grade 2 do not provide guidance to encourage teachers to draw upon student cultural and social backgrounds to facilitate learning.

The materials do not make connections to linguistic and cultural diversity to facilitate learning. There is no teacher guidance on equity or how to engage culturally diverse students in the learning of mathematics.

##### Indicator {{'3u' | indicatorName}}

Materials provide supports for different reading levels to ensure accessibility for students.

The materials reviewed for Achievement First Mathematics Grade 2 provide supports for different reading levels to ensure accessibility for students.

Strategies used include: teacher reading the problem, visualizing, and creating “mind-movies.” For example:

• Unit 1 Lesson 2, Introduction, Pose the Problem, “Visualize the story problem as I read it: 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?”

##### Indicator {{'3v' | indicatorName}}

Manipulatives, both virtual and physical, are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

The materials reviewed for Achievement First Mathematics Grade 2 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

Manipulatives are accurate representations of mathematical objects and are connected to written methods. Examples include:

• Unit 2, Lesson 7, to address standard 2.NBT.5 (fluently add and subtract within 100 using strategies based on place value…), the materials include place value blocks/cubes. In the Overview, Identify the Narrative, “Throughout the unit, manipulatives and math drawings  allow students to see the numbers in terms of place value units and serve as a reminder that they must add like units (e.g. knowing that 74 + 38 is 7 tens plus 3 tens and 4 ones plus 8 ones.)” Introduce the Math, “You learned about two-digit numbers in first grade. Today, you get to show what you know about two-digit numbers and work to represent them with sticks and dots and expanded notation.”

• Unit 6, Lesson 2, to address standard 2.NBT.1(understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones), the materials include place value blocks. In Overview, Identify the Narrative, “They represent three-digit numbers  using flats, sticks, and dots, and they understand that ten ones can be grouped into a ten, ten tens can be made into a hundred, and ten hundreds are equal to a thousand. They work with hundreds, tens, and ones on a place value chart and then write numbers in standard form and base-ten numeral form.” Introduction, “A Flat is worth 100. Let’s try representing a 3-digit number using flats, sticks, and dots: Kaleb has 3 boxes of 100 crayons, 6 boxes of 10 crayons, and 2 single crayons. How many crayons does Kaleb have?”

#### Criterion 3.4: Intentional Design

The program includes a visual design that is engaging and references or integrates digital technology, when applicable, with guidance for teachers.

The materials reviewed for Achievement First Mathematics Grade 2 do not integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the grade-level standards, include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other, or provide teacher guidance for the use of embedded technology to support and enhance student learning. The materials have a visual design that supports students in engaging thoughtfully with the subject, and is neither distracting nor chaotic.

##### Indicator {{'3w' | indicatorName}}

Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the grade-level/series standards, when applicable.

The materials reviewed for Achievement First Mathematics Grade 2 do not integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the grade-level/series standards, when applicable.

The materials do not contain digital technology or interactive tools such as data collection tools, simulations, virtual manipulatives, and/or modeling tools. There is no technology utilized in this program.

##### Indicator {{'3x' | indicatorName}}

Materials include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other, when applicable.

The materials reviewed for Achievement First Mathematics Grade 2 do not include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other, when applicable.

The materials do not provide any online or digital opportunities for students to collaborate with the teacher and/or with other students. There is no technology utilized in this program.

##### Indicator {{'3y' | indicatorName}}

The visual design (whether in print or digital) supports students in engaging thoughtfully with the subject, and is neither distracting nor chaotic.

The materials reviewed for Achievement First Mathematics Grade 2 have a visual design that supports students in engaging thoughtfully with the subject, and is neither distracting nor chaotic.

The student-facing printable materials follow a consistent format. The lesson materials are printed in black and white without any distracting visuals or an overabundance of graphic features. In fact, images, graphics, and models are limited within the materials, but they do support student learning when present. The materials are primarily text with white space for students to answer by hand to demonstrate their learning. Student materials are clearly labeled and provide consistent numbering for problem sets. There are several spelling and/or grammatical errors within the materials.

##### Indicator {{'3z' | indicatorName}}

Materials provide teacher guidance for the use of embedded technology to support and enhance student learning, when applicable.

The materials reviewed for Achievement First Mathematics Grade 2 do not provide teacher guidance for the use of embedded technology to support and enhance student learning, when applicable.

There is no technology utilized in this program.

## Report Overview

### Summary of Alignment & Usability for Achievement First Mathematics | Math

#### Math K-2

The materials reviewed for Achievement First Mathematics Grades K-2 meet expectations for Alignment to the CCSSM. In Gateway 1, the materials meet expectations for focus and coherence. In Gateway 2, the materials meet expectations for rigor and practice-content connections. The materials reviewed for Achievement First Mathematics Grades K-2 do not meet expectations for Usability, Gateway 3.

##### Kindergarten
###### Alignment
Meets Expectations
###### Usability
Does Not Meet Expectations
###### Alignment
Meets Expectations
###### Usability
Does Not Meet Expectations
###### Alignment
Meets Expectations
###### Usability
Does Not Meet Expectations

#### Math 3-5

The materials reviewed for Achievement First Mathematics Grades 3-5 meet expectations for Alignment to the CCSSM. In Gateway 1, the materials meet expectations for focus and coherence. In Gateway 2, the materials meet expectations for rigor and practice-content connections. The materials reviewed for Achievement First Mathematics Grades 3-5 do not meet expectations for Usability, Gateway 3.

###### Alignment
Meets Expectations
###### Usability
Does Not Meet Expectations
###### Alignment
Meets Expectations
###### Usability
Does Not Meet Expectations
###### Alignment
Meets Expectations
###### Usability
Does Not Meet Expectations

#### Math 6-8

The materials reviewed for Achievement First Mathematics Grades 6-8 meet expectations for Alignment to the CCSSM. In Gateway 1, the materials meet expectations for focus and coherence. In Gateway 2, the materials meet expectations for rigor and practice-content connections. The materials reviewed for Achievement First Mathematics Grades 6-8 do not meet expectations for Usability, Gateway 3.

###### Alignment
Meets Expectations
###### Usability
Does Not Meet Expectations
###### Alignment
Meets Expectations
###### Usability
Does Not Meet Expectations
###### Alignment
Meets Expectations
###### Usability
Does Not Meet Expectations

## Report for {{ report.grade.shortname }}

### Overall Summary

###### Alignment
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###### Usability
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##### Gateway {{ gateway.number }}
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