The instructional materials reviewed for Achievement First Mathematics Grade 1 meet expectations for assessing grade-level content. Above-grade-level assessment questions are present but could be modified or omitted without a significant impact on the underlying structure of the instructional materials.

The series is divided into nine units, and each unit contains a Unit Assessment available online in the Unit Overview document and can also be printed for students. Unit Assessments contain suggestions for use of Post-Unit Assessment questions as Pre-Unit Assessment questions. Teachers are directed to adjust instruction according to the Pre-Assessment results. Some parts of the assessment may be read to the students or done orally in small groups. 

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

• In Unit 2, Geometry Unit Assessment, Question 2 states, “Cross out the shapes that have 4 corners.” Pictures of a variety of two-dimensional shapes are given. (1.G.1)
• In Unit 3, Story Problems 1 Unit Assessment, Question 1, “Maya had 3 books. Sean had 5 books. How many books did they all have?” (1.OA.1)
• In Unit 5, Addition & Subtraction Unit Assessment, Question 5 states, “a. Sally had 4 stickers in her sticker collection. Her teacher gave her some more. Now she has 12. How many stickers did her teacher give her? b. What subtraction problem could you use to solve this story problem?” (1.OA.6) 
• In Unit 6, Two-Digit Numbers 1 Unit Assessment, Question 2 states, “Shanaya had 47 cubes. How many towers of ten could she make and how many single cubes would be left over?” (1.NBT.2)
• In Unit 8, Measurement Unit Assessment, Question 1 states, “Which shows the flowers in order from shortest to tallest?” The item is followed by four choices, each displaying three flowers in different order by height. (1.MD.1)

There are examples of above-grade-level assessment questions. The Guide to Implementing AF Math: Grade 1 and the assessments do not consistently align questions to the same standards. The Guide to Implementing AF Math: Grade 1 states, “Teachers should remove these items or use them for extension purposes only.” For example:

• In Unit 8, Measurement Unit Assessment, Question 4 states, “Steven’s foot is two inches shorter than Jason’s foot. Jason’s foot is 7 inches long. How long is Steven’s foot?” According to the Guide for Implementing AF Math: Grade 1, “Problems 4, 9, and 10 align with standard 2.MD.5.”
• In Unit 8, Measurement Unit Assessment, Question 9 states, “Trout keepers are 10 inches long. Kim caught a trout that was 7 inches long. How much longer would her trout need to be to be a keeper?” According to the Guide for Implementing AF Math: Grade 1, “Problems 4, 9, and 10 align with standard 2.MD.5.”
• In Unit 8, Measurement Unit Assessment, Question 10 states, “Julie’s bike is longer than Dave’s bike. Sarah’s bike is shorter than Dave’s bike. Whose bike is longer Julie’s or Sarah’s?” According to the Guide for Implementing AF Math: Grade 1, “Problems 4, 9, and 10 align with standard 2.MD.5.”
• In Unit 9, Two-Digit Numbers 2 Unit Assessment, Question 7 states, “67-22.” In Grade 1, students subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (1.NBT.6). This question aligns to 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).
• In Unit 9, Two-Digit Numbers 2 Unit Assessment, Question 8 states, “88-54.” In Grade 1, students subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (1.NBT.6). This question aligns to 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).

The instructional materials reviewed for Achievement First Mathematics Grade 1 meet expectations for spending a majority of instructional time on major work of the grade. 

• The approximate number of units devoted to major work of the grade, including assessments and supporting work connected to the major work, is 6.5 out of 9, which is approximately 72%.
• The number of lessons devoted to major work of the grade, including assessments and supporting work connected to the major work, is approximately 113 out of 150, which is approximately 75%. 
• The instructional block includes a math lesson, math stories, and math practice components. The non-major component minutes were deducted from the total instructional minutes resulting in 9,200 major work minutes out of 12,750 total instructional minutes. As a result of dividing the major work minutes by the total minutes, approximately 72% of the instructional materials focus on major work of the grade. 

A minute-level analysis is most representative of the instructional materials because the minutes consider all components included during math instructional time. As a result, approximately 72% of the instructional materials focus on major work of the grade.

The instructional materials reviewed for Achievement First Mathematics Grade 1 meet expectations that supporting work enhances focus and coherence simultaneously by engaging students in the major work of the grade.

The publishers identify connections between supporting content and major work within the lesson plan in the “Standards in Lesson” section, as well as in the Guide to Implementing AF Math: Grade 1. Additional connections exist within the materials, although not always stated by the publisher. In addition, the publisher identifies the CCSSM clusters at the top of each lesson plan as the “CC Clusters in Unit.” However, the major clusters listed are not consistent throughout the unit, and, therefore, it is unclear how the publisher identified clusters connected to the unit. For example, in Unit 5, Lesson 7, the publisher identifies 1.OA.A, represent and solve problems involving addition and subtraction, as connected to Unit 5. However, the 1.OA.A standards are not identified in any Unit 5 lesson. Examples of the connections between supporting work and major work include the following:

• In Unit 2, Lesson 7, Exit Ticket, students engage with the supporting work of 1.G.2, compose two-dimensional shapes or three-dimensional shapes to create a composite shape and the major work of 1.OA.1, use addition and subtraction within 20 to solve word problems by having students determine how to use the fewest pattern block shapes to fill a larger shape, complete a table, and add to find the total number of shapes used. Problem 2 states, “Elijah is trying to figure out a way to fill the same pattern using more than 4 pattern blocks. What is a way that he can fill the shape that uses more than 4 pattern blocks? Fill in the table to show how you fill the shape.” The table provided includes pictures of the different pattern blocks available, a place to record the number used, and a place to provide the total number of blocks used. 
• In Unit 4, Lesson 4, Exit Ticket, students engage with the supporting work of 1.MD.4, interpret data with up to three categories and answer questions about the total number of data points. This lesson also addresses, although not stated, the major work of 1.OA.2, adding three whole numbers whose sum is less than or equal to 20. A bar graph is shown representing the favorite sport of 3rd graders. Problem 3 states, “How many kids took the survey?” 
• In Unit 7, Lesson 6, Exit Ticket, students engage with the supporting work of 1.MD.3, tell and write time in hours and half-hours and with the major work of 1.NBT.1, read and write numbers to 120. In this lesson, students tell and write time in hours and half-hours using analog and digital clocks. In Problem 1, students are shown a digital clock showing 10:30 as the time. They are given a clock face without hands on it and asked to, “Draw the hands to show the time.” 
• Practice Workbook E, students engage with the supporting work of 1.MD.4, interpreting data up to three categories, and the major work of 1.OA.2, solving word problems that call for addition of three whole numbers whose sum is less than or equal to 20. 1.MD.4 is the only standard identified for this problem. In Problem 2, students are presented with a table that shows the types of shoe ties with three categories: “velcro, laces, no ties.” Students are asked, “Write a number sentence to show how many total students are asked about their shoes.”

The instructional materials reviewed for Achievement First Mathematics Grade 1 meet expectations that the amount of content designated for one grade-level is viable for one school year. The Guide to Implementing AF, Grade 1 includes a scope and sequence which states, “Not every lesson is entirely focused on grade level standards, and, therefore, some lessons can be used for either remediation or enrichment. As designed, the instructional materials can be completed in 150 days. One day is provided for each lesson and one day is allotted for each unit assessment.

• Nine units with 141 lessons in total.
• The Guide identifies lessons as either R (remediation), O (on grade level), or E (enrichment).  There are 10 lessons identified as E (enrichment), 0 identified as R (remediation), and 131 identified as O (on grade level). 
• Nine days for unit assessments. 

When reviewing the materials for Achievement First, Grade 1, a difference in the number of total instructional days was found. Although the publisher states the curriculum will encompass 151 days, there are 150 days of lessons and unit assessments.  The Grade 1 Unit Overview for Unit 6 shows 24 days for the unit while the Guide to Implementing AF, Grade 1 provides 23 days for the unit. The unit has 23 lessons including the unit assessment.

The publisher recommends 85 minutes of mathematics instruction daily.

• There are two lesson types, Game Introduction Lesson or Task Based Lesson. Each lesson is designed for 45 minutes.
• Math stories are designed for 25 minutes. 
• Calendar/practice is designed for 15 minutes.

The instructional materials reviewed for Achievement First Mathematics Grade 1 partially meet expectations for being consistent with the progressions in the Standards. Overall, the materials do not provide all students with extensive work on grade-level problems. The instructional materials develop according to the grade-by-grade progressions in the Standards. Content from future grades is clearly identified and relates to the grade-level work. The instructional materials relate grade-level concepts explicitly to prior knowledge from earlier units. Within the overview for each unit, there is “Identify the Narrative” component, which provides a description of connections to concepts in prior and future grade levels.

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

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

• In Unit 1, Counting Unit Overview, Identify the Narrative, Linking states, “Continuing through the rest of elementary school, students will use the counting sequence in all grades. In 2nd grade they’ll be using the counting and place value patterns to count to 1,000 and add and subtract within 1,000. This becomes fluent in 3rd grade. By fourth grade, they’ve generalized the counting and place value patterns to all numbers and can add and subtract any size and number.”
• In Unit 3, Story Problems Unit Overview, Identify The Narrative, Linking states, “In the rest of elementary school, students will continue to work with story problems following the protocol taught and practiced in this unit. In second grade, students will master the start unknown, compare-bigger unknown-fewer, and compare-smaller unknown-more problem types that they were exposed to in this unit, and they will begin to solve two-step story problems. They will continue to expand their bank of representation and solution strategies.”
• In Unit 5, Addition and Subtraction Unit Overview, Identify the Narrative, Linking states, “Looking ahead to the remainder of first grade, students will continue to use the strategies taught in this unit to efficiently solve addition and subtraction problems and story problems within 20. They will build on these strategies to solve problems beyond 20 and up to 100, especially using count on and count back to add and subtract multiples of 10 to two-digit numbers.”
• In Unit 8, Measurement Unit Overview, Identify The Narrative, Linking states, “In the remainder of first grade, comparing lengths of objects help support students in understanding and solving compare-difference unknown story problems. Moving into second grade, students begin to use standard units of measurements such as rulers, yardsticks, meter sticks, and measuring tapes to measure and estimate length. They relate the length of a unit of measurement to the length of the object being measured with that unit. (For example, students recognize that a table would be more inches long than feet because inches are shorter than feet.) Second graders also build on the compare work they did in first grade to determine how much longer one object is than another, expressing the difference in terms of a standard length unit.”
• In Unit 9, Two Digit Numbers 2 Unit Overview, Identify the Narrative, Structural Overview outlines the concepts of addition and subtraction across grades K to 4. The visual shows that addition and subtraction within 10 occurs in Kindergarten, while within 100 occurs in First Grade and within 1,000 occurs in Second through Fourth Grades. It also identifies that Properties of Addition and Subtraction are learned from First Grade through to Fourth Grade, while the Standard Algorithm for addition and subtraction is taught in Fourth Grade.

Overall, the materials do not provide all students with extensive work on grade-level problems. The majority of the lessons implement 45 minutes of math workshop with a whole group introduction, workshop in pairs or small groups, mid-workshop interruption, whole group discussion, and closing with an exit slip. As it is unclear if students are working together or individually, workshop lessons may not provide enough opportunity for students to independently demonstrate mastery. The Guide to Implementing AF, Grade 1, describes the workshop component as, “Collaborative processing time to continue to develop understanding of prioritized concept and strategy.” The lessons include a teacher-directed introduction to the workshop “game” and follows up with students tasked to participate in the “game.” Most lessons include an exit ticket with one or two questions for the students to complete individually. 

Beyond the lesson component of the math time, the Guide to Implementing AF Math, Grade 1 suggests 15 minutes of daily calendar and practice. Each unit indicates the Grade 1 Practice Workbook pages to be implemented during this time. However, the practice workbook pages contain a limited number of practice items and are recommended to be used repeatedly in different units. As a result of the limited number of opportunities to practice grade-level standards, the materials do not give students extensive work with grade-level problems. 

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

• In Unit 2, Lesson 13, Task (labeled Lesson 12 task in the lesson), students engage 1.G.1 as they build and draw shapes to possess defining attributes. Students do not experience the full intent or extensive work of 1.G.1 because they are only asked to draw four-sided shapes. Within the six Exit Tickets, students are not asked to independently build or draw shapes that possess defining attributes. For example, “Leah is learning about shapes. Leah makes for different shapes on her paper. The shapes are different colors and sizes, but all the shapes have four corners. What four different shapes can Leah make on her paper? Show all your mathematical thinking and be ready to explain your answer.”
• Unit 3, Lesson 11 is the only lesson students engage with the standard 1.OA.2 and practice solving word problems that call for addition of three whole numbers. The independent practice can be found on the Exit Ticket. The Practice Workbook for the grade-level provides zero problems to provide additional independent practice for this standard outside of the Exit Ticket. Extensive work is not provided for 1.OA.2. It is part of the major work of the grade, but is only addressed in 1 of the 141 lessons. In the Exit Ticket, Problem 1 states, “Chardae was drawing pictures of dogs. She drew 2 brown dogs, 3 black dogs, and 2 spotted dogs. How many dogs did she draw?”
• In Unit 5, Lesson 22, Workshop, students engage with 1.OA.8 as they determine the unknown whole number in an addition or subtraction equation relating three whole numbers. Students find the unknown numbers to make addition and subtraction equations true. This standard is addressed in only three lessons, and 16 Practice workbook problems. Exit Ticket states, “Fill in the blank to make the equations true.  $$10 -$$ ____ = $$8 - 2$$, $$2 +$$ ____ $$= 3 + 4$$, $$4 + 5 = 5 +$$ ____”.
• In Unit 6, Lesson 17, Workshop, students engage with a portion of 1.NBT.5 as they are given a two-digit number and asked to mentally find 10 more or 10 less than the number, without having to count and explain the reasoning used. Students do practice the skill of mentally finding 10 more or 10 less through a card game of Leapfrog. Students roll dice telling them how many spaces to move forward on a game board, then draw a card telling them how many tens to leap ahead. However, the lesson does not address the full intent of the standard because students are not asked to explain their reasoning, as is part of the standard. In the Exit Ticket, Problem 2 states, “Solve. $$58 + 30 =$$ ______.” 
• In Unit 7, Lesson 3, Workshop and Practice Workbook E, students engage with 1.G.3 as they partition circles and rectangles into two and four equal shares. However, only three lessons address the standard, 1.G.3, with four Exit Ticket problems and six Practice Workbook problems to independently practice partitioning. While students experience some practice during Workshop using the words halves, fourths, and quarters, they do not experience the use of the phrases “half of,” “fourth of,” and “quarter of.” As a result, students do not have the opportunity to meet the full intent of the standard. In addition, there are only three lessons that address 1.G.3 with limited independent practice; therefore, students do not get extensive work with this standard. During Workshop, students solve a story problem comparing halves and fourths of two same sized pies to determine who gets the biggest piece of pie.
• In Unit 8, Lesson 6, students engage with 1.MD.1 as they order three objects by length and compare the lengths of two objects indirectly by using a third object. Students are only provided four problems within Practice Workbook D and three Exit Tickets to meet the full intent of this standard. All other independent work does not require students to use three objects to compare lengths. This is not sufficient to meet extensive work of the standard. The Exit Ticket states, “Shaquan’s crayon was shorter than Tyra’s crayon. Jesse’s crayon is longer than Tyra’s crayon. Whose crayon is longer - Jesse or Shaquan? How do you know?”

The Unit Overview supports the progression of First 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 First Grade unit and/or concepts previously learned in Kindergarten. 

Each Unit Overview also contains an Identify Desired Results: Identify the Standards section that makes connections to supporting standards learned prior to the unit. In addition, some lessons make connections to previous grade-level learning in the Narrative section. Examples include: 

• In Unit 1, Lesson 2, Narrative, What is new and/or hard about the lesson? states, “Students will be familiar with counting by tens and ones from kindergarten, and many will recall that it is useful to group objects into sets of tens and ones from their work with teen numbers.”
• In Unit 2, Geometry Unit Overview, Identify the Narrative states, “Throughout the unit, students identify the defining characteristics, or attributes, of two- and three-dimensional shapes, building on their Kindergarten experiences of sorting, analyzing, comparing, and creating various two- and three-dimensional shapes and objects (1.G.1).”
• In Unit 3, Story Problems Unit Overview, Identify Desired Results states, “K.OA.1 Represent addition and subtraction with objects, fingers, mental images, drawings (no detail), sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations,” and “K.OA.5 Fluently add and subtract within 5” as previous grade level standards related to “1.OA.1 Use addition and subtraction within 20 to solve world problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol of the unknown number to represent the problem.”
• In Unit 5, Addition and Subtraction Unit Overview, Identify the Narrative states, “Make 10 is a valuable strategy in the base-ten system because it allows students to work flexibly with numbers to solve more challenging problems by breaking them down into easier problems that they can solve fluently. The building blocks for the make ten strategy are built in Kindergarten, as students become familiar with number partners for numbers 1-10, decompose teen numbers into a group of ten and some more ones. If students are struggling to use the make ten strategy, teachers should ensure that the students solidly understand K.OA.4, K.OA.3, and K.NBT.1 because they are foundational for the make ten strategy.” 
• In Unit 8, Measurement Unit Overview, Identify Desired Results: Identify the Standards, 1.MD.2 (Express the length of an object as a whole number of length unit, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.) is identified as a Unit 8 standard. The Kindergarten standard identified as foundational is K.MD.1 (Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object.)

The instructional materials reviewed for Achievement First Mathematics Grade 1 meet expectations for fostering coherence through connections at a single grade, where appropriate and required by the Standards. The publisher identifies the CCSSM clusters at the top of each lesson plan as the “CC Clusters in Unit.” However, the major clusters listed are not consistent throughout the unit, and, therefore, it is unclear how the publisher identified clusters connect to each lesson. 

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

• In Unit 2, Lesson 4, Aim is shaped by 1.G.A, reason with shapes and their attributes. The materials state, “SWBAT decompose a shape by asking, Which smaller shapes could be put together to make the larger shape?” 
• In Unit 4, Lesson 6, Aim is shaped by 1.MD.C, represent and interpret data. The materials state, “SWBAT solve comparison problems using a data set by using the graph as a representation or creating their own representation to match the graph.” 
• In Unit 5, Lesson 10, Aim is shaped by 1.OA.B, understand and apply properties of operations and the relationship between addition and subtraction. The materials state, “SWBAT solve subtraction problems by decomposing a number leading to a ten.”
• In Unit 7, Lesson 6, Aim is shaped by 1.MD.B, tell and write time. The materials state, “SWBAT show time to the half hour by drawing the hour hand and minute hand.”
• In Unit 9, Lesson 4, Aim is shaped by 1.NBT.C, use place value understanding and properties of operations to add and subtract. The materials state, “SWBAT add a two-digit and a one-digit number with regrouping by using a strategy that makes sense to them (count on-cubes, sticks and dots, fingers… make ten).”

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

• In Unit 3, Lesson 7, Exit Ticket, students engage with 1.OA.B, understand and apply properties of operations and the relationship between addition and subtraction, and 1.OA.C, add and subtract within 20, as they are provided different strategy options to solve and write an addition equation. Problem 1 states, “Solve for the unknown. Write an addition equation that shows the parts and whole (You may use the number line but do not have to.)” Students are provided with a number bond with 8 and 5 in two of the circles, a space to write the addition equation, and a number line to use. 
• In Unit 5, Lesson 18, Workshop, students engage with 1.OA.B, understand and apply the properties of operations and the relationship between addition and subtraction, 1.OA.C, add and subtract within 20, and 1.OA.D, work with addition and subtraction equations. During Workshop, students play a game called “True Match” in which they use the strategies explored in recent lessons to solve efficiently. They have two sets of cards with equivalent matches and are to use the following strategies: solve for the total by counting on, solve for the total by making ten, just know the total, and just know the equivalent expression without solving either expression (compensating). 
• In Unit 6, Lesson 5, Workshop, students engage with 1.NBT.B, understand place value, and 1.NBT.A, extend the counting sequence, as they draw numbers 10-90 using sticks and dots and write the numeral. Exit Ticket, Problem 2 states, “If you have 4 tens and 2 ones, how many do you have? Represent with sticks and dots and write the numeral.”
• In Unit 7, Lesson 5, Worksheet Packets engage students with 1.MD.B, tell and write time, and 1.G.A, reason with shapes and their attributes, as they identify a clock with a given time. Problem 1 states, “Circle the correct clock. 1. Half past 10 o’clock.” Students are provided with three clocks (10:30, 11:30, and 12:30).

The instructional materials for Achievement First Mathematics Grade 1 meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

Materials include problems and questions that develop conceptual understanding throughout the grade-level. Examples include: 

• In Unit 3, Lesson 5, Introduction, students engage with 1.OA.3, apply properties of operations as strategies to add and subtract, 1.OA.4, understand subtraction as an unknown-addend problem, and 1.OA.5, relate counting to addition and subtraction, as they represent addition and subtraction scenarios with number bonds. The teacher models how to play Roll and Record: Mixed Operations. The materials state, “Step 1: Pick a card and roll 2 cubes. (pick addition operation card first - for planning purposes, assume you roll 4 and 6), Step 2: solve and record equation. What’s the total?, Step 3 says record with a number bond; label parts and whole, Step 4 says record with other operation equation. We already wrote an addition equation… so now we need to record with subtraction.” 
• In Unit 5, Lesson 6, Introduction, students engage with K.CC.6, identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, as they play a game called Compare. During the Introduction, two cards are drawn (example, 7 and 9) and students are asked to pictorially show which is more or less by drawing circles on their whiteboards. The teacher asks, “How do you know from the picture?” A sample student response might be, “I know because in the picture you can see that there are extra circles in the row of 9 and the row of 7 is missing some.” 
• In Unit 6, Lesson 2, Workshop, students engage with 1.NBT.2, understand that the two digits of a two-digit number represent amounts of tens and ones, as they select a bag with 10-90 cubes and draw a representation of the two-digit numbers, showing tens and ones. The teacher is provided support in the Assessment and Criteria for Success portion of the lesson, “Students will pick a bag that is filled with ten sticks and loose ones. They will determine how many by counting by tens and ones and draw a literal picture and writ a numeral to match. Students should be able to explain why they are counting by tens and ones and what their picture and numeral represents. For example, ‘In my picture I drew 7 ten sticks and can count them by ten because there are ten cubes in each stick. Then I draw 4 loose ones and I would count on by ones. So “10, 20, 30, 40, 50, 60, 70, 71, 72, 73, 74.” There are 74. I would write 7 to show 7 groups of ten and 4 to show 4 loose ones.’”
• In Unit 6, Lesson 20, Introduction, students engage with 1.NBT.3, by comparing two two-digit numbers based on the meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, <. Step 5 states, “Is 65 greater or less than 63? How can we figure it out? Strategy 1: Sticks and dots (intervention). SMS: We could look at the sticks and dots and it looks like 65 has the same number of sticks/tens as 63 but 65 has more ones than 63. Therefore, 65 is greater than 63.”
• In Unit 9, Lesson 12, Workshop Worksheet, students engage with 1.NBT.6, subtract multiples of 10 in the range 10-90, using models, drawings, and strategies based on place value, as they subtract multiples of 10 from a two-digit number using strategies that work for them. Problem 4 states, “$$50 - 30 =$$ _____. How did you solve?”

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

• In Unit 3, Lesson 11, Exit Ticket, students engage with 1.OA.1, use addition and subtraction within 20 to solve word problems, as they solve story problems by visualizing and representing in a way that makes sense to them. Problem 2 states, “Tony was collecting buttons. He had 4 buttons and then his grandmother gave him 3 more buttons. How many buttons does he have now?”
• In Unit 6, Lesson 9, Exit Ticket, students engage with 1.NBT.4, add within 100, including adding a two-digit number and a one-digit number and adding a two-digit number and a multiple of ten, as they combine two multiples of ten by using a strategy that makes sense to them (cubes, literal pictures, sticks and dots, count all/on by tens, use place value). Problem 1 states, “Solve. $$30 + 20 =$$ ____.”
• In Unit 7, Practice Workbook D, students engage with 1.NBT.2, by understanding that the two digits of a two-digit number represent amounts of tens and ones. Problem 12 states, “Show the number 39 in tens and ones.”
• In Unit 7, Practice Workbook D, students engage with 1.NBT.6, subtract multiples of 10 in the range 10-90, using models, drawings, and strategies based on place value, as they independently subtract multiples of 10 from a two digit number using strategies that work for them. Problem 1 states, “$$80 - 60 =$$ _______”
• In Unit 9, Lesson 12, Exit Ticket, students engage with 1.NBT.6, subtract multiples of 10 in the range 10-90 from multiples of 10 in the range of 10-90, as they use strategies that work for them (count what’s left, count back, uses known facts). Problem 1 states, “Solve. 50 - 30 = __.” Additional guidance for the teacher is found in Assessment and Criteria for Success which states, “Students should be able to describe their work by saying, ‘I solved 50 - 30. First I drew 5 sticks and 0 dots to show 50 because there are 5 tens and 0 ones. Then I need to take away 30, which is 3 tens and 0 ones. So as I crossed out the sticks I counted back like this. 50 -- 40, 30, 20. The difference is 20.’”

The instructional materials for Achievement First Mathematics Grade 1 meet expectations that they attend to those standards that set an expectation of procedural skill and fluency. These skills are delivered throughout the materials in the use of games, workshop, practice workbook pages and independent practice, such as exit tickets. 

The instructional materials develop procedural skill and fluency throughout the grade level. Examples include but are not limited to: 

• In Unit 3, Lesson 9, Assessment and Criteria for Success, students engage with 1.OA.3, apply properties of operations as strategies to add and subtract, and 1.OA.4, understand subtraction as an unknown-addend problem, as they explain how they found the parts of their total. Workshop Written Assessment states, “Students find the number pairs to make a total by guessing and checking, counting up, counting back, or using known facts. Exemplar Student Response states, “My total is 6. I found the parts by picking a card and then counting up to the total. So I picked a 4. Then I counted up until I got to 6 because that’s the whole. I got 2 so that’s the other part. Then I recorded by putting 6 here because it’s the whole. Then I put 4 and 2 here because they are the parts. I wrote the equation $$4 + 2$$ because I’m combining the parts = 6 because they make the whole.”
• In Unit 3, Practice Workbook B, Activity: X-Ray Vision, students engage with 1.OA.6, adding and subtracting within 20, demonstrating fluency for addition and subtraction within 10, as they calculate the missing addend using counters. Partners to 10 states, “Place 10 counters on the floor next to a container. Tell students to close their eyes. Put one of the items into the container. Tell students to open their eyes and identify how many counters were put inside it. Continue the game, eliciting all partners to 10.”
• In Unit 4, Practice Workbook B, Activity: Ten and Tuck, students engage with 1.OA.6, add and subtract within 20, as they use their fingers to make 10. The materials state, “Directions: Tell students to show 10 fingers. Instruct them to tuck three (students put down the pinky, ring finger, and middle finger on their right hands). Ask them how many fingers are up (7) and how many are tucked (3). Then, ask them to say the number sentence aloud, beginning with the larger part (7 + 3 = 10), beginning with the smaller part $$(3 + 7 = 10)$$, and beginning with the whole $$(10 = 3 + 7 or 10 = 3 + 7)$$.”
• Unit 5, Lesson 5, Workshop, Intro Packet, students engage with 1.OA.6, add and subtract within 20, demonstrating fluency for addition and subtraction within 10, as they add three numbers rolled with number cubes, using the strategy of grouping facts they know or can easily figure out. Problem 1 states, “$$5 + 3 + 5$$.”

The instructional materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade-level. Examples include but are not limited to:

• In Unit 3, Lesson 4, Exit Slip, students engage with 1.OA.6, add and subtract within 20, as they use number bonds to help them solve. The materials state, “Find the difference between the number cubes. Represent by completing the number bond and equation.” Cubes show the numbers 9 and 6. 
• In Unit 4, Practice Workbook B, Math Sprint A, students engage with 1.OA.6, add and subtract within 20, demonstrating fluency for addition and subtraction within 10, as they practice addition and subtraction facts on a Math Sprint. Problem 23 states, “___ $$- 6 = 3$$” 
• In Unit 5, Lesson 21, Exit Slip, students engage with 1.OA.6, adding and subtracting within 20, demonstrating fluency for addition and subtraction within 10, as they find the missing subtrahend of a subtraction equation. The materials state, “Fill in the blank to make the equations true. $$10 -$$ ___ $$= 8 - 2$$.”
• In Unit 6, Lesson 17, Exit Ticket, students engage with 1.NBT.5, given a two-digit number, mentally find 10 more without having to count, as they independently add ten to a number. Problem 1 states, “$$32 + 10 =$$____.”

The instructional materials for Achievement First Mathematics Grade 1 partially meet expectations that the materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied. The series includes limited opportunities for students to independently engage in the application of routine and non-routine problems due to lack of independent work during Workshop and lack of non-routine problems. 

The instructional materials present opportunities for students to engage in application of grade-level mathematics; however, the problems are scaffolded through teacher led questions and partner work. According to the Guide to Implementing AF Grade 1, “Task Based Lesson Purpose: Students make sense of the mathematics they’ve learned by  working on a problem solving task and leveraging the knowledge they bring to math class to apply their math flexibly to non-routine, unstructured problems, both from pure math and from the real world. To shift the heavy lifting to scholars.” However, most of the task based lessons are considered enrichment and teachers may opt to not incorporate these non-routine opportunities into their math lessons. In addition, the task based lessons are not independent as they “encourage discussions between students about alternate methods or possibly incorrect solution paths.”

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

• In Unit 3, Lesson 15, Exit Slip, students engage with 1.OA.1, use addition and subtraction within 20 to solve 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. The materials state, “There were 17 jackets hanging in the hallway. Some of the jackets were black. 11 of the jackets were yellow. How many black jackets were hanging in the hallway?”
• In Unit 3, Lesson 26, Exit Ticket, students engage with 1.OA.1, use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, as they solve compare story problems by using pictures to represent. Problem 1 states, “Mrs. Sutherland had 10 kids in her math group. Mr. Alese had 2 fewer. How many kids did Mr. Alese have?” 
• In Unit 4, Workbook E, students engage with 1.MD.4, interpreting data with up to three categories and answering questions, and 1.OA.2, solving addition problems of three whole numbers with a sum less than 20, as they calculate add to problems with the results unknown of three addends. The materials state, “The class has 18 students. On Friday, 9 students wore sneakers, 6 students wore sandals, and 3 students wore boots. Use squares with no gaps or overlaps to organize the data.” Problem 11 states, “Write a number sentence to tell how many students were asked about their shoes on Friday.”
• In Unit 8, Lesson 8, Exit Ticket, students engage with 1.MD.2, express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps. The materials state, “How much longer is pencil B than pencil A? Use your inch tiles to help you. Pencil B is ___ inch tiles LONGER than pencil A.” 

Math Stories provide opportunities for students to engage in routine applications of grade- level mathematics. Students engage with Math Stories for 25 minutes, five days per week. The Guide to Implementing AF Grade 1 page four states the purpose of Math Stories, “Purpose: 

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

Examples of routine Grade 1 Math Stories:

• In Unit 2, Guide to Implementing AF Math, Math Stories, October, students engage with 1.OA.1, use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, as they complete math story problems. Sample Problem 13 states, “There were 20 kids at the haunted house. Then, some ran out screaming. Now there are only 5 kids in the haunted house. How many kids ran out screaming?”
• In Unit 3, Guide to Implementing AF Math, Math Stories, November, students engage with 1.OA.1, use addition and subtraction to solve word problems with unknowns in all positions, as they solve add to/start unknown word problems. Sample Problem 3 states, “There were some monkeys eating bananas on the tree. 11 more swung over. Now there are 14 monkeys on the tree. How many monkeys were there on the tree at first?”
• In Unit 4, Guide to Implementing AF Math, Math Stories, January, students engage with 1.OA.1, use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, as they solve story problems. Sample Problem 5 states, “Delilah was bird watching. She saw 8 more red birds than black birds. She saw 13 black birds. How many red birds did she see?”
• In Unit 5, Guide to Implementing AF Math, Math Stories, February, students engage with 1.OA.1, solve addition and subtraction word problems within 20, as students calculate take apart problems with both addends unknown. Sample Problem 3 states, “Ms. Russo had 20 awards to pass out to her class. Her class has boys and girls. How many could she pass out to the girls? (after they represent: Find at least 4 different solutions) ($$0 + 20$$ is a solution).”

The instructional materials have few opportunities for students to engage in non-routine application throughout the grade level. For example:

• In Unit 3, Unit Assessment, students engage with 1.OA.1, use addition and subtraction to solve word problems with unknowns in all positions, as they select a model from a list of given models, that does not represent a word problem. Problem 12 states, “Jason had some books in his backpack. He got 4 more from the library. Now he has 9 books. How many books did he have in his backpack to start? Which of the following does NOT represent the story?” Students are provided with four models with unknowns in various positions, including two equations, a Part-Part-Total Model, and a number model. 
• In Unit 5, Lesson 23, Lesson 15 Task, students engage with 1.OA.7, understand the meaning of the equal sign and determine if equations involving addition are true or false, as they solve a task-based word problem asking them to determine if the total of two groups are equal. Problem 1 states, “Ben has 9 ladybugs and 5 crickets in his jar. Jill has 8 ladybugs and 7 crickets in her jar. Dad thinks they have the same amount of insects in each jar. Is Dad correct? Show and tell how you know.” 

It is important to note that many of the recommended Math Stories go beyond the standard, 1.OA.1, as they incorporate addition and subtraction beyond 20. The Guide to Implementing AF Grade 1 page four states, “Some of the daily problems provided will be beyond the standard in magnitude. Teachers may opt to adjust the magnitude in these problems to reflect the Grade 1 standard and be within 20.” Additionally, Units 6-9 Math Stories incorporate two-step word problems which is beyond the scope of the first grade standard, 1.OA.1, use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions. For example:

• In Unit 3, Math Stories, November/December, Problem 10 states, “(AT-SU) Jerome has some cranberries to make a nice cranberry sauce. His mom gives him 2 more cranberries to add to his sauce. Now he has 23 cranberries. How many cranberries did he have at the beginning?” This word problem goes beyond the standard, 1.OA.1, as it goes beyond 20. 
• In Unit 5, Math Stories, February/March, Problem 4 states, “(AU-SU) Some bunnies were hopping in the cage. Farmer Brown put 13 more bunnies into the cage. Now there are 38 bunnies hopping in the cage. How many bunnies did Farmer Brown to add to the cage?” This word problem goes beyond the standard, 1.OA.1, as it goes beyond 20. 
• In Unit 8, Math Stories, May, Problem 2 states, “(2-step TF-RU with an embedded AT-RU) The movie theater has 22 seats. 8 kids and 9 adults went to the movie. How many seats at the movie theater were empty?” This word problem goes beyond the standard, 1.OA.1, as it is a two-step word problem.

The instructional materials for Achievement First Mathematics Grade 1 meet expectations that the materials reflect the balance in the standards and help students meet the standards’ rigorous expectations by helping students develop conceptual understanding, procedural skill and fluency, and application. The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade.

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

Conceptual understanding

• In Unit 6, Lesson 13, Exit Ticket, students engage with 1.NBT.6, subtract multiples of 10 in the range of 10-90 from multiples of 10 in the range 10-90, as they represent and solve two subtraction problems on an exit ticket. Problem 1, “Represent and solve. 50-30” Problem 2, “Represent and solve. 90-40” Assessment and Criteria for Success, “Students will find the difference of two multiples of ten. They may use any strategy that works. If counting back with fingers, they should be able to explain, ‘I started with 90 and then counted back 40 by counting back by tens 4 times because 40 is 4 tens.’”
• In Unit 7, Practice Workbook D, students engage with 1.NBT.2, understand that the two digits of a two-digit number represent amounts of tens and ones, as they write the number represented by images of sticks and dots. Problem 5 states, “Which number is represented?” Four rods are shown with five dots.
• In Unit 9, Lesson 9, Introduction, students engage with 1.NBT.4, add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of ten, as they use a strategy that makes sense to them (cubes, sticks and dots, count on by tens, expanded notation addition). Students pick two cards such as 47 and 50 and find the total. Students are to choose a strategy such as “count all by tens and ones.” The students might say, “We can count all of the tens by ten and the ones by one. Like this, 10, 20, 30, 40, 50, 60, 70, 80, 90, 91, 92, 93, 94, 95, 96, 97).” Students may also choose to count on by tens and ones. A student might explain, “I just know that we have 47 right there, so then I can just count on by tens like this 47 -- 57, 67, 77, 87, 97 (can use tens sticks with cubes or sticks to help count on).” 

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

• In Unit 3, Lesson 2, Exit Slip, students engage with 1.OA.6, adding and subtracting within 20, as they use number bonds to build addition equations. Problem 1 states, “Find the total of the number cubes. Represent by completing the number bond and the equation.” Two cubes are shown with 5 and 6 on them. A number bond frame is provided and “___ + ___ = ___.”
• In Unit 4, Practice Workbook B, Number Bond Roll, students engage with 1.OA.6, add and subtract within 20, demonstrating fluency for addition and subtraction within 10, as they review number bonds allowing students to build and maintain fluency with addition and subtraction facts within 10. The materials state, “Match partners of equal ability. Each student rolls one die. Students use the numbers on their own die and their partner’s die as the parts of a number bond. They each write a number bond, addition sentence, and subtraction sentence on their personal white boards.”    
• In Unit 5, Lesson 13, Exit Slip, students engage with 1.OA.6, add and subtract within 20, as they solve addition and subtraction problems by creating a fact family. The materials state, “Find the rest of the fact family. $$13 - 6 = 7$$” 

Application

• In Unit 2, Guide to Implementing AF Math, Math Stories, October, students engage with 1.OA.1, adding and subtracting within 20 to solve word problems. Sample Problem 12 states, “Zamira read 18 books. Some were about bugs. 2 were about snakes. How many books about bugs did she read?”
• In Unit 3, Lesson 13, Exit Ticket, students engage with 1.OA.1, use addition and subtraction within 20 to solve 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 2 states, “I made 3 yellow paper chains and 5 blue paper chains. How many paper chains did I make?”
• In Unit 4, Guide to Implementing AF Math, Math Stories, January, students engage with 1.OA.1, adding and subtracting within 20 to solve word problems. Sample Problem 10 states, “Diego wrapped 24 presents. Jessica wrapped 9. How many fewer presents did Jessica wrap than Diego?”
• In Unit 5, Guide to Implementing AF Math, Math Stories, February, students engage with 1.OA.1, adding and subtracting within 20 to solve word problems, as they solve compare problems with the smaller number unknown. Sample Problem 13 states, “Shayla has 12 fewer pencils than Matthew. Matthew has 19 pencils. How many pencils does Shayla have?”

Multiple aspects of rigor are engaged in simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. For example:

• In Unit 3, Lesson 9, Exit Ticket, students engage with 1.OA.1, using addition and subtraction within 20 to solve word problems, and 1.OA.6, adding and subtracting within 20, as they solve take apart problems (application) with both addends unknown (conceptual understanding). Problem 2 states, “There were 7 animals on the farm. Some were sheep and some were pigs. How many could be sheep and how many could be pigs? Show one combination using a number bond and an equation.”
• In Unit 3, Lesson 19, Exit Ticket, students engage with 1.OA.1, use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions and 1.OA.6, add and subtract within 20, as they solve a story problem (application), as they solve problems within 20 (procedural skill). Problem 5 states, “Sarah has 9 pennies. Michael has 6 pennies. How many more pennies does Sarah have than Michael? You can use your cubes to help you solve.” Assessment and Criteria for Success, Exemplar Response states, “Sara has 3 more pennies than Michael. I know because I built 9 cubes and 6 cubes and put them next to each other. They both have 6 cubes but Sarah has 3 more pennies than Michael.” 
• In Unit 4, Lesson 5, Exit Ticket, students engage with 1.MD.4, organize, represent, and interpret data; and answer questions about the data points, as they interpret the data presented on a pictograph (conceptual understanding) and use it to solve compare/difference unknown word problems (application). Problem 1 states, “How many more rainy days than sunny days?” Students are provided with a weather pictograph showing sunny days, rainy days, and cloudy days.  
• In Unit 9, Lesson 5, Exit Ticket, students engage with 1.NBT.4, add within 100 including a two-digit number and a one-digit number, using concrete models or drawings; understand that it is sometimes necessary to compose a ten, as they add a two-digit number by compose a ten (procedural skill), and explain how they solved the problem (conceptual understanding). Problem 1 states, “Solve. Show your work. $$57 + 6 =$$ ________. “ Problem 2 states, “How did you solve? Why?”

The instructional materials reviewed for Achievement First Mathematics Grade 1 partially meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade-level. All MPs are clearly identified throughout the materials, with few or no exceptions. However, there are inconsistencies between the identified MPs in the Unit Overview and the MPs identified in the Lesson Plans. The materials provide little direction as to how the MPs enrich the content and do not make connections to enhance student learning. The MPs are not treated separately from the content.

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

• In the Unit Overviews, the bolded MPs are the Focal MPs for the unit.
• In Unit 3, Unit Overview, Standards for Mathematical Practice identifies MP4, model with mathematics, as embedded in the story problems lessons of Unit 3.
• In Unit 9, Unit Overview, Standards for Mathematical Practice identifies MP8, look for and express regularity in repeated reasoning, as embedded in the two-digit addition and subtraction lessons. 
• The MPs are listed at the beginning of each lesson in the Standards section. For example, in Unit 3, Lesson 13, the following MPs are identified as in the lesson: MP 1, MP 2, MP 3, MP 4, MP 5 and MP 6. 
• The Mathematical Practices are not always identified accurately. For example: 
• In Unit 1, MP 3 is not bolded as a focus MP. However, it is identified in 15 out of 16 lessons. MP 7 is identified as a focus MP but is only identified in 8 out of 16 lessons. 
• At the unit level for Unit 2, MP 3 is not identified as a focus MP. However, at the lesson level, 9 out of 13 lessons identify it as connected. At the unit level, MP 5 is listed as a focus, but it is only connected to one of the thirteen lessons. 
• All MPs are represented throughout the materials, though lacking balance. For example, MP 8 is the focus of none of the first grade units, while MP 5 is the focus of eight of the nine units. 
• There are no stated connections to the MPs to the Math Stories component, Math Practice component, or Assessments. 

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

• It is left to the teacher to determine where and how to connect the emphasized mathematical practices within each lesson. 
• There are connections to the content described in the Unit Overview. However, if a teacher is not familiar with the MPs, the connection may be overlooked as there are no connections within the specific lesson content to any MPs. Examples include: 
• In Unit 2, Unit Overview, Standards for Mathematical Practice identifies MP7, look for and make use of structures, as embedded in the geometry lessons of Unit 2. The materials state, “Students use the defining attributes of shapes to identify, sort, and compare shapes.”
• In Unit 4, Unit Overview, Standards for Mathematical Practice identifies MP6, attend to precision, as embedded in the data lessons of Unit 4. The materials state, “Students must be precise in their representations of data so that others can interpret the data accurately.”

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

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

• In Unit 3, Lesson 14, Narrative states, “Acting out will be a part of the Introduction today to provide support for students who may struggle to visualize these types of problems on their own immediately. Creating a representation that depicts what is happening in the story can be challenging, so teachers should be ready to support students with questions like “What do you know? What do you need to find out?” Introduction, Step 1 states, “There were some butterflies in my net. Five of the butterflies escaped. There are still 7 butterflies in my net. How many butterflies were in my net to start?”
• In Unit 5, Lesson 22, Exit Slip, students revise false equations to make them true by changing one number. Students must recognize that there are multiple ways to revise the equation to make it true. The materials state, “Change one number to make the equation true. $$10 - 6 = 8 - 2$$.”
• In Unit 8, Lesson 6, Narrative states, “What do the students have to get better at today? Today scholars will apply their understanding of indirect comparison to story problems. Scholars will draw a picture to represent the problem and use the picture to help them solve the problem.” Exit Ticket states, “Shaquan’s crayon was shorter than Tyra’s crayon. Jesse’s crayon is longer than Tyra’s crayon. Whose crayon is longer- Jesse or Shaquan? How do you know?”

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

• Unit 1, Lesson 10, Materials, “Dot cubes/2 student” students are given dot cubes to roll and create addition equations, not other tools are provided or offered to students. 
• Unit 3, Lesson 1, Step 2, “How many are in that whole? (Record in the number bond and label “whole/cubes in all”) Look closely at your tower. Do you seen any parts? What parts do you see?” Students are not given a choice of tools but instead are given cubes to use. 
• Unit 5, Lesson 4, Materials, “Number cubes (3/student)” Students use the number cubes to create addition equations of three whole numbers, the students do not have a choice in the tools they use. 
• Unit 8, Lesson 2, Materials, The materials list centimeter cubes. The students use the centimeter cubes to measure various objects.  There is no other choice for students to use when measuring. 

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

• MP2: In Unit 3, Lesson 5, Introduction states, “How does the number bond show both addition and subtraction? It shows that two parts can be put together to make a whole (reference specific numbers from game), which is addition, and that the whole can be separated into two parts (reference numbers from game), which is subtraction.” Exit Ticket, Problem 1 states, “Write an addition equation and a subtraction equation that match the number bond.” (Number bond with 11, 4, 7)
• MP4: In Unit 6, Lesson 4, Narrative, students will decompose numbers 10-99 into tens and ones by using cubes, pictures, or knowledge of place value. The materials state, “Students will look at a number, decompose into tens and ones, and represent using literal pictures, sticks and dots or an equation.” 
• MP6: In Unit 4, Lesson 3, Workshop Practice Page, students must accurately organize and sort data using manipulatives. Problem 1 states, “Choose a bag of pattern blocks. Represent the data using a bar graph with categories and a scale.” 
• MP7: In Unit 7, Lesson 6, Workshop Worksheet, students use their understanding of the structure of a circle and fractions to tell time to the nearest half hour. Page 2, Problem 3 states, “Draw the missing hands on the clock. Half past 4.” Students are provided a picture of a clock with no hands.
• MP8: In Unit 9, Lesson 2, Exit Ticket, students use regular repeated reasoning to relate what they have learned about adding a two-digit number and a multiple of ten using sticks and dots, to adding them using the strategy of expanded notation. Problem 2 states, “Solve using expanded notation. $$37 + 50 =$$ ____”

The instructional materials reviewed for Achievement First Mathematics Grade 1 meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. The student materials prompt students to both construct viable arguments and analyze the arguments of others even though mathematical dialogue is mainly between the teacher and individual students.

Examples of constructing viable arguments include: 

• In Unit 3, Lesson 26, Introduction, students solve a compare story problem during the workshop introduction. The materials state, “Tamara found 12 ladybugs. Willie found 4 fewer ladybugs than Tamara. How many ladybugs did Willie find?” The teacher looks for accurate representations and then calls up students to explain their representations. The materials state, “SMS: First, I drew the 12 ladybugs because I knew that Tamara had 12 lady bugs. Then I connected circles until I got to 4 fewer because in the story Willie had 4 fewer. I drew an X to show that those were the 4 lady bugs that Willie didn’t have.” 
• In Unit 5, Lesson 6, Introduction, during a game, students determine if making a ten to solve is a strategy and explain why or why not. The materials state, “Play another round and check for understanding. Draw $$5 + 4$$. Students build with cubes. TT: Can we make ten to solve? Why or why not?” 
• In Unit 6, Lesson 8, Exit Ticket, Problem 1, students explain how they solved a problem. The materials state, “Solve using mental math. Explain how you figured it out. $$52 - 10 =$$ ___.”
• In Unit 7, Lesson 1, Workshop Worksheet, students construct a viable argument for partitioning  shapes into halves. Page 5 states, “Draw a line to split each of these shapes into halves. How do you know they are half and half?” Students are provided with pictures of nine shapes: two triangles, one hexagon, two circles, one rectangle, two squares, and one oval.

Examples of analyzing the arguments of others include: 

• In Unit 2, Lesson 2, Introduction, students analyze the thinking of their partner to see if they determined the correct shape or not. The materials state, “Step 3, my partner checks my shape and tells me if I did it correctly or not and how he/she knows. Can you all play my partner and help me check my shape?”
• In Unit 2, Lesson 9, Exit Ticket, Problem 2 states, “Fallon put together her puzzle like this. She says she made a square. Is she correct? How do you know?” An image of a rhombus made up of 4 triangles is shown.
• In Unit 5, Lesson 8, Introduction, students combine 2 numbers by using make 10 and match with an equivalent equation. The teacher draws the card with 7 + 5 on it and asks, “Sammy says he can make ten out of 7 by taking 3 from 5. Will says he can make 10 out of 5 by taking 5 from the 7. Who is correct? Why? They are both correct! We can think of it as $$2 + 5 + 5 = 2 + 10$$ like Will or we can think of it as $$7 + 3 + 2$$ like Sammy. Both are $$10 + 2 or 12$$!” 
• IN Unit 8 Assessment, students analyze the mathematical reasoning of others as they determine whether a fictional student measured the height of a tree correctly. Item 11 states, “Bobby says the tree is 4 toothpicks tall. Do you agree or disagree? Why?”

The instructional materials reviewed for Achievement First Mathematics Grade 1 meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. Examples of the materials assisting teachers in engaging students to construct viable arguments and analyze the arguments of others include:

• In Unit 2, Lesson 11, Introduction, a sentence starter is provided for teachers to give to students when describing why they chose to sort a shape into a particular group. The materials state, “‘Once you have all of the shapes sorted into the correct categories and glued down, you have to do one more thing. Step 6 is to write a sentence to describe one of the objects in each category using this sentence frame. Read it with me.’ ‘These are all_______. It is/has _______, just like all ______.’”
• In Unit 3, Lesson 4, Introduction, Step 3 states, “TT: How could we record what we just did? (hunt for students discussing using number bonds and equations; as they share whole group, use questions below to facilitate discussion). TT: Why do both these representations work? They both work because when you are subtracting you are starting with a whole and separating it into two parts/ you are starting with a whole, removing a part and you are left with the other part.”
• In Unit 3, Lesson 5, Introduction, students represent addition and subtraction scenarios with number bonds. The teacher asks, “How are addition and subtraction relate? What’s the same about them? What’s different? They both involve parts and wholes. In addition, two parts come together to make a whole, and in subtraction, a whole is separated into two parts.” The materials state, “Potential scaffold (if needed): How does the number bond show both addition and subtraction? It shows that two parts can be put together to make a whole (reference specific numbers from game), which is addition, and that the whole can be separated into two parts (reference numbers from game), which is subtraction.” 
• In Unit 6, Lesson 4, Introduction, students decompose numbers into tens and ones. The materials state, “Consolidate the Learning: How are you going to figure out how many tens and ones? SMS: I’m going to build it OR I’m going to look at the tens place and that will tell me how many tens. I’ll look at the ones place and that will tell me how many ones.”
• In Unit 7, Lesson 2, Introduction states, “T&T: Is there another way I can break this circle in quarters? Ask a student to come and model. Do you agree? Why? SMS: Yes that’s broken in quarters because there are 4 parts and they are equal (the same).” 
• In Unit 9, Lesson 4, Share/Discussion, guiding questions are provided for teachers to lead students to analyze the reasoning of other students as they share how they added a two-digit number and a one-digit number. The materials state, “2-3 students share their work/strategies in CPA order (count on, make ten). What is the same about these strategies? What is different? Which strategy is more efficient?”

The instructional materials reviewed for Achievement First Mathematics Grade 1 meet expectations that materials explicitly attend to the specialized language of mathematics.

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

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

• In Unit 2, Lesson 1, Introduction, teachers make visual aids for shapes with attributes states, “Repeat for circle, rectangle, square, trapezoid, and hexagons. Be sure to name each shape and record on posters, but you may not be able to record attributes for all of the shapes because of time; for shapes that have right angles, use the language of square corners to describe them.” 
• In Unit 3, Lesson 10, Introduction, students are provided explicit instruction in determining all combinations of numbers to make ten. The materials state, “Our last step, Step 4, is to record the combination we found, using a tape diagram. A tape diagram is a special tool that we can use to help us represent totals and parts of those totals, just like how we have used number bonds to help us represent totals and parts of totals!”
• In Unit 5, Lesson 1, Introduction, students are provided explicit instruction in the meaning of the commutative property as they play a game of roll and record and look for ways to efficiently add two numbers. The materials state, “TT: What happens to the total when we move around the addends or amounts we are combining? Why does this work? The total stays the same amounts together and no matter how you move the cubes around, the total number of dots doesn’t change because there are x dots on this cube and y dots on this cube and $$x$$ and $$y$$ makes $$z$$. (Drive this point home by moving cubes around and showing that there are still $$z$$ dots altogether no matter how you arrange the cubes). That’s right. Anytime you add two amounts, you can rearrange or move the amounts and still get the same total because you are putting the amounts together. This is called the commutative property. (record on VA)”
• In Unit 6, Lesson 20, Introduction, students are provided with explicit instruction in the meaning of the > and < symbols. The materials state, “Today we will also compare numbers using symbols.We have two new symbols to talk about. This symbol, >, means greater than. So I would write $$65 > 63$$ (model). What do you notice about the greater than symbol? SMS: I notice that one side on the left is big and open and the other side on the right is small and pointy. The big side is closer to the bigger number. The pointy side is pointing to the smaller number. (post on VA). Today you may also need to use the less than sign. It looks like this <. So I could write $$10 < 60$$. What do you notice about the less than symbol? SMS: I notice that one side on the right side is big and open and the other side on the left is small and pointy. It looks similar to the greater than sign. The big side is closer to the bigger number. The pointy side is pointing to the smaller number (post on VA).”
• In Unit 7, Lesson 5, Introduction, Introduce the math states, “Let’s do some quick review. (No more than 2 minutes.) What is this hand called - how do you know? (Point to hour hand/minute hand.) That’s the hour hand because it’s shorter. That’s the minute hand because it’s longer.”

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

• In Unit 2, Lesson 1, Introduction, accurate terminology is used when identifying and describing 2D shapes by making shape posters to articulate defining attributes. “Step 3 and the last step is to list the attributes, or things that are true about that shape. When I was drawing the different shapes, you all helped me to remember what I needed to have for all of my triangles to make sure I had 3 triangles. What were those attributes?SMS: 3 sides and 3 corners. Ok, so that is what I’m going to write down. These are things that are true of all of my triangles.” 
• In Unit 3, Lesson 4, Exit Slip, accurate terminology is used in the directions as students find the difference between two numbers and write an equation to match the problem. Problem 1 states, “Find the difference between the number cubes. Represent by completing the number bond and the equation.”
• In Unit 4, Lesson 1, Introduction, Introduce the Math states, “Today we are starting a brand new unit in math! During this unit we’ll get to use all of the things we know about counting and numbers to help us read and make graphs and charts! Graphs and charts are really cool because they help us organize and understand information. So we’re going to start today by collecting some DATA (choral response). Data is a collection of information. One way that we collect data is through surveys. So we’re going to do a quick survey in our class!”
• In Unit 5, Lesson 4, Skeleton VA, accurate terminology is used on the visual aid provided to support students as they learn to use the associative property. The materials state, “Associative Property what happens when we are combining amounts and we group the amounts differently. We get the same total! When I add, it doesn’t matter how I group the numbers--the total is the same.” Students are provided with pictures of three dice showing four, three, and two dots, and group the numbers in three different ways to demonstrate that they always have nine dots in all.
• In Unit 7, Lesson 2, Introduction, during a game the teacher develops vocabulary. The materials state, “Step 1 says Look at the shape. Step 2 says Break it in Quarters. T&T: How can I break this rectangle in quarters? SMS: You should draw a line down the middle so that it’s in two equal parts. Then draw another line down the middle so it’s four equal parts.”