## Achievement First Mathematics

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### Overall Summary

The materials reviewed for Achievement First Mathematics Grade 3 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 3 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 3 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.

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

The materials reviewed for Achievement First Mathematics Grade 3 meet expectations for assessing grade-level content and, if applicable, content from earlier grades. Each unit contains a Post-Assessment which is a summative assessment based on the standards designated in that unit.

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

• Unit 3, Post-Assessment, Item 1, “Round 452 to the nearest ten.” (3.NBT.1)

• Unit 5, Post-Assessment, Item 10, “Mark and his family were eating pizza for dinner. The pizza was split into 8 parts. Mark, his mom and his dad each ate one slice. What fraction of the pizza was NOT eaten?” (3.NF.1)

• Unit 7, Post-Assessment, Item 1, “Tyshio spent \$48 on gifts for her friends. She bought gifts for 6 friends. How much did each gift cost?” (3.OA.3)

• Unit 8, Post-Assessment, Item 3, “Mike runs 2 miles a day. His goal is to run 25 miles. After 5 days, how many miles does Mike have left to run in order to reach his goal?” (3.OA.8)

There is one above grade level assessment item that can be omitted or modified without impacting the underlying structure of the materials. For example:

• Unit 4 Post- Assessment, Item 2, “Ophelia had 64 ounces of milk. She wants to pour an equal amount of milk into 8 glasses for her children. How many ounces will Ophelia pour into each glass?” (4.MD.1)

Achievement First Mathematics Grade 3 has assessments linked to external resources in some Unit Overviews; however there is no clear delineation as to whether the assessment is used for formative, interim, cumulative, or summative purposes.

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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 3 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards.

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 4, Lessons 4 and 5, Independent Practice, students use addition, subtraction, multiplication, and division to solve one-step word problems involving masses or values that are given in the same units (3.MD.2). Most items require students to use addition and subtraction, though practice is provided for all operations. Lesson 5, Independent Practice, Problem 6, students are shown a picture of a scale balance with a mass of 18g on one side and two oranges on the other. “Julian placed two oranges on the balance scale below with the weight shown. What is the weight of one orange?”

• Unit 5, Lesson 2, Independent Practice, students partition shapes into parts with equal areas and express the area of each part as a unit fraction of the whole (3.G.2). Students practice partitioning fraction strips or shapes constructed with pattern blocks and record a specified portion as a fraction in 13 problems. In the Exit Ticket, Problem 2, students “Build the shaded shape below with your pattern blocks, tiles, or fraction strips. Then identify the unit fraction that represents the piece below that the arrow is pointing to.” An additional 13 items are provided in Practice Workbook D (pages 65-69) addressing this standard.

• Unit 7, Lesson 4, Workshop, students fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division or properties of operations (3.OA.7). During Independent Practice, students solve multiplication facts using the number 9. The Exit Ticket also provides an opportunity to engage with 3.OA.7 as students solve four more problems. Problem 4, “Carter thinks that the product of 9×8 is 63. Use what you know about patterns of multiples of 9 to explain why you agree or disagree with Carter.”

#### Criterion 1.2: Coherence

Each grade’s materials are coherent and consistent with the Standards.

The materials reviewed for Achievement First Mathematics Grade 3 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.

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When implemented as designed, the majority of the materials address the major clusters of each grade.

The materials reviewed for Achievement First Mathematics Grade 3 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each grade.

• The number of lessons devoted to major work of the grade (including assessments and supporting work connected to the major work) is 109 out of 127, which is approximately 86%.

• The number of days devoted to major work (including assessments and supporting work connected to the major work) is 117 out of 132, which is approximately 89%.

• The instructional minutes were calculated by taking the number of minutes devoted to the major work of the grade (10,380) and dividing it by the total number of instructional minutes (11,390), which is approximately 91%.

A minute-level analysis is most representative of the materials because the units and lessons do not include all of the components included in the math instructional time. The instructional block includes a math lesson, math stories, and math practice components. As a result, approximately 91% of the materials focus on major work of the grade.

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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 3 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.

There are opportunities in which supporting standards/clusters are used to support major work of the grade and are connected to the major standards/clusters of the grade. Examples include:

• Unit 2, Lesson 5, Workshop, Problem 2, students analyze a pictograph where one picture represents 4 cars. The pictograph shows the following, Honda = 3 car pictures, Ford = 4 car pictures, Toyota = 6 car pictures, and Chevrolet = 4 car pictures. Students are asked, “a. Were there more Honda and Toyota cars or Ford and Chevrolet cars in the neighborhood? b. Eight of the Fords moved away, and 2 more families with Toyotas moved in. How many Ford and Toyotas are in the neighborhood now? c. Jiang is interpreting the pictograph and says there are 6 more Toyotas than Hondas in the neighborhood. Is he correct?” This problem connects the major work of 3.OA.1, interpret products of whole numbers, and 3.OA.5, apply properties of operations as strategies to multiple and divide, to the supporting work of 3.MD.3, solve “how many more” and “how many less” problems using information presented in the pictograph.

• Unit 5, Lesson 2, Independent Practice, Problem 2, “Build a model of the unit fraction below with your fraction strips. Then, record the shape you made on the rectangle and label one unit fraction \frac{1}{6}.” This problem connects the major work of 3.NF.1, understanding a fraction \frac{1}{b} as the quantity formed by one part when a whole is partitioned into b parts, to the supporting work of 3.G.2, partition shapes into parts with equal areas.

• Unit 6, Lesson 6, Exit Ticket, “Heather is measuring the length of glue sticks to the nearest \frac{1}{4} inch. The lengths she’s measured so far are in the table below. Measure the remaining glue sticks and add their lengths to the table. Use the data to draw a line plot below.” This problem connects the major work cluster of 3.NF.A to the supporting work standard of 3.MD.4, as students develop understanding of fractions as numbers by generating data and representing it on a line plot.

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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 3 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 include:

• Unit 2, Lesson 5 connects the supporting work of 3.MD.B to the supporting work of 3.NBT.A, as students interpret data and use the properties of operations to perform multi-digit arithmetic. Independent Practice, Problem 1a, using a bar graph students answer, “How many fewer visitors were there on the least busy day than on the busiest day?”

• Unit 3, Lesson 12 connects the major work of 3.MD.A, solve problems involving measurements and estimations of intervals of time, liquid volumes and masses of objects, to the major work of 3.OA.A, represent and solve problems involving multiplication and division. In Workshop, Problem 5, “Laila is practicing her new step routine. It takes her 8 seconds to do the routine once. How long will it take her to do the routine 5 times?”

• Unit 8, Lesson 8 connects the major work of 3.OA.D, solve problems involving the four operations, and identify and explain patterns in arithmetic, to the major work of 3.OA.A, as students represent and solve problems involving multiplication and division. In Workshop, Problem 4, “Joel needs highlighter and pencils for his classroom. He buys 6 packs of highlighters with 5 in each pack. He also buys 7 packs of pencils with 4 in each pack. How many more highlighters doesJoel buy than pencils?”

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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 3 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.

Each unit has a Unit Overview and a section labeled “Identify Desired Results” where the standards for the unit are provided as well as a correlating section “Previous Grade Level Standards/Previously Taught & Related Standards” where prior grade-level standards are identified. Examples include:

• Unit 2, Unit Overview, Identify Desired Results: Identify the Standards lists 3.MD.3 as being addressed in this unit and identifies 2.MD.9, 2.NBT.2, 2.MD.10, and 3.OA.1 as Previous Grade Level Standards/Previously Taught & Related Standards. In the Linking Section, a brief description of the progression of the standards is given. “In grade 2, students draw a picture graph and a bar graph (with single-unit scale and including a title, axis labels, and category labels) to represent a data set with up to four categories. Using the information, students solve simple put-together, take-apart, and compare problems using information presented in a bar graph. Later in grade 3, students will return to generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch.”

• Unit 8, Unit Overview, Identify Desired Results: Identify the Standards lists 3.OA.9 as being addressed in this unit and identifies 2.OA.1 as Previous Grade Level Standards/Previously Taught & Related Standards connected to it. In the Linking section, a brief description of the progression of the standards is given. “By the end of second grade they’ve mastered all of the addition/subtraction story problem types within 100 - and even tackled two step story problems. In third grade, they begin multiplication and solve equal groups/array story problem types within 100 and solve for two-step story problems with all four operations.”

• Unit 9, Unit Overview, Identified Desired Results: Identify the Standards lists 3.G.1 as being addressed in this unit and identifies 2.G.A.1 under Previous Grade Level Standards/ Previously Taught and Related Standards. In the Linking section, a brief description of the progression of the standard is given. “In second grade, scholars learn to recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Specifically, they are taught to identify triangles, quadrilaterals, pentagons, hexagons, and cubes, many of which they have been able to recognize for years. In this unit, scholars use their same understanding of shape characteristics to classify different kinds of quadrilaterals. By the end of elementary school, scholars continue their work with classifying shapes based on their attributes. Specifically, scholars learn to classify shapes based on the presence or absence of parallel or perpendicular lines. They also learn to identify types of right triangles.”

The materials develop according to the grade-by-grade progressions in the Standards. Content from future grades are clearly identified and are related to grade-level work within each Unit Overview. Each Unit Overview contains a narrative that includes a “Linking” section that describes in detail the progression of the standards within the unit. Examples include:

• Unit 4, Unit Overview, Linking (p.6), “Measurement as it pertains to estimating liquid volume and the masses of objects using standard units of grams, kilograms, liters, and milliliters is introduced in grade 3. Students will then have to apply this information to answer one- and two-step story problems about the measurements they collect. In grade 2, students followed a similar trajectory with length. They explored standard units of length and then related this to addition and subtraction. In fourth grade, students continue to solve word problems involving liquid volumes and masses of objects. However grade 4 scholars are also expected to be able to convert units within a single system (from a larger unit to a smaller unit) of units including km, m, cm; kg; g; lb., oz; l, ml; hr., min, sec.”

• Unit 9, Unit Overview, Linking (p.5), “In second grade, scholars learn to recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Specifically, they are taught to identify triangles, quadrilaterals, pentagon, hexagons, and cubes, many of which they have been able to recognize for years. In this unit, scholars use their same understanding of shape characteristics to classify different kinds of quadrilaterals. By the end of elementary school, scholars continue their work with classifying shapes based on their attributes. Specifically, scholars learn to classify shapes based on the presence or absence of parallel or perpendicular lines. They also learn to identify types of right triangles.”

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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 3 foster coherence between grades and can be completed within a regular school year with little to no modification.

The Guide to Implementing AF, Grade 3 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 134 days. One day is provided for each lesson and one day is allotted for each unit assessment.

• There are nine units with 127 lessons in total.

• The Guide to Implementing Achievement First Mathematics Grade 3 identifies lessons as either R (remediation), E (enrichment), or O (on grade level). There are zero lessons identified as R (remediation), one lesson identified as E (enrichment), and 126 lessons identified as O (on grade level).

• There are seven days for Post-Assessments.

According to The Guide to Implementing Achievement First Mathematics Grade 3, each lesson is designed to be completed in 90 minutes. Each lesson consists of three parts:

• Math Lesson (60 min)

• Math Stories (20 min)

• Practice/Cumulative Review (10 min)

### 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 3 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.

The instructional materials reviewed for Achievement First Mathematics Grade 3 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

• Unit 1, Lesson 13, students develop conceptual understanding of 3.OA.2, interpret whole- number quotients of whole numbers. During the Workshop, students are provided with a variety of sharing situations and representations. In the Workshop, Problem 3, “Mr. Ziegler bought a pack of 18 markers. He wants to split them equally between himself and his niece, Sarah. How many markers will each person get?” Students are shown two picture representations, one showing two groups with nine items each and the other showing nine groups with two items each and asked, “Which drawing represents Mr. Ziegler’s problem? Why?”

• Unit 5, Lesson 10, students develop conceptual understanding of 3.NF.3, explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. During Pose the Problem students compare fractions using models to support their answer, “Lily and Jasmine each bake a chocolate cake. Lily puts \frac{3}{8} of a cup of sugar in her cake. Jasmine puts \frac{5}{8} of a cup of sugar into her cake. Who uses less sugar? Draw a model to support your answer.”

• Unit 7, Lesson 6, students develop conceptual understanding of 3.OA.B, as they draw arrays and write equations to model the distributive property of multiplication. In the Independent Practice, Problem 2, “Draw an array to match the equation 6 x 9 then use the distributive property to break apart the array and solve it. Array: Equation: ( )+( )”

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

• Unit 2, Cumulative Review 2.2, students demonstrate conceptual understanding of 3.MD.C, as they determine the area of shapes and solve problems based on provided grids with unit squares. Problem 6, “Anna’s garden is 7 feet long and 7 feet wide. Noah’s garden is 8 feet long and 6 feet wide. Which garden has a smaller area? ______’s garden is smaller.”

• Unit 5, Lesson 2, Independent Practice, students demonstrate conceptual understanding of 3.NF.1, as they build a model of a unit fraction. Problem 3, “Build a model of the unit fraction below with your fraction strips. Then, record the shape you made on the rectangle and label one unit fraction \frac{1}{8}.”

• Unit 7, Lesson 12, Problem of the Day, Let’s Try One More, students demonstrate conceptual understanding of 3.OA.5, as they create equations based on their knowledge of the distributive model. “Write three different equations that we could use to find the area of the following rectangle. Then, find the area of the rectangle.”

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

Materials give attention throughout the year to individual standards that set an expectation for procedural skill and fluency.

The materials for Achievement First Mathematics Grade 3 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. The materials include opportunities for students to build procedural skill and fluency in both Math Practice and Cumulative Review worksheets. The materials do not include collaborative or independent games, math center activities, or non-paper/pencil activities to develop procedural skill and fluency.

Math Practice is intended to “build procedural skill and fluency” and occurs four days a week for 10 minutes. There are six Practice Workbooks in Achievement First Mathematics, Grade 3. Two workbooks, B and F, contain resources to support the procedural skill and fluency standards 3.NBT.2: Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction; and 3.OA.7: Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division or properties of operations. In the Guide To Implementing Achievement First Mathematics Grade 3, teachers are provided with guidance for which workbook to use based on the unit of instruction. Examples include:

• Practice Workbook B, Problem 11, “Calculate. 605 - 327 = ; 708 - 439 = ; 875 - 218 = ; 575 + 219 = ; 238 + 573 = ; 117 + 582 = .” (3.NBT.2)

• Practice Workbook B, Problem 2, “$$303 - 165 =$$ .” (3.NBT.2)

• Practice Workbook F contains 25 independent practice problems that allow students to build procedural skill and fluency with multiplication and division within 100. Problem 6, “$$8 × ______ = 56$$.” (3.OA.7)

Cumulative Reviews are intended to “facilitate the making of connections and build fluency or solidify understandings of the skills and concepts students have acquired throughout the week to strategically revisit concepts, mostly focused on major work of the grade.” Cumulative Reviews occur every Friday for 20 minutes. Examples include:

• Unit 5, Cumulative Review 5.2, Problem 2, students practice adding and subtracting within 1000. “Solve. 909 - 690.” (written vertically). (3.NBT.2)

• Unit 5, Cumulative Review 5.4, Problem 4, students practice solving division problems within 100. “Solve. 40 ÷ 8 = ; 16 ÷ 8 =; 7 ÷ 7 =; 72 ÷ 8 =; 27 ÷ 3 =.” (3.OA.7)

• Unit 7, Cumulative Review 7.2, Problem 2, students practice solving multiplication problems. “Solve. 6 × 7; 8 × 4= ; 9 × 3= ; 3 × 6= ; 8 × 3 = ; 5 × 5 = .” (3.OA.7)

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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 3 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 3, Lesson 11, Problem of the Day, students engage with 3.OA.8 as they solve a two-step word problem in a non-routine format. "Monday morning, Ashley starts a wood pile by stacking 215 pieces of wood. Monday afternoon, Dad takes 17 pieces of wood from the pile to burn in the fireplace. Tuesday morning, Ashley stacks 118 pieces of wood on the pile. Tuesday afternoon, Dad takes 26 pieces of wood to burn in the fireplace. Ashley wants to have exactly 350 pieces of wood on the pile on Wednesday. Does Ashley have to stack more wood on the pile or does Dad have to burn more wood in the fireplace? Show all your mathematical thinking.”

• Unit 4, Guide to Implementing AF, Math Stories, students engage with 3.OA.3 as they use multiplication and division within 100 to solve routine word problems in situations involving equal groups, arrays, and measurement quantities. Sample Problem 8, “Carla has 12 stuffed bears, 18 stuffed rabbits, and 6 stuffed elephants to donate to the local charity shop. The shop wants to arrange them into equal groups. What are two different ways the shop could arrange Carla’s donated stuffed animals?”

• Unit 8, Lesson 13, Math Stories, Perimeter Robot Project, students engage with 3.MD.7 as they apply their knowledge of solving problems involving area in a non-routine format. "You have worked on so many different kinds of word problems in this unit, and the last few days we have been focusing on area and perimeter. Today we will use what we know about the perimeter formula and our addition patterns from our addition table to help us create our own Perimeter Robot! We are going to use this table to help us brainstorm dimensions for the different body parts for our robot. Turn and talk with your partner, what do you notice about the table? We have to find the length and width for each of the perimeters, we have to make dimensions for all the different body parts.”

Materials provide opportunities for students to independently demonstrate routine and non-routine applications of the mathematics throughout the grade level. Examples include:

• Unit 2, Lesson 7, Independent Practice, students engage with 3.MD.3 as they draw a scaled picture graph and a scaled bar graph to represent a data set with several categories in a non-routine problem. Question 5, “Mr. Park spent 38 minutes grading exit tickets on Friday night. Ms. Duke spent 44 minutes. Ms. Ervey spent 26 minutes grading exit tickets. Ms. Negron spent 30 minutes. Ms. Max-McCarthy spent the most time grading exit tickets with 46 minutes. Use this data to create a pictograph and bar graph. Write 3 questions for your partner to solve.”

• Unit 7, Lesson 1, Exit Ticket, students demonstrate application of 3.OA.3 as they use multiplication to solve routine word problems. Question 3, “Gretta says there would be 17 hands on 9 people. Use what you know about the patterns for multiples of 2 to explain why you agree or disagree with Gretta.”

• Unit 8, Lesson 8, Independent Practice, students demonstrate application of 3.OA.8 as they solve a routine two step word problem involving the four operations. Question 2, “Marlon buys 9 packs of hot dogs. There are 6 hot dogs in each pack. After the barbeque, 35 hot dogs are left over. How many hot dogs were eaten?”

##### 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 3 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.

The instructional materials include opportunities for students to independently demonstrate the three aspects of rigor. Examples include:

• Unit 2, Cumulative Review 2.2, students demonstrate conceptual understanding as they interpret products of whole numbers as the total number of objects in groups by comparing two grouping strategies used to evaluate the same expression. Problem 7, “Finn and Sadie are both solving the problems 4 x 5 x 2. Their teacher said they are both correct. Their work is below. Why are they both correct?” (3.OA.1)

• Practice Workbook B, Problem 29, students demonstrate procedural skill and fluency related to addition and subtraction as they solve problems. It, “Solve to find the missing numbers. 142 + \_ = 225, 506 - \_ = 329\_ + 344 = 764.” (3.NBT.2)

• Unit 8, Lesson 10, Independent Practice, students apply their understanding of the four operations as they solve a word problem. Problem 7, “Tajah washes 4 loads of laundry each week. Each load requires 2 ounces of washing powder. If she washes laundry for 50 weeks this year, how many ounces of washing powder will she use?” (3.OA.8)

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 3, Lesson 7, Independent Practice, students demonstrate conceptual understanding and procedural skills as they solve problems using strategies based on place value with multiple addends. Problem 3, “Ryan, Dominic, and Brittney were collecting acorns. Ryan gathered 109 in his bag. Dominic collected 87 in his bag. Brittney picked up 132 acorns. At the end of the day, they put all the acorns into a cardboard box. How many acorns were in the box?” (3.NBT.2)

• Unit 4, Cumulative Review 4.1, students demonstrate conceptual understanding and application of multiplication within 100 as they write an equation and solve problems with an array. “Write a multiplication equation to match the picture below. Use p to represent the unknown number. How many paint cans are there?” (3.OA.7)

• Unit 7, Lesson 6, Independent Practice, students demonstrate conceptual understanding and application of multiplication as they create arrays and apply their knowledge of multiplication to solve problems. Problem 3, “Franklin collects stickers. He organizes his stickers in 5 rows of four. a. Draw an array to represent Franklin’s stickers. Use an x to show each sticker. b. Solve the equation to find Franklin’s total number of stickers. 5 × 4 = \_.” (3.OA.3)

#### 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 3 meet expectations for practice-content connections. The materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs).

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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 3 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:

• The Unit 2 Overview outlines the intentional development of MP1. “Students look for entry points and strategies to solve story problems. Students annotate the problem and its graph to identify the information given and the information needed to solve the problem. Students assess the reasonableness of their solutions against their representations and strategies.”

• Unit 3, Lesson 10, How will embedded MPs support and deepen the learning? describes the intentional development of the MP within the lesson. “Students engage with MP 1 as students solve story problems and make sense of what the problem is asking. They preserve when solving problems by annotating, representing the ‘big questions’ and ‘little questions,’ and using calculation strategies to help solve and make sense of the problem. As students solve, they must assess the reasonableness of their solutions against their representations and strategies.”

• Unit 6, Lesson 12, Independent Practice, students access relevant knowledge and work through a task with multiple entry points. “Directions: The zookeepers are designing a habitat for their newest animal, the pandas! They know they need a pen with an area of 24 square meters, but they want to know all of the possible options. Find the four different pens they could make with an area of 24 square meters. Record the details about each shape in the chart below!”

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

• The Unit 4 Overview outlines the intentional development of MP2. “Students decontextualize metric measurements as they solve problems involving addition, subtraction, multiplication, and division. They round to estimate and then precisely solve, evaluating solutions with reference to units and with respect to real world contexts. Students use real-world benchmarks of metric units to estimate measures for weight, determining whether an object is greater than or less than the benchmarks. Students use intervals on scales to determine the most precise measurement. Students represent their final answers using appropriate units from the problem context (e.g., dollars and coins or g versus kg).”

• The Unit 7 Overview describes development of MP2. “Students make sense of quantities and their relationships as they explore the properties of multiplication and division and the relationship between them. Students decontextualize when representing equal group situations as multiplication, and when they represent division as partitioning objects into equal shares or as unknown factor problems. Students contextualize when they consider the value of units and understand the meaning of quantities as they compute. Students will build towards abstraction by composing and decomposing tiled shapes first and then working with just given dimensions or the area of a shape.”

• Unit 8 Lesson 4, Pose the Problem, students engage with MP2 as they predict patterns between factors and products, and explore the strategy of predicting whether a product is odd or even. “Draco says that 4 times a number will always give you an even product. His partner Harry isn’t quite sure. Is he correct? Discuss with your partner, you may use the multiplication table to prove your thinking.”

##### 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 3 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 3, Lesson 2, Exit Ticket, students critique the reasoning of others based on their knowledge of estimation and rounding. Problem 3, “There are 525 pages in a book. Julia and Kim round the number of pages to the nearest hundred. Julia says it is 600. Kim says it is 500. Who is correct? Explain your thinking.”

• Unit 3, Lesson 2, Try One More, students are asked to determine if an answer is reasonable. “Samantha solved this problem: 472 + 371 She got an answer of 743. Round to the nearest hundred and then solve to determine whether or not her answer is reasonable.” The teacher’s guidance includes, “How did you round 472? How did you round 371? Is Samantha’s answer reasonable? Why or why not?”

• Unit 4, Lesson 4, Workshop, Problem 7, “The capacity of a pitcher is 3 liters. What is the capacity of 9 pitchers? John says that 9 ÷ 3 = P represents this story. Do you agree or disagree? Explain.”

• Unit 5, Lesson 24, Problem of the Day, students analyze the reasoning of others and use models to explain their reasoning. “Treasure and Shianne are having an argument. Shianne thinks that \frac{3}{3} is larger than \frac{3}{1}. Treasure disagrees. She thinks that \frac{3}{3} is smaller than \frac{3}{1}. Use models to show both numbers and explain which is larger.”

• Unit 8, Lesson 2, Try One More, students critique the reasoning of others and construct a viable argument based on their knowledge of the properties of addition. “Khallel also says that when you add an even number and an odd number you will get an odd sum. Is Khaleel correct? Work with your partner to explain why; prove your answer using a visual model or the addition table.”

• Unit 9, Lesson 4, Exit Ticket, students construct a viable argument based on their understanding of shapes. Problem 3, “In the grid below, draw a rhombus that is also a rectangle. Explain how your shape fits in both categories.”

##### 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 3 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 2 Lesson 2, students engage with MP4 as they create and use tape diagrams to collect and organize data. Pose the Problem, “Reisha plays in three basketball games. She scores 12 points in Game 1, 8 points in Game 2, and 16 points in Game 3. Each basket that she made was worth 2 points. Represent each game with a tape diagram.”

• Unit 3, Lesson 10, Narrative assists the teacher with intentional development of MP4 within the lesson. “Students also engage in MP 4 as they solve story problems by using mathematical models and connecting it back to the story problem. Students model the story problem with an appropriate representation (tape diagrams or equations) and use an appropriate strategy (expanded notations, add by place, or number lines) to solve. 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.”

• Unit 7, Lesson 11, Exit Ticket Question 1, “students engage with MP 4 as students compose smaller areas of rectangles to determine the area of a larger rectangle, ‘Heather has two rugs. One rug is 5 feet by 6 feet. The other rug is 6 feet by 3 feet. She puts the two rugs next to each other on her floor. a) Draw the rugs in the grid. Then write a number sentence to find the area covered by both rugs on the floor.’”

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

• Unit 2 Overview, “Students use cubes or other items to concretely illustrate the values represented in graphs. Students use the problem context to define how the representation matches the problem situation in creating a key and scale for the representation. Students define each value of a pictograph using multiplication facts.”

• Unit 5 Overview, “Students use pattern blocks, fraction tiles, and circles to illustrate unit fractions and non-unit fractions as copies of unit fractions as the basis for reasoning about comparing fractions with the same denominators or numerators as well as using the foundation of non-unit fractions as a basis for creating equivalent fractions by partitioning part size in models or intervals using number lines.”

• Unit 6, Lesson 4, Narrative assists the teacher with intentional development of MP5 within the lesson. “Students practice SMP 5 as they consider how the title, range, and number of X indicate the unit of measurement, range of the measurements represented, and the number of times each measurement occurred.”

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 3 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 3, Lesson 5, Exit Ticket, Question 3, students accurately calculate and explain how to use expanded notation to find a sum. “Your friend at another school wants to combine 684 and 134. Show and explain how they could add with expanded notation.”

• Unit 7, Lesson 6, Exit Ticket Question 1, precision is used as students are expected to draw an array and write an expression. “Mrs. Stern roasts cloves of garlic. She places 9 rows of 6 cloves on a baking sheet. Part A: Draw an array to show the total number of cloves. Write an expression to describe the number of cloves Mrs. Stern bakes. Part B: Use the distributive property to solve this problem. Draw a model and write an equation to show your thinking.”

• The Unit 9 Overview, “Students use geoboards to model and name polygons (specifically quadrilaterals). Students use the corner of a piece of paper or square tile to compare the sizes of the angles on the page to verify which are and are not square angles; Students measure lengths of sides with rulers; Students extend sides with rulers to check to see if sides are parallel or not; Use geoboards to model and name polygons (specifically quadrilaterals); Students attend to precision when drawing polygons and quadrilaterals (ensuring straight lines, right angles, parallel lines, equal length sides).”

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

• Unit 1, Lesson 6, State the Aim provides guidance for teachers in introducing the identified vocabulary word area. “For the last few days we have been studying multiplication using equal group pictures and arrays. Today we are going to use some of that knowledge to study a new topic called area...How many triangles did it take to fill shape A? What about shape B? Great work! We just found the area of these shapes. Area is the amount of flat space that an object takes up.”

• Unit 4, Lesson 1, State the Aim provides guidance for teachers in introducing mathematical terminology related to measurement. “You’ve worked on measuring in 2nd grade and this year we will continue our growth with measurement. By the end of today you will be able to estimate the weight of objects in grams, kilograms, and measure using scales. This is another day that units will be super important! Today we will be talking about two units of weight. The first is called a gram. A gram is about the weight of 1 paper clip. The second is called a kilogram and weighs about as much as a textbook.”

• Unit 9, Lesson 1, State the Aim identifies new vocabulary used in the lesson and provides specific guidance for teachers in introducing the terminology, including counterexamples. “Today we are starting a new unit on geometry. Geometry is the study of shapes and their attributes, or the way we describe them. We will start our units describing polygons. Polygons are closed shapes made up of line segments or straight lines. Take a look at the top of your page. The first shape is not a polygon because it has an open-space and is not closed, the second shape is not a polygon because the top is curved so it is not made up of line segments or straight lines. The last shape is a polygon because it is a closed shape made up of straight lines. Let’s get started on our study of polygons and how we can talk about them.”

##### 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 3 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 3 Lesson 17, Independent Practice, Question 6, students make sense of structure as they solve problems to find start time by counting back on a number line in hour and minute intervals. Problem 6, “Mr. Wellborn arrives at work at 8:30. He leaves for work 50 minutes before. What times does Mr. Wellborn leave for work?”

• Unit 5, Lesson 1, Independent Practice, Questions 1-4, students engage with MP7 as they use unit fractions as the basic building blocks of all fractions on the number line. “DIRECTIONS: Build the unit fraction below with your pattern blocks. Then, record the shape you made on the dot paper space and label one unit fraction. \frac{1}{2}, \frac{1}{6}, \frac{1}{9}, \frac{1}{5}.”

• Unit 8, Lesson 5, Independent Practice, students extend a pattern based on a provided pattern. Problem 2, “Marc-Anthony wrote the number pattern below. 15, 19, 23, ____, 31. Part A: What is the missing number in Marc-Anthony’s patterns? e) 24; f) 29; g) 27; h) 30. Part B: What is the rule for this pattern? Part C: What would the next three numbers in his pattern be? __,___,___.”

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

• Unit 1, Lesson 20, Independent Practice, Question 2c, students engage with MP8 as they use the commutative and associative properties of multiplication to solve problems. “Find the products for each. First solve the part in parenthesis and write a new multiplication fact on the first line. Then write the product on the bottom line. (4\times5)\times2=4\times(5\times2).”

• Unit 3, Lesson 5, Independent Practice students add two and three digit numbers using expanded notation. Problem 3, “Juan added 375 and 128 and got 503. Do you agree with his answer? Why or why not? Explain your thinking on the lines below.”

• Unit 5, Lesson 9, Pose the Problem, students reason about the size of fractions and look for strategies they can use when comparing fractions with the same numerator or same denominator. “Genesis cut his hot dog into thirds and ate \frac{1}{3}. Ameera cut her hot dog into fourths and ate \frac{1}{4}. Who ate more? Use >, < or = to write a comparison statement. Use your fraction strips or a picture to show and explain your thinking.”

### Usability

The materials reviewed for Achievement First Mathematics Grade 3 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 3 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 3 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 5, Fractions, Lesson 25 include:

• “What do students have to get better at today? Students need to combine all of the skills they have been building over the unit to solve problems today. Scholars will work to place fractions on number lines, as well as compare fractions and find equivalent fractions using number lines. What is new and/or hard about that? This is challenging because scholars may still be struggling to accurately place fractions on number lines, therefore finding equivalent fractions and comparing fractions may be difficult. In addition, number lines require precision, particularly for comparing and finding equivalent fractions. So where will time be focused/funneled? The Intro and Interruption will focus on creating accurate number lines and reiterating the key points of placing, comparing, and finding equivalencies.”

• “Exemplar Student Response: “I can use a number line to solve many different types of fraction problems. To use a number line correctly I need to remember to partition the number line first into the wholes, then partition each whole into the parts shown in the denominator of my fraction. Then I can count by my unit fractions to place my fractions accurately. Once I have done this, I can either place a second fraction to compare by thinking about which fraction is further away from zero and therefore larger, or I can create more smaller partitions or fewer, larger partitions to find an equivalent fraction for the one I have placed.”

• “Mid-Workshop Interruption What is the next level for the skill in the objective? What do you want most of your students to start doing?”

• “Discussion: Discuss relevant student work samples to push thinking to the next level. What questions will you ask to get students to articulate the big ideas before moving on?”

• “Closure: How can we get kids to summarize what was learned today and connect back to the aim? Exit ticket.”

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 5, Fractions, Lesson 25 include:

• “What Key Points What should students know and be able to do? I can place fractions on a number line. I can compare fractions using a number line. I can compare fractions using reasoning. I can find equivalent fractions using a number line.”

• “How Key Points How will they do it? We can place fractions accurately on a number line by drawing a number line and partitioning it into the parts shown by the denominator, then counting by the fraction until we reach the fraction shown by the numerator and marking that spot. We can compare fractions with the same denominator using a number line by drawing a number line and partitioning it into the number of parts shown by the denominator, then marking both fractions on the line. The fraction closer to 0 is the lesser fraction. We can compare fractions with the same numerator using a number line by drawing two number lines and partitioning them into the parts shown by the two different denominators, then counting tick marks until we get to the amount shown in the numerator. The fraction closest to 0 is the lesser fraction. We can compare fractions by reasoning about their size. When fractions have the same denominator we compare their numerators, the larger numerator is the larger fraction because that means it has more parts or more of the whole would be shaded. When fractions have the same numerator, we compare their denominators. The fraction with the smaller denominator is the greater fraction because that means the whole has been broken up into fewer pieces, and therefore the pieces are larger. We can find equivalent fractions by drawing a number line and then partitioning the intervals further in order to create more smaller parts, or grouping partitions together to create less, larger parts.”

##### 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 3 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. In the Unit 6 Overview when discussing a Key Strategy for Finding the length of a missing side. “Scholars can split the shape into squares and rectangles so as to use common attributes (i.e. all of the sides of the square will be 2cm) to find the lengths of missing sides. Parallel sides on a rectilinear figure follow a sort of part-part-whole relationship (i.e. the horizontal sides 4cm and 2cm are parts of the longer horizontal side, which is the whole). Scholars can look at sides parallel to the side that’s missing. They can decide if the missing side is a part or whole in the relationship and use that to find the missing side length (i.e. 4cm + 2cm = 6cm).”

• 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 3 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.

• Guide to Implementing AF Grade 3, 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 5 Overview, “Students begin their introduction to fractions (at least fractional language) in first grade when students partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Students are also primed by being taught to recognize that equal shares of identical wholes need not have the same shape. In fourth grade, scholars extend their understanding of fractions by learning how to explain a/b as being equal to (n x a)/ (n x b). This allows scholars to generate equivalent fractions. Scholars also extend their ability to compare fractions with different numerators and denominators. Scholars also learn to use the identity property to make either the numerators or denominators the same for the purpose of comparing. Scholars also learn how to add and subtract, and multiply fractions by thinking about joining and separating parts of the same whole. Furthermore, scholars applying these concepts for the first time in the form of story problems. Lastly, fourth grade introduces decimal notation for fractions with denominators 10 or 100. Scholars then compare these decimals to hundredths.”

• Unit 9 Overview, “In second grade, scholars learn to recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Specifically, they are taught to identify triangles, quadrilaterals, pentagons, hexagons, and cubes, many of which they have been able to recognize for years. In this unit, scholars use their same understanding of shape characteristics to classify different kinds of quadrilaterals. By the end of elementary school, scholars continue their work with classifying shapes based on their attributes. Specifically, scholars learn to classify shapes based on the presence or absence of parallel or perpendicular lines. They also learn to identify types of right triangles.”

##### 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 3 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 3 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 3 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 3, Lesson 12, the Lesson Overview includes, “Materials: student work packets, stopwatches for partners, VA.”

• Unit 4, Lesson 3, the Lesson Overview includes, “Materials: Student work packet, VA (stands for visual aid), Objects to model with: eye dropper, 1 liter water bottle, 500 mL water bottle, 1 liter beaker with milliliter intervals, Containers as models for Workshop, such as spoon, soda can, milk, or other containers to estimate and measure capacity.”

##### 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 3 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 3 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 some of the formal assessments, but do not include all standards for the grade or course level. Examples include:

• Most Unit Overviews provide 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.

• Units 6 and 9 do not identify standards for the Unit Assessment.

• Some standards are not included in Unit Assessments: 3.MD.4, 3.MD.5, 3.MD.8, and 3.G.2.

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 3 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 3 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:

• The Unit 3 Assessment contributes to the full intent of 3.NBT.1(use place value understanding to round whole numbers to the nearest 10 or 100) and 3.NBT.2 (fluently add and subtract within 1000). Items 1-6, “1) Round 452 to the nearest ten. 2) Round 452 to the nearest hundred. 3) When rounding to the nearest ten, what is the least whole number that will round to 50? Explain how you know. 4) Solve 205 - 186 = . 5) Compute 98 + 427 = . 6) Part A. First grade eats 242 bananas and second grade eats 183 bananas. Find the sum of the bananas they eat. Show your strategy to solve and explain your thinking. Part B. Mr. Teague estimates the sum of the bananas by rounding each amount to the nearest ten before adding. He decides to order the estimated number of bananas for tomorrow’s lunch. Will Mr. Teague have ordered enough bananas? Why or why not?”

• Unit 5, Lesson 25, Exit Ticket, Problem 1 contributes to the full intent of 3.NF.3 (Explain equivalence of fractions in special cases), “Martin says that if a fraction has an even denominator, its equivalent fractions will also always have an even denominator. Is Martin correct? Explain why or why not on the lines below.”

• The Unit 9 Assessment contributes to the full intent of 3.G.1 (recognize rhombuses, rectangles, and squares as examples of quadrilaterals). Item 4, “Frederick draws a quadrilateral with two sets of parallel sides. Draw what his shape could be on the grid below. Then, name the shape you drew.”

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

• Unit 3 Assessment, Item 10, supports the full development of MP1 as students look for entry points and information needed to solve a word problem. “Baseball practice starts at 6:30 in the evening. Nelson has a list of activities to do before going to baseball practice: 15 minutes to feed and walk the dog; half an hour to eat dinner; twenty-five minutes to do homework. It takes fifteen minutes to get to baseball practice. What time does Nelson need to start his activities in order to get to baseball practice on time? Show all your mathematical thinking.”

• Unit 7 Assessment, Item 9, supports the full development of MP3 as students justify their answer. “Mr. Yearwood is arranging desks in rows with a front section and a back section. The array below represents his desks. Fill in the blanks below with the numbers that make the equation true: (__ x __) + (__ x __) = 6 x 6.  Explain why his number sentence will work to find the total.”

• Unit 8 Assessment, Item 1, supports the full development of MP8 as students look for and explain patterns of odd and even factors. “The first row in a pattern of tiles had 5 tiles. Each row after the first had 2 more tiles than the row before it, as shown below. Which statement is true about the number of tiles in any row?  a. It is divisible by 10; b. It is an even number; c. It is a multiple of 3; d. It is an odd number.”

##### 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 3 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 3 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 3 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 3 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 3, Lesson 13, Extension extends students' understanding of telling time. “Encourage students to begin figuring out how much time has passed between two times. For example: if class starts at 8:31 and lasts for 45 minutes. What time does it end?”

• Unit 9, Lesson 3, Extension problem, extends students’ understanding of comparing quadrilaterals. “Encourage students to write quadrilateral riddles in which they describe a shape based on its attributes. They can then trade with a friend and try and figure out which shape is being described in the riddle.”

##### 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 3 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 3 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 2, Lesson 1, Introduction, “Okay, working with your partner and using what you know, create a pictograph to represent the data in the chart.”

• Unit 6, Lesson 5, Introduction, “Give students a few minutes to work with a partner and to come up with a solution. Circulate as they work, gathering data.”

##### 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 3 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 3 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 3 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 3 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 3 do not provide supports for different reading levels to ensure accessibility for students.

The materials do not include strategies to engage students in reading and accessing grade-level mathematics. There are not multiple entry points that present a variety of representations to help struggling readers to access and engage in grade-level mathematics.

##### 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 3 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 most commonly found in the Intervention suggestion at the end of Workshop time. However, there is little teacher guidance to explain how and when the intervention is intended to be used. Examples include:

• Unit 1, Lesson 2, “students interpret arrays as multiplication and use arrays to represent multiplication.” The Intervention includes the use of manipulatives, “Allow students to use counters to create their groups and then their arrays.”

• Unit 5 Overview, Lesson Sequence, Lesson 2, “Scholars will fold rectangular strips to create their own fraction strips. Scholars will continue their work with naming fractional parts and identifying where a whole (strip) is split equally.” In Lesson 2, Problem of the Day,  “Maya baked a rectangular cake. She wants to break it into fourths so that she can share it with her mom, dad and sister. Use your fraction strips to represent \frac{1}{4} and show your work below.”

• Unit 7, Lesson 20, Narrative, “students need to find the area of shapes by partitioning the shape into smaller rectangles.” The Intervention includes the use of manipulatives,“Transfer shapes onto grid paper. Have students cut shapes in order to see that the area of the shape remains the same even if you cut the shape into 2 or more rectangles. Guide students to break down shape into rectangles and squares. Have them write in the dimensions for the small shapes. Prompt them to use what they know about the irregular shape’s sides to help them figure out the side lengths for the small shapes.”

There are a few instances where manipulatives are not connected to written methods. Examples Include:

• Unit 2 Overview, “Students use cubes or other items to concretely illustrate the values represented in graphs. Students use the problem context to define how the representation matches the problem situation in creating a key and scale for the representation.”  However there is no use of cubes in any of the lessons of Unit 2.

#### 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 3 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 3 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 3 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 3 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 3 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

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### Overall Summary

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###### Usability
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