Achievement First Mathematics

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Report for 3rd Grade

Overall Summary

The instructional materials reviewed for Achievement First Mathematics Grade 3 partially meet expectations for alignment to the CCSSM. ​The instructional materials meet expectations for Gateway 1, focus and coherence, by assessing grade-level content, focusing on the major work of the grade, and being coherent and consistent with the Standards. The instructional materials partially meet expectations for Gateway 2, rigor and balance and practice-content connections. The materials meet the expectations for rigor and balance and partially meet the expectations for practice-content connections.

3rd Grade
Alignment
Partially Meets Expectations
Not Rated

Focus & Coherence

The instructional materials reviewed for Achievement First Mathematics Grade 3 meet expectations for Gateway 1, focus and coherence. The instructional materials meet the expectations for focus by assessing grade-level content and spending at least 65% of instructional time on the major work of the grade, and they also meet expectations for being coherent and consistent with the standards.

Gateway 1
Meets Expectations

Criterion 1.1: Focus

Materials do not assess topics before the grade level in which the topic should be introduced.

The instructional materials reviewed for Achievement First Mathematics Grade 3 meet expectations for not assessing topics before the grade level in which the topic should be introduced.

Indicator {{'1a' | indicatorName}}
The instructional material assesses the grade-level content and, if applicable, content from earlier grades. Content from future grades may be introduced but students should not be held accountable on assessments for future expectations.

The instructional materials reviewed for Achievement First Mathematics Grade 3 meet expectations for assessing grade-level content. Each unit of instruction 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:

• In Unit 3, Post-Assessment, Item 1 states, “Round 452 to the nearest ten.” (3.NBT.1)
• In Unit 5, Post-Assessment, Item 10 states, “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)
• In Unit 7, Post-Assessment, Item 1 states, “Tyshio spent $48 on gifts for her friends. She bought gifts for 6 friends. How much did each gift cost?” (3.OA.3) • In Unit 8, Post-Assessment, Item 3 states, “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: • In Unit 4 Post- Assessment, Item 2 states, “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. Criterion 1.2: Coherence Students and teachers using the materials as designed devote the large majority of class time in each grade K-8 to the major work of the grade. The instructional materials reviewed for Achievement First Mathematics Grade 3, when used as designed, spend approximately 91% of instructional time on the major work of the grade, or supporting work connected to major work of the grade. Indicator {{'1b' | indicatorName}} Instructional material spends the majority of class time on the major cluster of each grade. The instructional materials reviewed for Achievement First Mathematics Grade 3 meet expectations for spending a majority of instructional time on major work of the 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 instructional 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 instructional materials focus on major work of the grade. Criterion 1.3: Coherence Coherence: Each grade's instructional materials are coherent and consistent with the Standards. The instructional materials reviewed for Achievement First Mathematics Grade 3 meet expectations for being coherent and consistent with the standards. The instructional materials have supporting content that engages students in the major work of the grade and content designated for one grade level that is viable for one school year. The materials also foster coherence through connections at a single grade. Indicator {{'1c' | indicatorName}} Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade. The instructional materials reviewed for Achievement First Mathematics Grade 3 meet expectations that supporting work 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: • In 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. • In Unit 5, Lesson 2, Independent Practice, Problem 2 states, “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. • In Unit 6, Lesson 6, Exit Ticket states, “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. Indicator {{'1d' | indicatorName}} The amount of content designated for one grade level is viable for one school year in order to foster coherence between grades. The instructional materials reviewed for Achievement First Mathematics Grade 3 meet expectations that the amount of content designated for one grade-level is viable for one school year. The Guide to Implementing Achievement First Mathematics Grade 3 includes a scope and sequence and states, “Not every lesson is entirely focused on grade level standards, and, therefore some lessons can be used for remediation or enrichment.” As designed, the instructional materials can be completed in 134 days. • 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) Indicator {{'1e' | indicatorName}} Materials are consistent with the progressions in the Standards i. Materials develop according to the grade-by-grade progressions in the Standards. If there is content from prior or future grades, that content is clearly identified and related to grade-level work ii. Materials give all students extensive work with grade-level problems iii. Materials relate grade level concepts explicitly to prior knowledge from earlier grades. The instructional materials reviewed for Achievement First Mathematics Grade 3 meet expectations for being consistent with the progressions in the Standards. Content from prior or future grades is identified and connected to grade-level work, and students are given extensive work with grade-level problems. Overall, the materials develop according to the grade-by-grade progressions in the Standards. Content from prior or 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: • In Unit 4, Unit Overview, Linking (p.6) states, “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.” • In Unit 9, Unit Overview, Linking (p.5) states, “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.” The instructional materials for Achievement First Mathematics Grade 3 give all students extensive work with grade-level problems. 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: • In 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. For example in 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. They are asked “Julian placed two oranges on the balance scale below with the weight shown. What is the weight of one orange?” • In 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. • In 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. facts using 9 during Independent Practice. The Exit Ticket also provides an opportunity to engage with 3.OA.7 as students solve four more problems. For example, Problem 4 states, “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.” Achievement First Mathematics Grade 3 relates 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: • In 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. The materials state, “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.” • In 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. The materials state, “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.” • In 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. The materials state, “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 {{'1f' | indicatorName}} Materials foster coherence through connections at a single grade, where appropriate and required by the Standards i. Materials include learning objectives that are visibly shaped by CCSSM cluster headings. ii. Materials include problems and activities that serve to connect two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important. The instructional materials reviewed for Achievement First Mathematics Grade 3 meet expectations for fostering coherence through connections at a single grade, where appropriate and required by the Standards. The materials include learning objectives, identified as Aims, that are visibly shaped by the CCSM cluster headings. The instructional materials utilize the acronym MWBAT to stand for “Mathematicians will be able to” when identifying the lesson objectives. Examples include: • In Unit 3, Lesson 6, the Aim states, “MWBAT add two-and three-digit numbers using add by place or a strategy that works for us (focus on add by place),” which is shaped by 3.NBT.A, “Use place value understanding and properties of operations to perform multi-digit arithmetic.” • In Unit 5, Lesson 14, the Aim states, “MWBAT identify and represent fractions on a number line between 0 and 1 by partitioning the line into equal intervals and labeling endpoints,” which is shaped by 3. NF.A, “Develop understanding of fractions as numbers.” • In Unit 8, Lesson 3, the Aim states “MWBAT identify and describe patterns in the multiplication and division table and explain why they work,” which is shaped by 3.OA.D, “Solve problems involving the four operations, and identify and explain patterns in arithmetic.” The materials include problems and activities that connect two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important. Examples include: • In 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. In the 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?” • In 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 states, “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?” • In 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 states, “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?” Overview of Gateway 2 Rigor & Mathematical Practices The instructional materials reviewed for Achievement First Mathematics Grade 3 partially meet the expectations for rigor and the Mathematical Practices. The materials meet the expectations for rigor as they develop conceptual understanding and procedural skill and fluency and balance the three aspects of rigor. The instructional materials partially meet the expectations for practice-content connections. The Standards for Mathematical Practice (MPs) are identified. The materials also prompt students to construct viable arguments and analyze the arguments of others and attend to the specialized language of mathematics. Gateway 2 Partially Meets Expectations Criterion 2.1: Rigor Rigor and Balance: Each grade's instructional materials reflect the balances in the Standards and help students meet the Standards' rigorous expectations, by helping students develop conceptual understanding, procedural skill and fluency, and application. The instructional materials reviewed for Achievement First Mathematics Grade 3 meet the expectations for rigor and balance. The materials meet the expectations for rigor as they develop conceptual understanding and procedural skill and fluency and balance the three aspects of rigor. The materials partially meet the expectations for application due to a lack of independent practice with non-routine problems. Indicator {{'2a' | indicatorName}} Attention to conceptual understanding: Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. The instructional materials for Achievement First Mathematics Grade 3 meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. The materials include problems and questions that develop conceptual understanding throughout the grade level. For example: • In 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 states, “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 then 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?” • In 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.” • In 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 states, “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. For example: • In 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 states, “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.” • In 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 states, “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 $$\frac{1}{8}$$. • In 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. The materials state, “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}} Attention to Procedural Skill and Fluency: Materials give attention throughout the year to individual standards that set an expectation of procedural skill and fluency. The instructional materials for Achievement First Mathematics Grade 3 meet expectations that they attend to those standards that set an expectation of procedural skill and fluency. The instructional 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. For example: • In Practice Workbook B, Problem 11 states, “Calculate. $$605 - 327 =$$ ; $$708 - 439 =$$ ; $$875 - 218 =$$ ; $$575 + 219 =$$ ; $$238 + 573 =$$ ; $$117 + 582 =$$ .” (3.NBT.2) • In Practice Workbook B, Problem 2 states, “$$303 - 165 =$$ .” (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. For example, Problem 6 states, “$$8×$$__$$=56$$.” (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. For example: • In Unit 5, Cumulative Review 5.2, Problem 2, students practice adding and subtracting within 1000. It states, “Solve. 909 - 690.” (written vertically). (3.NBT.2) • In Unit 5, Cumulative Review 5.4, Problem 4, students practice solving division problems within 100. It states, “Solve. $$40÷8=$$; $$16÷8=$$; $$7÷7=$$; $$72÷8=$$; $$27÷3=$$.” (3.OA.7) • In Unit 7, Cumulative Review 7.2, Problem 2, students practice solving multiplication problems. It states, “Solve. $$6×7$$; $$8×4=$$ ; $$9×3=$$ ; $$3×6=$$ ; 8 x 3 = ; 5 x 5 = .” (3.OA.7) Indicator {{'2c' | indicatorName}} Attention to Applications: Materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade The instructional materials for Achievement First Mathematics Grade 3 partially meet expectations that the materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied. The instructional materials include limited opportunities for students to independently engage in the application of routine and non-routine problems due to teacher heavily scaffolded tasks and the lack of non-routine problems. The instructional materials present opportunities for students to independently demonstrate routine application of mathematics; however, there are few opportunities for students to independently demonstrate application of grade-level mathematics in non-routine settings. Routine problems are found in the Independent Practice and Exit Tickets components of the materials. Examples of routine applications include: • In Unit 3, Lesson 10, Independent Practice, students demonstrate application of 3.OA.8 as they solve a two step word problem involving addition and subtraction. Problem 1 states, “Janie is in charge of organizing the school dance. She has$786 to spend. The DJ costs $399 and the snacks cost$157. How much money will Janie have left over for decorations?”
• In Unit 7, Lesson 1, Exit Ticket, students demonstrate application of 3.OA.3 as they use multiplication to solve word problems. Problem 3 states, “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.”
• In Unit 8, Lesson 8, Independent Practice, students demonstrate application of 3.OA.8 as they solve a two step word problem involving the four operations. Problem 2 states, “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?”

Achievement First Mathematics Grade 3 provides limited opportunities for students to engage in non-routine problems. Additionally, the non-routine problems are often heavily scaffolded for students with directed teacher questioning techniques. Non-routine problems are found within the Workshop, Math Stories, and Problem of the Day components of the materials. For example:

• In Unit 3, Lesson 11, Problem of the Day, students engage with 3.OA.8 as they solve a two-step word problem. The materials state, “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.”
• In 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. The materials state, “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.”
Indicator {{'2d' | indicatorName}}
Balance: The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the 3 aspects of rigor within the grade.

The instructional materials for Achievement First Mathematics Grade 3 meet expectations for balancing the three aspects of rigor. Overall, within the instructional materials the three aspects of rigor are not always treated together and are not always treated separately.

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

• In 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 states, “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)
• In Practice Workbook B, Problem 29, students demonstrate procedural skill and fluency related to addition and subtraction as they solve problems. It states, “Solve to find the missing numbers. $$142+\_=225$$, $$506-\_=329$$,  $$\_+344=764$$.” (3.NBT.2)
• In Unit 8, Lesson 10, Independent Practice, students apply their understanding of the four operations as they solve a word problem. Problem 7 states, “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. For example:

• In 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 9 states, “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)
• In 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. The materials state, “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)
• In 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 states, “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

Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice

The instructional materials reviewed for Achievement First Mathematics Grade 3 partially meet the expectations for practice-content connections. The Standards for Mathematical Practice (MPs) are identified. The materials also prompt students to construct viable arguments and analyze the arguments of others and attend to the specialized language of mathematics.

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The Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout each applicable grade.

The instructional materials reviewed for Achievement First Mathematics Grade 3 partially meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade-level.

There are discrepancies between the Unit and Lesson Overviews as to which MPs are areas of focus for the instruction. While the MPs are identified for each lesson, clear guidance on how the MPs are used to enrich the mathematics content is not generally provided within the lesson or lesson components. The materials do not indicate where teachers might focus student attention or make specific connections between practices and content. Throughout the materials there are some MPs that are bolded; however, there is not an explanation available for teachers to determine the reason.

The MPs are identified throughout the materials in the Unit and Lesson Overviews, and they are not treated as separate from the grade-level, mathematics content. Examples of the identification for MPs within the Unit Overview and a general connection to the learning of the unit include:

• Unit 3, Unit Overview, MP1, “Students will tackle word problems in this unit involving time. They will have to work independently to solve these problems, determining what is being asked of them, using various representations, and calculating strategically. The problems will test their thinking and encourage them to be creative with their problem solving.”
• Unit 7, Unit Overview, MP5, “Scholars may use hundreds charts to discern patterns in products with given factors.”
• Unit 9, Unit Overview, MP6, “Scholars attend to precision when drawing polygons and quadrilaterals (ensuring straight lines, right angles, parallel lines, equal length sides).”
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Materials carefully attend to the full meaning of each practice standard

The instructional materials reviewed for Achievement First Mathematics Grade 3 partially meet expectations for carefully attending to the full meaning of each practice standard. The Mathematical Practices (MPs) are represented in each of the nine units in the curriculum and labeled in each lesson. Math Practices are represented throughout the year and not limited to specific units or lessons. The materials do not attend to the full meaning of MPs 1 and 5.

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

• In Unit 2, Lesson 16, Exit Ticket, students analyze information presented in a word problem and choose an appropriate addition strategy to solve it. Problem 2 states, “Sue is in charge of the budget for her company. This month the company plans to spend $768,000 on employee salaries and$5,098 on office supplies. About how much will the company spend on these two things this month?”
• In Unit 3, Lesson 15, Independent Practice, students make sense of information to solve a word problem involving elapsed time. Problem 9 states, “Ronald wants to spend at least two hours and 40 minutes practicing the trumpet each day. If he plans to begin at 2:45pm and end at 3:30, will he have practiced for enough time? Explain your thinking.”
• In Unit 6, Lesson 12, Independent Practice, students solve a problem requiring them to access relevant knowledge and work through a task with multiple entry points. The materials state, “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!”

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

• In Unit 2, Lesson 3, Exit Ticket, students follow step-by-step directions to create a bar graph from a table of information. Problem 1 states, “Create a bar graph below shows the students’ favorite ice cream flavors. Remember to label all the parts of your graph.”
• In Unit 6, Lesson 2, students use provided broken rulers to measure lengths of given objects in the materials. Independent Practice, Problem 4 states, “Find the length of each object to the nearest $$\frac{1}{4}$$ of an inch. The doll is __ inches long.”
• In Unit 6, Cumulative Review 6.3, students use a provided grid to create a shape that matches a given surface area. Problem 1 states, “Use the grid below to make a shape that has an area of 24 square units.”

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

• MP2: In Unit 4, Lesson 5, Day 2, Exit Ticket, students reason abstractly to solve problems based on units of measurement. Problem 1 states, “Tyreese is weighing objects using a balance scale. On one side he places a 400 gram weight, and on the other side he places a statue that weighs 176 grams. How many more grams will he need to place on the statue side to balance the scale?”
• MP4: In Unit 7, Lesson 11, Exit Ticket, students use a grid model to represent relationships between quantities. Problem 1 states, “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.”
• MP6: In Unit 6, Lesson 1, Exit Ticket, students attend to precision as they make their own ruler and measure objects to the nearest quarter inch. Problem 3 states, “Explain how you used your ruler to measure the line above.”
• MP7: In Unit 8, Lesson 5, Independent Practice, students extend a pattern based on a provided pattern. Problem 2 states, “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? __,___,___.”
• MP8: In Unit 3, Lesson 5, Independent Practice students add two and three digit numbers using expanded notation. Problem 3 states, “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.”
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Emphasis on Mathematical Reasoning: Materials support the Standards' emphasis on mathematical reasoning by:
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Materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards.

The instructional materials reviewed for Achievement First Mathematics Grade 3 meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

The instructional materials provide opportunities for students to engage in constructing viable arguments and analyzing the work and understanding of others. Examples of opportunities for students to construct viable arguments and/or critique the reasoning of others include:

• In Unit 3, Lesson 2, Exit Ticket students critique the reasoning of others based on their knowledge of estimation and rounding. Problem 3 states, “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.”
• In 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. The materials state, “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.”
• In Unit 9, Lesson 4, Exit Ticket students construct a viable argument based on their understanding of shapes. Problem 3 states, “In the grid below, draw a rhombus that is also a rectangle. Explain how your shape fits in both categories.”
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Materials assist teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards.

The instructional materials reviewed for Achievement First Mathematics Grade 3 meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. There are opportunities within the teacher materials that guide teachers in assisting students to construct viable arguments and analyze the arguments of others, through the use of questioning.

Examples of the instructional materials assisting teachers in engaging students to construct viable arguments and analyze the arguments of others include:

• In Unit 3, Lesson 2, Try One More, students are asked to determine if an answer is reasonable. The materials state, “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?”
• In Unit 4, Lesson 4, Workshop, Problem 7 states, “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.” The teacher's guidance includes, “Is the equation 9 divided by 3 correct? Why or why not? What equation represents the story?
• In Unit 5, Lesson 24, Problem of the Day, students analyze the reasoning of others and use models to explain their reasoning. The materials state, “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.” The teacher is guided to support students as they represent the problem and analyze the reasoning of others. Teacher guidance includes, “How did you represent this problem? First, I represented $$\frac{3}{3}$$ with a visual model by drawing one whole that I partitioned into three parts. I labeled each part with $$\frac{1}{3}$$ because that is the unit fraction since there are three parts in one whole. Then, I shaded in three parts because the numerator is 3 which means that there are 3 parts being referred to. Add to VA. What do you notice about $$\frac{3}{3}$$? What is it equivalent to? $$\frac{3}{3}$$ is equivalent to 1 whole. Hmm. The numerator and the denominator are the same. Why do we have 1 whole when the numerator and the denominator are the same? The total number of parts and the number of parts being referred to are the same. That makes one whole, and we can see it here because the whole shape is shaded in.”
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Materials explicitly attend to the specialized language of mathematics.

The instructional materials reviewed for Achievement First Mathematics Grade 3 meet  expectations that materials explicitly attend to the specialized language of mathematics.

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

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

• In Unit 1, Lesson 6 provides guidance for teachers in introducing the identified vocabulary word area. State the Aim states, “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.” Students then attempt to fill a shape using pattern blocks to determine which one takes up more space. The materials state, “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.”
• In Unit 4, Lesson 1 provides guidance for teachers in introducing mathematical terminology related to measurement. State The Aim, “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. Students pick up a paperclip in their hand; add benchmark to VA. Place paperclip on the scale. 1 gram The second is called a kilogram and weighs about as much as a textbook. Students lift the textbook; add benchmark to VA. Place the text book on the scale. ____ grams. Note: should be close to 1000 grams. ____ grams is ABOUT 1,000 grams. 1,000 grams is the same as 1 kilogram so let’s say that a textbook is ABOUT 1 kilogram. A textbook can be our benchmark for kilograms. Add to VA.”
• In Unit 9, Lesson 1 identifies new vocabulary used in the lesson and provides specific guidance for teachers in introducing the terminology, including counterexamples. State the Aim states, “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. Reveal KP on VA. Take a look at the top of your page. The first shape is not a polygon because it has an open-spaced 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.”

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

• In Unit 4, Lesson 3, Independent Practice, accurate terminology is used as students solve problems related to volume. Problem 2 states, “Mrs. Goldstein pours 3 juice boxes into a bowl to make punch. Each juice box holds 236 milliliters. How much juice does Mrs. Goldstein pour into the bowl?”
• In Unit 7, Lesson 6, Exit Ticket, accurate terminology is used as students are expected to draw an array and write an expression. Problem 1 states, “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.”
• In Unit 8, Lesson 3, Pose the Problem, accurate terminology is used as students work with partners to solve a problem. The materials state, “Khaleel says that whenever you add an odd number plus an odd number, you should get an odd sum because both addends are odd. Is Khaleel correct? Work with your partner to explain why; prove your answer using a visual model or the addition table.”

Usability

This material was not reviewed for Gateway Three because it did not meet expectations for Gateways One and Two
Not Rated

Criterion 3.1: Use & Design

Use and design facilitate student learning: Materials are well designed and take into account effective lesson structure and pacing.
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The underlying design of the materials distinguishes between problems and exercises. In essence, the difference is that in solving problems, students learn new mathematics, whereas in working exercises, students apply what they have already learned to build mastery. Each problem or exercise has a purpose.
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Design of assignments is not haphazard: exercises are given in intentional sequences.
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There is variety in what students are asked to produce. For example, students are asked to produce answers and solutions, but also, in a grade-appropriate way, arguments and explanations, diagrams, mathematical models, etc.
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Manipulatives are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.
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The visual design (whether in print or online) is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.

Criterion 3.2: Teacher Planning

Teacher Planning and Learning for Success with CCSS: Materials support teacher learning and understanding of the Standards.
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Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.
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Materials contain a teacher's edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning.
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Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that contains full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge of the subject, as necessary.
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Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that explains the role of the specific grade-level mathematics in the context of the overall mathematics curriculum for kindergarten through grade twelve.
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Materials provide a list of lessons in the teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials), cross-referencing the standards covered and providing an estimated instructional time for each lesson, chapter and unit (i.e., pacing guide).
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Materials contain strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.
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Materials contain explanations of the instructional approaches of the program and identification of the research-based strategies.

Criterion 3.3: Assessment

Assessment: Materials offer teachers resources and tools to collect ongoing data about student progress on the Standards.
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Materials provide strategies for gathering information about students' prior knowledge within and across grade levels.
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Materials provide strategies for teachers to identify and address common student errors and misconceptions.
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Materials provide opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.
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Materials offer ongoing formative and summative assessments:
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Assessments clearly denote which standards are being emphasized.
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Assessments include aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.
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Materials encourage students to monitor their own progress.

Criterion 3.4: Differentiation

Differentiated instruction: Materials support teachers in differentiating instruction for diverse learners within and across grades.
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Materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.
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Materials provide teachers with strategies for meeting the needs of a range of learners.
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Materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.
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Materials suggest support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics (e.g., modifying vocabulary words within word problems).
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Materials provide opportunities for advanced students to investigate mathematics content at greater depth.
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Materials provide a balanced portrayal of various demographic and personal characteristics.
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Materials provide opportunities for teachers to use a variety of grouping strategies.
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Materials encourage teachers to draw upon home language and culture to facilitate learning.

Criterion 3.5: Technology

Effective technology use: Materials support effective use of technology to enhance student learning. Digital materials are accessible and available in multiple platforms.
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Digital materials (either included as supplementary to a textbook or as part of a digital curriculum) are web-based and compatible with multiple internet browsers (e.g., Internet Explorer, Firefox, Google Chrome, etc.). In addition, materials are "platform neutral" (i.e., are compatible with multiple operating systems such as Windows and Apple and are not proprietary to any single platform) and allow the use of tablets and mobile devices.
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Materials include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology.
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Materials can be easily customized for individual learners. i. Digital materials include opportunities for teachers to personalize learning for all students, using adaptive or other technological innovations. ii. Materials can be easily customized for local use. For example, materials may provide a range of lessons to draw from on a topic.
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Materials include or reference technology that provides opportunities for teachers and/or students to collaborate with each other (e.g. websites, discussion groups, webinars, etc.).
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Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices.

Report Overview

Summary of Alignment & Usability for Achievement First Mathematics | Math

Math K-2

The instructional materials reviewed for Achievement First Mathematics Grades K-2 partially meet the expectations for alignment. At all grade levels, the assessments are focused on grade-level standards and devote at least 65% of class time to major clusters of the grade. The materials make connections between major and supporting work of the grade and between prior and future work. The materials do not provide all students with extensive work with grade-level problems, nor do the materials meet the full intent of all grade-level standards. Grades K-2 partially meet expectations for Gateway 2, rigor and mathematical practices. The lessons include conceptual understanding and procedural skill and fluency and balance the aspects of rigor. The Standards for Mathematical Practice (MPs) are identified and partially used to enrich the learning; however, the full intent of all MPs is not met.

Kindergarten
Alignment
Partially Meets Expectations
Not Rated
1st Grade
Alignment
Partially Meets Expectations
Not Rated
2nd Grade
Alignment
Partially Meets Expectations
Not Rated

Math 3-5

The instructional materials reviewed for Achievement First Mathematics Grades 3-5 partially meet the expectations for alignment. At all grade levels, the assessments are focused on grade-level standards and devote at least 65% of class time to major clusters of the grade. The materials make connections between major and supporting work of the grade and between prior and future work. The materials do not provide all students with extensive work with grade-level problems, nor do the materials meet the full intent of all grade-level standards. Grades 3-5 partially meet expectations for Gateway 2, rigor and mathematical practices. The lessons include conceptual understanding and procedural skill and fluency and balance the aspects of rigor. The Standards for Mathematical Practice (MPs) are identified and partially used to enrich the learning; however, the full intent of all MPs is not met.

3rd Grade
Alignment
Partially Meets Expectations
Not Rated
4th Grade
Alignment
Partially Meets Expectations
Not Rated
5th Grade
Alignment
Partially Meets Expectations
Not Rated

Math 6-8

The instructional materials reviewed for Achievement First Mathematics Grades 6-8 partially meet the expectations for alignment. At all grade levels, the assessments are focused on grade-level standards and Grades 7 and 8 devote an appropriate amount of class time to major clusters of the grade. Grade 6 does not meet the expectation for focus, as it devotes less than 58% of class time to major clusters of the grade. The materials make connections between major and supporting work of the grade and between prior and future work. The materials provide all students with extensive work with grade-level problems, and meet the full intent of all grade-level standards. Grades 6-8 partially meet expectations for Gateway 2, rigor and mathematical practices. The lessons include conceptual understanding and procedural skill and fluency and balance the aspects of rigor. The Standards for Mathematical Practice (MPs) are partially identified and the materials partially meet the full intent of all of the MPs. The MPs are used to enrich the learning.

6th Grade
Alignment
Partially Meets Expectations
Not Rated
7th Grade
Alignment
Partially Meets Expectations
Not Rated
8th Grade
Alignment
Partially Meets Expectations
Not Rated

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Overall Summary

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Alignment
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Usability
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