## Achievement First Mathematics

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## Report for 4th Grade

### Overall Summary

The materials reviewed for Achievement First Mathematics Grade 4 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.

##### 4th Grade
###### Alignment
Meets Expectations
###### Usability
Does Not Meet Expectations

### Focus & Coherence

The materials reviewed for Achievement First Mathematics Grade 4 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 4 meet expectations for focus as they assess grade-level content and provide all students extensive work with grade-level problems to meet the full intent of grade-level standards.

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

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

The materials reviewed for Achievement First Mathematics Grade 4 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 1, Post-Assessment, Item 5, “The cost of buying a movie is 4 times the cost of renting a movie. It costs $30 to buy a movie. Write two equations that can be used to determine the cost, r, or renting a movie.” (4.OA.1) • Unit 4, Post-Assessment, Item 16, “Divide. 7,285 ÷ 4. Answer choices include: A. 1,801, B. 1,801 R1, C. 1,821, D. 1,821 R1.” (4.OA.3) • Unit 9, Post-Assessment, Item 15, “Explain why a square is also a rectangle and a rhombus.” (4.G.2 ) • Unit 10, Post-Assessment, Item 13, “A circular pizza was cut into 5 equal slices from the vertex at the center of the pizza. 3 of the slices of pizza get eaten. What is the measurement of the angle formed at the vertex of the slices that are left?” (4.MD.5) Reviewers noted that in the Achievement First Mathematics Grade 4 materials, there was not a Unit 2 Overview therefore an assessment was not available to be reviewed. Examples of above-grade-level assessments or assessment items which can be omitted or modified: • Unit 6, Post-Assessment, Item 2, “For each of the following sums, decide which ones are equal to \frac{22}{17}. For the sums that are equal to \frac{22}{17}, circle YES. For the sums that are not equal to \frac{22}{17}, circle NO. ...; YES or NO \frac{1}{17}+\frac{1}{17}+\frac{9}{17}+\frac{3}{17}.” (4.NF.2, expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, 100.) • Unit 6, Post-Assessment, Item 14, “Each day Milo reads 18of his new book. Which number sentence best represents the fractions of his book that Milo has read after 7 days? Answer choices include: A. \frac{1}{8}×\frac{1}{7}=\frac{1}{56}, B. \frac{1}{8}×\frac{1}{7}=\frac{7}{8}, C.$$\frac{1}{8}×7=\frac{1}{56}$$, D.$$\frac{1}{8}×7=\frac{7}{8}$$.” (4.NF.4, students are expected to “solve word problems involving multiplication of a fraction by a whole number.” Answer choices A and B do not meet the criteria outlined by the standard.) Achievement First Mathematics Grade 4 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. ##### Indicator {{'1b' | indicatorName}} Materials give all students extensive work with grade-level problems to meet the full intent of grade-level standards. The materials reviewed for Achievement First Mathematics Grade 4 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 3, Lessons 2 through 9, students fluently add and subtract multi-digit whole numbers using the standard algorithm, as they solve more than 100 problems in independent practice opportunities (Independent Practice, Exit Ticket, and Practice Workbook) and explain their use of the algorithm (4.NBT.5). Lesson 6, Independent Practice, Problem 5, “$$50,019-12,877$$” and Problem 6, “In the last problem, what place values did you need to regroup and how did you do it? Explain on the lines below.” • Unit 5, Lessons 1 through 10, students solve multi-step word problems posed with whole numbers and having whole number answers using the four operations, including problems in which the remainders must be interpreted, and represent these problems using equations with a letter standard for the unknown quantity (4.OA.3). There are 83 Independent Practice problems and 15 Exit Tickets that require students to solve multi-step word problems. Lesson 6, Independent Practice, Problem 4, “Mia represented the above problem like this: (437 pies x 9 wards) x 14 = Total pie Pieces,” and asks students, “Is this representation reasonable? Tell why or why not on the lines below.” While there are 98 opportunities for students to engage with this standard, there are less than 10 opportunities for students to represent problems with a letter standard for the unknown quantity. Unit 10, Lesson 4, students engage with 4.MD.6, measure angles in whole number degrees using a protractor, as they solve problems requiring them to use a protractor to accurately identify the angle measurement. Exit Ticket 1, “For numbers 1-2, use your protractor to create an angle of the given size. Be sure to check if your drawn angle matches the type of angle indicated by the measurement. 1. 67\degree.” #### Criterion 1.2: Coherence Each grade’s materials are coherent and consistent with the Standards. The materials reviewed for Achievement First Mathematics Grade 4 meet expectations for coherence. The materials: address the major clusters of the grade, have supporting content connected to major work, make connections between clusters and domains, and have content from prior and future grades connected to grade-level work. ##### Indicator {{'1c' | indicatorName}} When implemented as designed, the majority of the materials address the major clusters of each grade. The materials reviewed for Achievement First Mathematics Grade 4 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 114 out of 125, which is approximately 91%. • The number of days devoted to major work (including assessments and supporting work connected to the major work) is 121 out of 132, which is approximately 92%. • The instructional minutes were calculated by taking the number of minutes devoted to the major work of the grade (10,425) and dividing it by the total number of instructional minutes (11,475), 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. ##### Indicator {{'1d' | indicatorName}} Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade. The materials reviewed for Achievement First Mathematics Grade 4 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 4, Lesson 25, Independent Practice, Problem 2, “Leonard bought 4 liters of orange juice. How many milliliters of juice does he have?” This problem connects the major work of 4.NBT.5, multiply a whole number of up to four digits by a one-digit whole number, to the supporting work of 4.MD.A, as students solve a problem involving measurement conversion from a larger unit to a smaller unit. • Unit 8, Lesson 7, Independent Practice, Problem 2, “Julio starts school at 7:45 am and finishes school at 3:30 pm. He has 25 minutes of recess, 32 minutes of lunch, and he has a 46 minute free period in the afternoon. The rest of the time he is in classes. How many hours and minutes does Julio spend in class?” This problem connects the major work standard 4.OA.3 and the supporting work standard 4.MD.2, as students solve multi-step word problems involving intervals of time. • Unit 10, Lesson 2, Independent Practice, Problem 2, “Megan has a very large round table. In order for her to seat her guests, she divided it into 10 equal sections. What is the angle measure of each section of the table?” This problem connects the major work of 4.NF.3 and the supporting work of 4.MD.5, as students find the measurement of angles when given a fraction of a circle. ##### Indicator {{'1e' | indicatorName}} Materials include problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade. The materials reviewed for Achievement First Mathematics Grade 4 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 1, Assessment, students connect the work of 4.OA.B, gain familiarity with factors and multiples, to 4.OA.C, generate and analyze patterns, as they complete a pattern involving multiples. Item 1, “Alfonzo applies numbers on the back of football jerseys. Below are the first five numbers he applies. If the pattern continues, what are the next three numbers he will apply? 9, 18, 27, 36, 45, ___,___,____ a. 54, 63, 72 b. 54, 63, 71 c. 63, 64, 72 d. 63, 72, 8.” • Unit 5, Lesson 9, students connect the work of 4.OA.A, use the four operations with whole numbers, to solve problems to 4.NBT.B, use place value understanding and properties of operations to perform multi-digit arithmetic, as students use estimation strategies and a visual model to solve multi-step problems. Exit Ticket, Problem 1, “Katie and her sister are saving up money to build a tree house. Each month they have saved 46 each from their allowance. In the last month, they each did some extra jobs in order to get the total amount they needed for the house. They spend 13 months saving and an extra month doing more jobs. If the cost of the tree house was 1299, how much money did they earn in the last extra month? Represent, estimate and solve.” • Unit 7, Assessment, students connect the work of 4.NF.A, extend understanding of fraction equivalence, and ordering to 4.NF.C, understand decimal notation for fractions, and compare decimal fractions, as students locate and label points on a number line. Item 19, “Locate and label the following points on the number line below: \frac{130}{10},13.21, 13\frac{12}{100}.” ##### Indicator {{'1f' | indicatorName}} Content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades. The materials reviewed for Achievement First Mathematics Grade 4 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 3, Overview, Identify Desired Results: Identify the Standards, identifies 3.NBT.2 under Previous Grade Level Standards/Previously Taught and Related Standards for 4.NBT.5. In Enduring Understandings - What it Looks Like in This Unit, a connection is made between the addition and subtraction work to place value skills in prior grades. “In previous grades, students used place value blocks and pictures of place value blocks to add and subtract numbers. Place value relationships help them regroup. When they need to take away more than they have of a certain place value, they regroup one of a greater place value to ten of that place value.” • Unit 6, Unit Overview, Identify Desired results: Identify the Standards identifies 4.NF.1 as being addressed in this unit and 3.NF.1 as Previous Grade Level Standards/ Previously Taught & Related Standards connections. In Identify the Narrative, a description is provided, “In third grade they recognized equivalent fractions using visual models and number lines in ‘special cases’ such as \frac{2}{4}=\frac{1}{2} and \frac{1}{3}=\frac{2}{6}. In fourth grade, they use visual models of equivalent fractions to understand how to use the identity property to find equivalent fractions.” 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 3, Unit Overview, Linking, “Later in the year students will add and subtract mixed units of measurement which will again call upon regrouping concepts -- in this case, from one unit of measurement to another.” An additional reference is made to fifth grade, “When students move to fifth grade, they will continue to solve multi-step word problems with all four operations, so they will be relying on their abilities to add and subtract with the standard algorithm.” • Unit 6, Unit Overview, Identify the Narrative, refers to prior work students engaged in with fractions. “The fourth grade unit on fractions combines students’ prior knowledge on fractions, the meaning of operations, logical reasoning and new learning experiences to elaborate their understanding of fractions and allow them to operate with fractions.” • Unit 10, Unit Overview, Linking, “Angle measurements will not come up as a formal part of the math curriculum again until seventh grade. Although there is a large gap in time between this unit and seventh grade, the seventh grade standards rely heavily on students’ skill and knowledge from this unit in fourth grade.” ##### Indicator {{'1g' | indicatorName}} In order to foster coherence between grades, materials can be completed within a regular school year with little to no modification. The instructional materials reviewed for Achievement First Mathematics Grade 4 foster coherence between grades and can be completed within a regular school year with little to no modification. The Guide to Implementing AF, Grade 4 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 135 days. One day is provided for each lesson and one day is allotted for each unit assessment. • There are 10 units with 131 lessons in total. • The Guide to Implementing Achievement First Mathematics Grade 4 identifies lessons as either R (remediation), E (enrichment), or O (on grade level). There is one lesson identified as R (remediation), zero lessons identified as E (enrichment), and 130 lessons identified as O (on grade level). • There are 4 days for Post-Assessments. According to The Guide to Implementing Achievement First Mathematics Grade 4, 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) ###### Overview of Gateway 2 ### Rigor & the Mathematical Practices The materials reviewed for Achievement First Mathematics Grade 4 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 4 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 for Achievement First Mathematics Grade 4 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. The materials include problems and questions that develop conceptual understanding throughout the grade level. Examples include: • Unit 4, Lesson 8, students develop conceptual understanding of 4.NBT.5, as they use place value blocks to help them solve multi-digit multiplication problems. In Problem of the Day, “Problem: A video store display shelf has DVDs stacked in 3 rows. There are 246 videos in each row. How many videos can the shelves hold? TT: How can we represent this problem with an equation? We can write 246 videos x 3 rows = K total videos. Add to VA. Why does that work? It works because this is a problem about equal groups. In this problem, we have 3 groups—the rows—with 246 DVDs in each. We need to figure out the total number of DVDs. We can do this with multiplication. Today we’re going to solve 2, 3, and 4-digit multiplication with place value blocks. Work with your partner to solve this equation with place value blocks.” • Unit 6, Lesson 6, students develop conceptual understanding of 4.NF.1, as they use tape diagrams and number lines to find equivalent fractions. In Workshop, Problem 2, “Markette is using a number line to figure out how many sixths are equal to \frac{2}{3}. She tells her partner, “We should partition each interval on our number line into 3 new parts, because 3+3 is 6. Is Markette’s strategy reasonable? Explain on the lines below. You may use pictures, number sentences, or number lines to help you.” • Unit 7, Lesson 6, students develop conceptual understanding of 4.NF.7, compare two decimals to the hundredths by reasoning about their size. In the Workshop, Problem 1, students use visual models and number lines to support their reasoning about comparisons. “For each problem, shade each decimal amount on the given grids and plot them on the number line. Then use those models to compare the decimals using <, > or =0.2__$$0.19$$.” The materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Examples include: • Unit 2, Lesson 4, students demonstrate conceptual understanding of 4.NBT.2, as they use a provided place value chart and their knowledge of place value to determine the reasonableness of a provided answer. Independent Practice, Problem 2, “Kate used the place value chart to write the number below in standard form. Is Kate’s work correct? Explain why or why not on the lines below.” • Unit 6, Lesson 1, students demonstrate conceptual understanding of 4.NF.3, as they draw a visual model and write an equation to solve a problem. Independent Practice, Problem 1, “Terrell is keeping track of his running for the week. Draw a visual model and write an addition equation to model Terrell’s running plan. How far will he have to run at the end of the week?” • Unit 10, Lesson 3, students demonstrate conceptual understanding of 4.MD.5, as they use manipulatives to find the measure of a given angle. In the Exit Ticket, Problem 3, “Using pattern blocks, how can you find the measure of the angle below? Use pictures, words and numbers to show how you found your answer.” ##### 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 4 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 eight Practice Workbooks in Achievement First Mathematics, Grade 3. One workbook, C, contains resources to support the procedural skill and fluency standard 4:NBT.4: Fluently add and subtract multi-digit whole numbers using the standard algorithm. In the Guide To Implementing Achievement First Mathematics Grade 4, teachers are provided with guidance for which workbook to use based on the unit of instruction. Examples include: • Practice Workbook C, Problem 1, students solve subtraction problems. “Find the difference. 51,348 and 22,122. Use the standard algorithm to solve.” (4.NBT.4) • Practice Workbook C, Problem 6, students solve subtraction problems. “Use a strategy that makes sense to you to solve. 59,637 – 34,721 = .” (4.NBT.4) • Practice Workbook C, Problem 9, students practice subtraction. “$$56432 - 33224 =$$ _____.“ (4.NBT.4) 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 Review occurs every Friday for 20 minutes. Examples include: • Unit 4, Cumulative Review 4.5, Problem 4, students solve subtraction problems. “Find the difference. Show your work. 6,241 - 1,366 = _____.” (4.NBT.4) • Unit 4, Cumulative Review 4.5, Problem 2, students solve addition and subtraction problems using the standard algorithm. “Camden Yard sold 5,864 tickets and 2,549 students tickets to last Friday's Baltimore Orioles baseball game. How many total tickets were sold for last Friday’s game?”(4.NBT.4) • Unit 6, Cumulative Review 6.4, Problem 5, students solve a subtraction problem. “Find the difference. 2,301 - 1,976 = ___” (4.NBT.4) ##### Indicator {{'2c' | indicatorName}} Materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. The materials reviewed for Achievement First Mathematics Grade 4 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 2, Guide to Implementing AF, Math Stories, October, students engage with 4.OA.3 as they solve a multi-step word problem posed with whole numbers and having whole-number answers using the four operations in a non-routine format. Sample problem 13, “Jamal and Sarah are playing a game with 5 counters. On each person’s turn they can take either 1or 2 counters from the pile. The player with the last turn loses. If Jamal starts the game and takes 1 counter away, what are two possible outcomes for the game?” • Unit 7, Lesson 11, Problem of the Day, students engage in a routine problem with 4.NF.5 as they apply their understanding of fractions. "Victoria finds a multicolored quilt that exactly matches the colors in her bedroom. Victoria is so excited that she phones her mom to tell her about the quilt. This is what Victoria tells her mom: The quilt is a rectangle with one hundred squares, \frac{36}{100} of the quilt is made of red and yellow squares \frac{5}{10} of the quilt is blue squares, \frac{14}{100} of the quilt is green squares. Victoria's mom is very excited about the new quilt. She asks Victoria what total fraction of the quilt is made of blue and green squares. What fraction should Victoria tell her mom is the total fraction of the quilt made of blue and green squares? Show all your mathematical thinking.” • Unit 9, Guide to Implementing AF, Math Stories, May, students engage with 4.NF.1 as they apply the use of a visual fraction model to generate equivalent fractions and solve a non-routine problem. "Akilah draws two rectangles of the same size, and divides them into a different number of total parts. In the first rectangle, she colors in 4 parts. In the second rectangle, she colors in 5 parts. The colored areas are equal. What fractions could she have divided her rectangles up into?” • Unit 10, Guide to Implementing AF, Math Stories, June, students engage with 4.MD.2 as they use the four operations to solve a routine word problem involving money. Sample Problem 1, “There are 3,418 students at Brookside Elementary and 2,192 students at La PLaya Elementary. All of the students are going on a field trip to the Natural history museum, where tickets for children are$3 each. The schools have a budget of 20,000 to spend on field trips. How much money will they have left over?” Materials provide opportunities for students to independently demonstrate routine and non-routine applications of the mathematics throughout the grade level. Examples include: • Unit 1, Lesson 11, Independent Practice, students engage with 4.OA.2 as they solve a routine word problem involving multiplicative comparisons. Problem 2, “Kenny is 56 years old. His sister is 7 years old. How many times younger is Kenny’s sister than him?” • Unit 2, Cumulative Review 2.2, Problem 4, students engage in non-routine application of 4.OA.4 as they use their knowledge of factor pairs to solve a problem in more than one way. "Yvette is making bracelets for her friends. Each bracelet will have an equal number of charms. She has 24 charms and she wants each bracelet to have at least 2 charms, but no more than 8 charms. Part A: Which is NOT a way that Yvette can make her bracelets? a) 8 bracelets with 3 charms on each bracelet. b) 6 bracelets with 4 charms on each bracelet. c) 4 bracelets with 6 charms on each bracelet. d) 4 bracelets with 8 charms on each bracelet. Part B: Are there any other ways that work for Yvette to make her bracelets? Show your work below.” • Unit 6, Lesson 22, Independent Practice, students engage with 4.NF.3 as they solve routine word problems involving addition and subtraction of fractions. Problem 1, “A cabinet has shelves that are 11\frac{1}{4} inches tall. Mike stacked a speaker that is 4\frac{3}{4} inches tall on top of a DVD player that is 3\frac{2}{4} inches tall. How much space is left between the objects and the top of the shelf?” ##### 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 4 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 of factors by determining whether there are additional factors for a number within 100. Problem 7, “Marco and Desiree made 56 cookies for a bake sale. They will put an equal amount of cookies into bags. Marco and Desiree want to put more than 2 cookies but fewer than 10 cookies into each bag. Desiree says that they can only put 7 cookies into 8 bags or 8 cookies into 7 bags. Marco thinks there are more ways to put an equal number of cookies into bags. Who is right? Why are they right?” (4.OA.4) • Practice Workbook D, students develop procedural skill and fluency as they multiply whole numbers. Problem 14, “Calculate the product of 64 × 35.” (4.NBT.5) • Unit 6, Lesson 23, Exit Ticket, students apply their understanding of fraction multiplication as they solve word problems. “Edwin uses \frac{3}{4} of a teaspoon of baking powder for each batch of muffins he makes. He needs to make 3 batches for his Cub Scout meeting and 4 batches for his study group. How many teaspoons of baking powder will Edwin need?” (4.NF.4) 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 7, Lesson 11, Independent Practice, students apply their conceptual understanding of adding decimals to solve a real-world problem. Part 1, “Mrs. Evans, the physical education teacher, is forming relay teams to help raise money for cancer research. There must be two students on each relay team. To determine the teams, Ms. Evans uses the students’ practice times from the last physical education class. Ms. Evans wants the teams to be as evenly matched as possible so they have a fair chance to win the race. What would the best combination of students be for each of the relay teams? Show all your mathematical thinking.” (4.NF.5, 4.NF.7) • Unit 5, Lesson 7, Exit Ticket, students apply their conceptual understanding of multiplication to solve a two-step word problem using tape diagrams and equations. Problem 1, “Draw a tape diagram to model the following equation. Create a word problem. Solve for the value of the variable. (A × 2) + 4,892 = 6,392.” (4.OA.3) • Unit 8, Lesson 4, Independent Practices, students apply their conceptual understanding of place value to solve a problem involving the value of coins. Problem 6, “Which is more, 68 dimes or 679 pennies? Prove with a place value chart and then explain on the lines below.” (4.MD.2) #### 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 4 meet expectations for practice-content connections. The materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs). ##### Indicator {{'2e' | indicatorName}} Materials support the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. The materials reviewed for Achievement First Mathematics Grade 4 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. The Standards for Mathematical Practice are identified and incorporated within mathematics content throughout the grade level. The Mathematical Practices are listed in the Unit Overviews as well at the beginning of each lesson. There is intentional development of MP1 to meet its full intent in connection to grade-level content. Examples include: • Unit 3, Lesson 3, Independent Work Question 8, students engage with MP1 as they make sense of a problem involving multi-digit addition of whole numbers. “Milos’s family is keeping track of their steps each week. So far his sister has walked 15,678 steps, his father has walked 123,098 steps, and his mother has walked 435,607 steps. If Milos has walked twice as many steps as his sister, how many total steps has the family walked altogether?” • The Unit 6 Overview outlines the intentional development of MP1. “Students apply the meaning of area and perimeter in order to interpret word problems in which area and perimeter are implicitly stated. Students practice division and multiplication strategies in the context of word problems. Calculations can be tedious and long, and students must continue to persevere through many steps in order to solve problems. Students employ a variety of problem solving skills in order to solve conversion problems. They must use ratios and calculations, but also determine which operation to use to convert.” • Unit 5, Lesson 4, “How will embedded MPs support and deepen the learning?”, teachers are provided with explanations of connections between content and practices, “Students continue to practice SMP 1 as they plan to represent and solve multi-step problems by identifying all of the values and relationships between the values in the word problem, paying close attention to the questions asked in the word problem.” • Unit 8, Lesson 6, Workshop Problem 3, students engage with MP1 as they work through multi-step word problems that require them to apply the concept of elapsed time. “Ms. Johnson has 20 minute meetings with students during the school day. She has a five-minute break between meetings. She does not have a break before her first meeting or after her last meeting. If she starts meetings at 7:45am, and has 4 meetings scheduled, what time will she be finished?” There is intentional development of MP2 to meet its full intent in connection to grade-level content. Examples include: • The Unit 2 Overview outlines the intentional development of MP2. “When students start to work with numbers in greater place values such as the hundred thousands and ten thousands, they use abstract reasoning to understand quantitative meanings. Since there are no place value blocks big enough to show a hundred thousand and they often can’t draw quantities this large, they must apply patterns of the place value system to logically understand the magnitude of larger numbers. SMP 2 is developed in lessons 2, 3, 5, & 9-11.” • Unit 2, Lesson 3, Independent Practice Question 5, students engage with MP2 as they read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. “Lee and Gary visited South Korea. They exchanged their dollars for South Korean bills. Lee received 5 thousand dollar bills, 6 hundred dollar bills, 9 ten dollar bills, and 5 one dollar bills. What was Lee’s total amount of money in standard, written, and expanded form?” • The Unit 6 Overview describes development of MP2. “The true understanding of the fundamental meaning of a fraction is abstract reasoning. Students must learn to take an abstract representation of two numbers (a numerator and a denominator) and give it a new meaning referring to a part of a whole – a value less than one. Through visual models and many examples, students should begin to understand that a fraction is a quantity in itself that has a position on a number line. This is extremely abstract quantitative reasoning. Students reason abstractly when they compare fractions of different wholes. The idea that a fraction can have a different value based on the size of its whole, but the same whole is implied when it is not specified, is a challenging abstract concept for students to grasp.” ##### 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 4 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. There is intentional development of MP3 to meet its full intent in connection to grade-level content. Examples include: • Unit 2, Lesson 16, Workshop, students critique the reasoning of others and construct a viable argument as they evaluate the estimation strategies used by other students to determine who is correct. Problem 2, “Patricia said the best way to estimate the solution to 8,421 - 462 is to round each number to the nearest hundred. Matthew said the best way to estimate is to round each number to the nearest thousand. Who is correct? Explain your answer.” • Unit 7, Lesson 4, Exit Ticket, Problem 3, “Patrice is measuring the rainfall for December. On Monday there was 0.09 of an inch of rainfall. On Tuesday there was 0.9 of an inch of rainfall. Patrice tells his sister that it rained the same amount on Monday and Tuesday. Tell whether or not Patrice is correct on the lines below.” • Unit 7, Lesson 8, Discussion, teachers are provided with guidance and questions to engage students in critiquing the reasoning of others. “Last year I had a scholar who told me that when you compare decimals it's like the opposite of comparing whole numbers. What do you think they meant by that? How are comparing whole numbers and comparing decimals similar?” • Implementation Guide, Unit 7, Math Stories, February/March, students are provided with an opportunity to share their math thinking as they solve problems involving fractions. Problem 12, “Yanira ran \frac{3}{4} miles each day for 6 days. How many miles did she run over the course of 6 days?” Teachers are provided with three specific protocols to assist them in helping students represent and/or solve the problem, including sentence stems, for example: “First I put ____ because the story ____. Then I put ____ because in the story ____. Finally, I put ____ because in the story/we need to figure out ____.” • Unit 10, Lesson 4, Exit Ticket, students construct an argument and critique the reasoning of others based on their knowledge of shapes. Problem 3, “Carlos is helping his brother with homework. He tells his brother that if you want to draw an obtuse angle, you should always use the bottom set of degrees on the protractor arc, and if you want to draw an acute angle you should always use the top set. Is Carlos’s reasoning accurate? Explain why or why not on the lines below.” ##### 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 4 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. • The Unit 1 Overview describes the intentional development of MP4. “Students model real-world mathematical situations using equations and tables with patterns. Students use tables, pictures, and mathematical formulas to solve problems involving patterns, multiplicative comparisons, and to determine factors, and classify numbers as prime or composite. SMP4 is developed in lessons 2 and 5 - 9.” • The Unit 4 Overview provides guidance for connecting MP4 with area, perimeter, and solving conversion problems. “Students interpret word problems referring to area and perimeter and represent the information. When students solve conversion problems, they use many different types of models to determine how to convert, or show how they converted. They use the ratio to draw appropriate pictures, create tables, and write equations that use mathematics to model how to convert from one unit of measurement to another. Students also use benchmarks to understand units of measurement, which is a way of modeling a mathematical concept with real-world objects.” • Unit 6, Lesson 18, Pose the Problem, students subtract mixed numbers by using fraction tiles and drawings. “Moira ordered 6 pizzas for the Student Council meeting. At the end of the meeting there were 2\frac{3}{4} pizzas left. How many pizzas did the student council eat during the meeting?” There is intentional development of MP5 to meet its full intent in connection to grade-level content. Examples include: • The Unit 2 Overview describes the intentional development of MP5. “Students choose between many methods when working with place value. In almost every aspect of this unit (place value relationships, expanding, reading, writing, comparing and rounding numbers, and non-standard partitioning) students have a variety of tools that could assist them. They can use place value charts, place value blocks, pictures of dots in each place value, pictures of place value blocks, organized lists, etc. to solve these types of problems. They must determine which tools are most effective for certain tasks.” • Unit 6, Lesson 6, Exit Ticket Question 2, students use various tools, models and representations to show the meaning of fractions and different ways of showing fractional quantities. “Delilah is using a number line to figure out how many eighths are equal to \frac{3}{4}. She tells her partner, ‘We should partition each interval on our number line into 4 new parts, because 4 + 4 is 8.’ Is Delilah’s strategy reasonable? Explain on the lines below. You may use pictures, number sentences, or number lines to help you.” • Unit 9, Lesson 4, Independent Practice Question 6, students use square corners and rulers to determine types of lines and angles. “Can a triangle have two right angles? Explain and draw an example to prove your thinking.” 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 4 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 2, Lesson 1, Exit Ticket, students solve problems requiring them to demonstrate an understanding of the term expanded form. Problem 3, “What two hundreds is seven hundred twelve between? Write seven hundred twelve in standard form. Write seven hundred twelve in expanded form.” • Unit 8, Lesson 2, Independent Practice Question 1 (Bachelor Level), students use precision to solve problems related to volume and capacity. “The capacity of each pitcher in the teacher work room is 3 quarts. Right now, each pitcher contains 1 quart 3 cups of liquid. If there are 3 pitchers in the room, how much more total liquid can the pitchers hold?” • In the Unit 9 Overview, “Students attend to precision when naming and identifying lines, angles and triangles based on names using points.” The instructional materials attend to the specialized language of mathematics. The materials use precise and accurate mathematical terminology. Examples include: • Unit 4, Lesson 15, Try One More, teachers are provided with instructions to explicitly teach the term remainder. “We do have 1 leftover in this problem. This is called our remainder. A remainder is the amount leftover after dividing a number when one number does not divide evenly into another number. What was our answer before the remainder and why?” Students might say, “114 because we have 1 hundred + 1 ten + 4 ones.” The teacher replies, “Yes. Now our answer becomes 114 R1 because our answer is 114 with a remainder of 1.” • Unit 6 Lesson 1, teachers are provided with guidance in reviewing vocabulary related to fractions. The introduction, “Before you get started, let’s review some key fraction vocabulary. In a fraction, what does the denominator tell us? The denominator tells us the total number of parts in a whole. In a fraction, what does the numerator tell us? The numerator tells us the total number of parts being referred to.” • Unit 9, Lesson 4, the introduction provides teachers with guidance in introducing terminology related to triangles through a series of questions. “We have learned about different angle types, which will help us in our work today. What are the different types of angles? Today you will use the different types of angles to help you classify triangles!” Students work to observe and note information about triangles. The teacher prompts, “What did you observe about triangles? Triangles are classified by two names, kind of like how you have a first name and a last name. One name tells us about their angles, and one name tells us about their angles, and one name tells us about their sides.” ##### 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 4 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards. There is intentional development of MP7 to meet its full intent in connection to grade-level content. Examples Include: • Unit 2, Lesson 5, Independent Practice, students solve problems by looking for structures based on place value. Problem 4, “Tiana drew 12 hundreds blocks on her paper. How many tens is that equal to? a. 1,200 b. 120 c. 12 d. 12,000.” • Unit 5, Lesson 4, Independent Practice, Question 2, students look for structure as they solve problems involving parts adding up to a whole by using tape diagrams to represent the situations. “Malia is keeping track of the subway riders on Saturday. At the first stop, some people got on the train. At the second stop, three times more people got on the train than at the first stop. At the last stop some people got off the train. How many people are on the train now?” • Unit 8, Lesson 5, Exit Ticket, Question 2, students solve word problems involving adding enough of a smaller unit in order to regroup in the context of money amounts and determining change. “Meiling needed5.35 to buy a ticket to a show. In her wallet, she found 2 dollar bills, 11 dimes, and 5 pennies. How much more money does Meiling need to buy the ticket?”

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

• Unit 5, Lesson 8, Exit Ticket, Question 1, students look for regularity in repeated reasoning as they solve problems by interpreting and labelling a representation such as a tape diagram. “Draw a tape diagram to model the following equation. Create a word problem. Solve for the value of the variable. (A\times2)+4,892=6,392.”

• Unit 6, Lesson 16, Independent Practice, Question 2, students see regularity in regrouping as they add and subtract mixed numbers. “Khalia and Jermaine are in a pie-eating contest. After 5 minutes, Khalia ate 3\frac{3}{4} pies and Jermaine at 4\frac{3}{4} . How much total pie did they consume altogether?”

• Unit 10, Lesson 2, Independent Practice, students look for repeated calculations as they solve a problem involving a circle divided into angles with given measurements. Problem 3, “Joanne cut a round pizza into equal wedges with angles measuring 30 degrees. How many pieces of pizza does she have?”

### Usability

The materials reviewed for Achievement First Mathematics Grade 4 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 4 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 4 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, etc.

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

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. Each lesson includes anticipated challenges, misconceptions, key points, sample dialogue, and exemplar student responses. Examples from Unit 7, Decimals, Lesson 10 include:

• “What do students have to get better at today? Students add fractions with denominators of 10 and 100 by changing both fractions to have like denominators or by relating them to decimals in expanded form. Students may still be relying on visual models today, though some may be able to ‘just know’ or use place value knowledge to add.”

• “What is new and/or hard about that? This is challenging because scholars may not have solidified adding or subtracting fractions from the previous unit. Scholars may struggle to regroup accurately, and when converting from mixed numbers to decimals may struggle to accurately change from fractions to decimals, particularly when needing to account for whole numbers. Lastly, scholars must be able to convert fractional tenths to hundredths (or understand how to accurately combine with tenths and hundredths) which could be another area for misunderstanding, as scholars can forget to multiply or divide the numerator by ten.”

• “Exemplar Student Response: ‘When I added \frac{9}{10} and \frac{18}{100}, I converted \frac{9}{10} to hundredths because we can only add fractions if they have the same denominator. I converted \frac{9}{10} to hundredths by multiplying the top and the bottom of the fraction by 10; I got \frac{90}{100}. Then, I added the numerators together and got \frac{108}{100}. I regrouped because \frac{108}{100} is a fraction greater than 1. I thought about how many wholes and how many fractional parts I have; I know that \frac{100}{100} makes 1 whole which leaves \frac{8}{100} leftover. I wrote my answer first as a fraction -$$\frac{108}{100}$$ and then as a decimal -1.08.’”

• “Introduction State the aim: Connect it to their lives and prior knowledge. Discuss how they will be working on it today. Plan a problem and questions to uncover key points and address common errors and misconceptions.”

• “Mid-Workshop Interruption: Share a strategy you’d like more students to use OR clarify a major misconception.”

• “Discussion: Discuss a major misconception OR have students share their work in CPA order OR ask students to apply their learning in a new way OR direct students to complete a pre-planned written response followed by a share from 1-2 students.”

• “Closing & Exit Ticket A quick debrief to clear up confusion OR cement a key point or big idea from the lesson.”

Each lesson includes both “What” and “How” Key Point sections that describe what students should know and be able to do and how they will do it. Examples from Unit 7, Decimals, Lesson 10 include:

• “What Key Points What should students know and be able to do? I can accurately add tenths and hundredths as fractions. When adding fractions, they must have the same denominator because we can only combine fractions with the same-sized pieces. I can accurately convert between fractions and decimals.”

• “How Key Points How will they do it? I can accurately add tenths and hundredths by converting tenths to hundredths (if needed) and then adding the numerators (number of pieces) of each fraction and writing my sum over the given denominator.”

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

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

• Unit Overviews provide thorough information about the content of the unit which often includes definitions of terminology, explanations of strategies, and the rationale about incorporating a process. Unit 3 Overview, “Research and practice in the field of mathematics education have shown that there are alternative algorithms and strategies that students develop, that help them maintain a focus on understanding place value and the operations and, at the same time, are easily generalized and efficient. Although each student may primarily use one strategy for each operation, in Investigations (and now in Common Core), students are expected to study more than one algorithm or strategy for each operation. Students study a variety of approaches for the following three reasons: Different algorithms and strategies provide access to analysis of different mathematical relationships. Access to different algorithms and strategies leads to flexibility in solving problems. One method may be better suited to a particular problem. Students learn that algorithms are ‘made objects’ that can be compared, analyzed, and critiqued according to a number of criteria.”

• 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 4 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.

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

• Guide to Implementing AF Grade 4, 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 4 Overview, “In 5th grade, students must explain patterns in the number of zeroes of the product when multiplying by multiples of 10. This directly relates to strategies and understanding students use in 4th grade to multiply multiples of 10. Students learn rules with zeroes based on place value. These rules also extend to multiplying and dividing decimals by multiples of 10 in 5th grade. They must use this same pattern of zeroes and changes in place value to explain why decimal points move certain amounts of places in certain directions when multiplying and dividing by multiples of 10.”

• Unit 10 Overview, “Angle measurements will not come up as a formal part of the math curriculum again until seventh grade. Although there is a large gap in time between this unit and seventh grade, the seventh grade standards rely heavily on students’ skills and knowledge from this unit in fourth grade. Understanding of the concept of angles, the additive property of angles, and measurements of benchmark angles are foundational for seventh grade angle content. In 7th grade, students use supplementary angles, complementary angles and adjacent angles in problems and write and solve equations to find measurements of unknown angles. They also start to work with vertical angles and use them to find angle measurements. The strong ties between the 4th grade and 7th grade angle measurement standards, and the fact that students will not revisit this content for three years makes it crucial for students to deeply understand the fourth grade angle measurement material.”

##### 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 4 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 4 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 4 meet expectations for providing a comprehensive list of supplies needed to support instructional activities.

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

• Unit 2, Lesson 2, the Lesson Overview includes, “Materials: all pages of this packet, 10 hundreds chart pags per student or per partner pair, colored pencils, scissors, staples, VA.”

• Unit 9 Lesson 1, the Lesson Overview includes, “Materials: All pages of this packet, VA (stands for visual aid).”

##### 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 4 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 4 partially meet expectations for having assessment information included in the materials to indicate which standards are assessed. There are connections identified for standards, but not the mathematical practices.

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

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

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

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

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

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

The materials reviewed for Achievement First Mathematics Grade 4 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 4 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:

• Unit 1, Lesson 13, Exit Ticket contributes to the full intent of 4.OA.5 (Generate a number or shape pattern that follows a given rule). “1) Look at the towers and table below and complete the table. 2) Explain how you figured out how many blocks are in tower 4.”

• The Unit 7 Assessment contributes to the full intent of 4.NF.5 (Express a fraction with denominator 10 as an equivalent fraction with denominator 100). Item 10, “Jenny split one cake into 10 equal pieces and ate 3 of the pieces. Then she split another cake into 100 equal pieces and ate 13 of the pieces. Write a sentence telling the total amount of cake(s) Jenny ate as a decimal.”

• The Unit 10 Assessment contributes to the full intent of 4.MD.6. Item 4 includes an image of a kite, “Use your protractor to answer the question below. Which measure is closest to the measure of angle H? A) 30° B) 35° C) 150° D) 155°” Item 7, “Which angle has a measure of 65°?” Item 11, “Draw an angle with a measure of exactly 147°.”

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

• Unit 4 Assessment, Item 15, supports the full development of MP2 as students convert between units of measurement, using abstract reasoning to determine which operations to use. “A race is 5 km long. How many meters long is the race? How many centimeters long is the race? a. 500 meters; 5000 centimeters; b. 5,000 meters; 50,000 centimeters; c. 5,000 meters;  500,000 centimeters; d. 500 meters; 50,000 centimeters.”

• Unit 1 Assessment, Item 6, supports the full development of MP4 as students write and solve an equation to represent a situation. “The length of the kitchen counter at Sal’s house is 9 times the length of a book. The length of a book is 8 inches. What is the length of the kitchen counter? Write an equation and solve.”

• Unit 10 Assessment, Item 2, supports the full development of MP7 as students look for and make sense of structure. “Which statement about angles is true? A) An angle is formed by two rays that do not have the same endpoint; B) An angle that turns through 1/360 of a circle has a measure of 360 degrees; C) An angle that turns through 5- 1 degree angles has a measure of 5 degrees D) An angle measure is equal to the total length of the two rays for the angle.”

##### 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 4 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 4 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 4 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 4 partially meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.

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

• Unit 1, Lesson 11, Extension problem, students extend their understanding of multiplication equations as a comparison. “Encourage students to pick a number and try and come up with a multiplicative clue for that number. Have them switch with a partner and try and figure out the solutions.”

• Unit 9, Lesson 3, Extension problem, students extend their understanding of right angles. “Challenge students to identify types of angles in figures or around the room, have students create their own figures with the different angle sizes, can have students start to experiment with protractors.”

##### 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 4 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 4 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 4, Lesson 3, Discussion, “Turn & Talk: What are we working on today? We are working on finding the perimeter of rectilinear figures by solving for unknown sides and adding them all up.”

• Unit 5, Lesson 7, Narrative, “Students continue to practice SMP 7 by considering "what smaller questions do I need to answer first" and SMP 3 by discussing the answer to this question as they discuss as a class or in partners which steps need to be completed and in what order these questions must be answered to arrive at a final answer.”

##### 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 4 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 4 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 4 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 4 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 4 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 4 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 4, Lesson 4, Narrative, “students will find the area of rectilinear figures by decomposing them into rectangles.” The Intervention includes the use of manipulatives, “Have scholars use tiles or counters to fill the shape, counting as they go to find area. Have scholars count squares inside and around shapes to determine area.”

• Unit 6, Lesson 1, Narrative, students will use “fraction tiles and visual models to break fractions down into their unit fractions.” Students use manipulatives to solve the Problem of the Day, “Park Ranger O’Hara is putting trail markers down on Mount Bonnell. On the Mockingbird trail, he puts down a trail marker at every \frac{1}{8} of a mile. If the trail is \frac{7}{8} of a mile long, how many trail markers will he use? Use your fraction tiles to help you.”

• The Unit 8 Overview, “Students use many different strategies to solve different types of measurement problems such as timelines (or clock manipulatives) to solve elapsed time problems.” The Lesson 6 materials list includes clocks and the Intervention suggests “students just use clocks to solve” but there is no additional guidance.

Manipulatives are not always connected to written methods. There are several instances where manipulatives are listed as materials but not incorporated into the lesson. Examples Include:

• Unit 8, Lesson 4, Materials list, “Coin/dollar manipulatives for intervention.” “Use a place value chart to represent each money/decimal/coin amount. In each column on the place value chart, write the decimal place value and the coin associated with that place value. Then have students put each problem from workshop into the place value chart.” However the Intervention does not include the use of such manipulatives.

#### 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 4 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 4 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 4 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 4 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 4 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
##### 1st Grade
###### Alignment
Meets Expectations
###### Usability
Does Not Meet Expectations
##### 2nd Grade
###### 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.

##### 3rd Grade
###### Alignment
Meets Expectations
###### Usability
Does Not Meet Expectations
##### 4th Grade
###### Alignment
Meets Expectations
###### Usability
Does Not Meet Expectations
##### 5th Grade
###### 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.

##### 6th Grade
###### Alignment
Meets Expectations
###### Usability
Does Not Meet Expectations
##### 7th Grade
###### Alignment
Meets Expectations
###### Usability
Does Not Meet Expectations
##### 8th Grade
###### Alignment
Meets Expectations
###### Usability
Does Not Meet Expectations

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### Overall Summary

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###### Alignment
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###### Usability
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