The instructional materials reviewed for Achievement First Mathematics Grade 4 meet expectations for assessing grade-level content. Each unit of instruction contains a Post-Assessment which is a summative assessment based on the standards designated in that unit. Examples of assessment items aligned to grade-level standards include: 

• In Unit 1, Post-Assessment, Item 5 states, “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)  
• In Unit 4, Post-Assessment, Item 16 states, “Divide. $$7,285÷4$$. Answer choices include:   A. 1,801, B. 1,801 R1, C. 1,821, D. 1,821 R1.” (4.OA.3)
• In Unit 9, Post-Assessment, Item 15 states, “Explain why a square is also a rectangle and a rhombus.” (4.G.2 )
• In Unit 10, Post-Assessment, Item 13 states, “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:

• In Unit 6, Post-Assessment, Item 2 states, “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.)
• In Unit 6, Post-Assessment, Item 14 states, “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, states that 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.

The instructional materials reviewed for Achievement First Mathematics Grade 4 meet expectations for spending a majority of instructional time on major work of the grade.

• The number of lessons devoted to major work of the grade (including assessments and supporting work connected to the major work) is 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 instructional materials because the units and lessons do not include all of the components included in the math instructional time. The instructional block includes a math lesson, math stories, and math practice components. As a result, approximately 91% of the instructional materials focus on major work of the grade.

The instructional materials reviewed for Achievement First Mathematics Grade 4 meet expectations that supporting work enhances focus and coherence simultaneously by engaging students in the major work of the grade. There are opportunities in which supporting standards/clusters are used to support major work of the grade and are connected to the major standards/clusters of the grade. Examples include:

• In Unit 4, Lesson 25, Independent Practice, Problem 2 states, “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.  
• In Unit 8, Lesson 7, Independent Practice, Problem 2 states, “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.  
• In Unit 10, Lesson 2, Independent Practice, Problem 2 states, “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.

The instructional materials reviewed for Achievement First Mathematics Grade 4 meet expectations that the amount of content designated for one grade-level is viable for one school year. The Guide to Implementing Achievement First Mathematics Grade 4 includes a scope and sequence and states, “Not every lesson is entirely focused on grade level standards, and, therefore some lessons can be used for remediation or enrichment.” As designed, the instructional materials can be completed in 135 days.

• 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)

The instructional materials reviewed for Achievement First Mathematics Grade 4 meet expectations for being consistent with the progressions in the Standards. Content from prior or future grades is identified and connected to grade-level work, and students are given extensive work with grade-level problems. 

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

• In Unit 3, Unit Overview, Linking states, “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.”
• In Unit 6, Unit Overview, Identify the Narrative, refers to prior work students engaged in with fractions. The materials state, “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.”
• In Unit 10, Unit Overview, Linking state, “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.” 

The instructional materials in Achievement First Mathematics Grade 4 provide opportunities for students to engage with grade-level problems. Each unit consists of lessons that are broken into four components: Introduction, Workshop/Discussion, Independent Practice, and Exit Ticket. In addition to lessons, there are Math Stories “to enable students to make connections, identify and practice representation and calculation strategies, and develop deep conceptual understanding through the introduction of a specific story problem type in a clear and focused fashion with deliberate questioning and independent work time,” and Math Practice (Practice Workbook) for students “to build procedural skill and fluency.” Examples include: 

• In Unit 3, Lessons 2 through 9, students engage with 4.NBT.5, 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. For example, Lesson 6, Independent Practice, Problem 5 states, “$$50,019-12,877$$” and Problem 6 states, “In the last problem, what place values did you need to regroup and how did you do it? Explain on the lines below.”
• In Unit 5, Lessons 1 through 10, students engage with 4.OA.3, 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, represent these problems using equations with a letter standard for the unknown quantity. There are 83 Independent Practice problems and 15 Exit Tickets that require students to solve multi-step word problems. For example, Lesson 6, Independent Practice, Problem 4 states, “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.
• In 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. For example, Exit Ticket 1 states, “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. $$67\degree$$.”

Achievement First Mathematics Grade 4 relates grade-level concepts explicitly to prior knowledge from earlier grades. Each unit has a Unit Overview and a section labeled “Identify Desired Results” where the standards for the unit are provided as well as a correlating section “Previous Grade Level Standards/Previously Taught & Related Standards” where prior grade-level standards are identified. Examples include:

• In Unit 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. The materials state, “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.” 
• In 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 instructional materials reviewed for Achievement First Mathematics Grade 4 meet expectations for fostering coherence through connections at a single grade, where appropriate and required by the Standards.

The materials include learning objectives, identified as Aims, that are visibly shaped by the CCSM cluster headings. The instructional materials utilize the acronym SWBAT to stand for “Students will be able to” when identifying the lesson objectives. Examples include: 

• In Unit 2, Lesson 7, the Aim states, “SWBAT evaluate many forms of numbers in non-standard partitioning and find various ways to show the same numbers by thinking about place value and decomposing/composing numbers to create values equal to a given number (4.NBT.1 & 4.NBT.2),” which is shaped by 4.NBT.A, “Generalize place value understanding for multi-digit whole numbers.”
• In Unit 5, Lesson 3, the Aim states, “SWBAT solve 2-step word problems involving addition and subtraction by visualizing, representing, estimating, and using a variety of strategies to calculate,” which is shaped by 4.OA.A, “Use the four operations with whole numbers to solve problems.”
• In Unit 9, Lesson 4, the Aim states, “SWBAT classify triangles as right, acute, or obtuse and equilateral, isosceles or scalene based on their side lengths and angle types,” which is shaped by 4.G.2, “Classify two-dimensional figures based on the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right angles.” 

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

• In Unit 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 states, “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.”
• In 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. In the Exit Ticket, Problem 1 states, “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.” 
• In 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 states, “Locate and label the following points on the number line below: $$\frac{130}{10},13.21, 13\frac{12}{100}$$.”

The instructional materials for Achievement First Mathematics Grade 4 meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. The materials include problems and questions that develop conceptual understanding throughout the grade level. For example: 

• In Unit 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 states, “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.”
• In 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 states, “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.”
• In 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. The materials state, “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. For example: 

• In 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. In Independent Practice, Problem 2 states, “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.”
• In 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. In Independent Practice, Problem 1 states, “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?” 
• In 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 states, “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.”

The instructional materials for Achievement First Mathematics Grade 4 meet expectations that they attend to those standards that set an expectation of procedural skill and fluency. The instructional materials include opportunities for students to build procedural skill and fluency in both Math Practice and Cumulative Review worksheets. The materials do not include collaborative or independent games, math center activities, or non-paper/pencil activities to develop procedural skill and fluency.

Math Practice is intended to “build procedural skill and fluency” and occurs four days a week for 10 minutes. There are 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. For example:

• In Practice Workbook C, Problem 1, students solve subtraction problems. It states, “Find the difference. 51,348 and 22,122. Use the standard algorithm to solve.” (4.NBT.4)
• In Practice Workbook C, Problem 6, students solve subtraction problems. It states, “Use a strategy that makes sense to you to solve. 59,637 – 34,721 = .” (4.NBT.4)
• In Practice Workbook C, Problem 9, students practice subtraction. The problem states, “$$56432 - 33224 =$$ _____.“ (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. For example:

• In Unit 4, Cumulative Review 4.5, Problem 4, students solve subtraction problems. It states, “Find the difference. Show your work. $$6,241 - 1,366 =$$ _____.” (4.NBT.4) 
• In Unit 4, Cumulative Review 4.5, Problem 2, students solve addition and subtraction problems using the standard algorithm. It states, “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?”
• In Unit 6, Cumulative Review 6.4, Problem 5, students solve a subtraction problem. It states, “Find the difference. $$2,301-1,976 =$$ ___” (4.NBT.4)

he instructional materials for Achievement First Mathematics Grade 4 partially meet expectations that the materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied. The instructional materials include limited opportunities for students to independently engage in the application of routine and non-routine problems due to teacher heavily scaffolded tasks and the lack of non-routine problems.  

The instructional materials present opportunities for students to independently demonstrate routine application of mathematics; however, there are few opportunities for students to independently demonstrate application of grade-level mathematics in non-routine settings. Routine problems are found in the Independent Practice and Exit Tickets components of the materials. Examples of routine applications include: 

• In Unit 1, Lesson 11, Independent Practice, students engage with 4.OA.2 as they solve a word problem involving multiplicative comparisons. Problem 2 states, “Kenny is 56 years old. His sister is 7 years old. How many times younger is Kenny’s sister than him?”
• In Unit 5, Lesson 10, Independent Practice, students engage with 4.OA.3 as they solve multi-step word problems with whole numbers. Problem 7 states, “Three friends have 3,498 water balloons to share equally. It takes 2 minutes to fill each water balloon. How much will one person spend filling up their balloons?”
• In Unit 6, Lesson 22, Independent Practice, students engage with 4.NF.3 as they solve word problems involving addition and subtraction of fractions. Problem 1 states, “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?”

Achievement First Mathematics Grade 4 provides limited opportunities for students to engage in non-routine problems. Additionally, the non-routine problems are often heavily scaffolded for students with directed teacher questioning techniques. Non-routine problems are found within the Workshop, Math Stories, and Problem of the Day components of the materials. For example: 

• In Unit 2, Cumulative Review 2.2, Problem 4, students engage in 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.”
• In Unit 5, Unit Assessment, students engage with 4.OA.3 as they solve a word problem involving the four operations. Item 15, “Jian’s family sells honey from beehives. They collected 3,311 ounces of honey from the beehives this season. They will use the honey to completely fill 4-ounce jars or 6-ounce jars. Jian’s family will see 4-ounce jars for $5 each or 6-ounce jars for $8 each. Will they make more money from selling 4-ounce jars or 6-ounce jars? How much more money?”
• In Unit 7, Lesson 11, Problem of the Day, students engage with 4.NF.5 as they apply their understanding of fractions. Part 1, 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.”

The instructional materials for Achievement First Mathematics Grade 4 meet expectations for balancing the three aspects of rigor. Overall, within the instructional materials the three aspects of rigor are not always treated together and are not always treated separately.

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

• In Unit 2, Cumulative Review 2.2, students demonstrate conceptual understanding of factors by determining whether there are additional factors for a number within 100. Problem 7 states, “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)
• In Practice Workbook B, students develop procedural skill and fluency as they multiply whole numbers. Problem 14 states, “Calculate the product of $$64×35$$.”  (4.NBT.5) 
• In Unit 6, Lesson 23, Exit Ticket, students apply their understanding of fraction multiplication as they solve word problems. The materials state, “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. For example:

• In Unit 4, Lesson 11, Independent Practice, students apply their conceptual understanding of adding decimals to solve a real-world problem. The materials state, “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) 
• In 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 states, “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) 
• In 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 states, “Which is more, 68 dimes or 679 pennies? Prove with a place value chart and then explain on the lines below.” (4.MD.2)

The instructional materials reviewed for Achievement First Mathematics Grade 4 partially meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade-level.

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

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

• Unit 4, Unit Overview, MP 2, “Students will have opportunities to use multiplication and divisions as an abstract representation to solve word problems in context. When students convert between units of measurement, they must use abstract reasoning to determine which operations to use along with ratios in order to convert.”
• Unit 7, Unit Overview, MP7, “Students use place value structures to interpret decimals. They relate the ten times greater/less relationship between whole number place values to decimal place values. The idea that ten of one place value creates one of the next greatest place value is a consistent structure in our number system which adds clarity to why 10 tenths are in 1, and 10 hundredths make 1 tenth. Structures for expanded form also apply to decimals. Values of digits are written in certain ways based on their place values and these values follow consistent structures and patterns in numbers of zeros, places away from the ones place, etc.”
• Unit 10, Unit Overview, MP5, “Students correctly use protractors when measuring and drawing angles. They must strategically follow and understand the process for using this tool.”

The instructional materials reviewed for Achievement First Mathematics Grade 4 partially meet expectations for carefully attending to the full meaning of each practice standard. The Mathematical Practices (MPs) are represented in each of the nine units in the curriculum and labeled in each lesson. Math Practices are represented throughout the year and not limited to specific units or lessons. The materials do not attend to the full meaning of MPs 1 and 5.

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

• In Unit 4, Lesson 7, Independent Practice, students analyze information to solve multi-step problems involving measurement. Problem 7 states, “Jeremiah walks 7 dogs when he volunteers at the Humane Shelter on Saturdays. Each dog must drink 30 mL of water for every 15 minutes they walk. If Jeremiah plans to walk each dog for 30 minutes, how many total mL of water will he need?”
• In Unit 5, Lesson 10, Independent Practice, students make sense of information to solve a multi-step word problem involving conversions. Problem 3 states, “Nick and his friends are working on their final thesis project. So far they have spent 2,029 minutes on the projects altogether. His friend Kara wrote 14 pages that each took 32 minutes, and his friend Jarred spent twice as many minutes as Kara interviewing people. How many minutes has Nick spent working on the project?”
• In Unit 8, Lesson 5, Independent Practice students select an appropriate strategy to solve multi-step problems involving money. Problem 2 states, “Bruson bought 2 concert tickets for a total of $38.75. He gave the salesperson 3 ten dollar bills and 2 five dollar bills. How much change should the salesperson give Bryson?”

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

• In Unit 7, Lesson 3, Exit Ticket, students use a provided visual model to represent coin values as a total decimal amount. Problem 2 states, “Doreen is cleaning out her purse. In the bottom she found 7 dimes and 2 pennies. Represent this amount using the visual model, as money, and as a decimal amount.” 
• In Unit 8, Cumulative Review 8.2, students use shaded fraction strips to generate and add fractions. Problem 3 states, “The shaded part of the fraction strips below represent two fractions. What is the sum of the two fractions?
• In Unit 10, Lesson 4, Independent Practice, students use a protractor to draw angles of a given measure. Problem 5 states, “Use the protractor below to draw angles of a given measure. 163°.”

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

• MP2: In Unit 9, Lesson 2, Independent Practice, students use reasoning skills to develop a claim based on their knowledge of angles. Problem 4 states, “True or false? Shapes that have at least one right angle also have at least one pair of perpendicular sides. Explain your thinking.”
• MP4: In Unit 6, Lesson 1, Independent Practice, students create a visual model and equation to represent an equivalent fraction. Problem 4 states, “Claire is baking a cake. The recipe calls for $$\frac{3}{4}$$ of a tablespoon of vanilla, but Claire only has a $$\frac{1}{4}$$ tablespoon measuring spoon. Draw a visual model and write an addition equation to model how many $$\frac{1}{4}$$ tablespoons Claire will need to equal the recipe.”
• MP6: In Unit 10, Lesson 6, Independent Practice, students attend to precision as they use a picture of a ray and create and measure supplementary angles. Problem 3 states, “Split each straight angle into two supplementary angles. Measure ONE of the supplementary angles with your protractor and write an equation and solve to find the second angle.” 
• MP7: In Unit 2, Lesson 5, Independent Practice, students solve problems by looking for structures based on place value. Problem 4 states, “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.” 
• MP8: In 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 states, “Joanne cut a round pizza into equal wedges with angles measuring 30 degrees. How many pieces of pizza does she have?”

The instructional materials reviewed for Achievement First Mathematics Grade 4 meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

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

• In Unit 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 states, “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.” 
• In Unit 7, Lesson 4, Exit Ticket, students use their knowledge of decimal comparisons to the hundredths place to critique the reasoning of others and construct a viable argument. Problem 3 states, “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.”
• In Unit 10, Lesson 4, Exit Ticket, students construct an argument and critique the reasoning of others based on their knowledge of shapes. Problem 3 states, “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.”

The instructional materials reviewed for Achievement First Mathematics Grade 4 meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. There are opportunities within the teacher materials that guide teachers in assisting students to construct viable arguments and analyze the arguments of others, through the use of questioning.

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

• In Unit 2, Lesson 12, Problem of the Day, teachers are provided with language to assist students in analyzing the work of others. Try One More states, “Tonya is building numbers in math class. She wants to use her place value blocks to make some numbers that are greater than her partner’s number, 150,600. However, she only has one hundred-thousand block and no hundreds blocks. Write some numbers that Tonya could build that would be greater than her partner’s. Work with your partner to solve. Circulate as students work, look for scholars who are using place value charts or other efficient strategies to brainstorm numbers that fit the criteria. What strategy did you use to create numbers that fit the problem? We started by drawing a place value chart and writing a 1 in the hundred-thousands place and a 0 in the hundreds place, then we chose digits that were bigger than the ones in the partner’s number for all the other places. I have an idea, help me see if it works. Jot down ’98,023’. No! Why not? That number doesn’t work because it doesn’t have any hundred-thousands, so it's smaller than Tonya’s partner’s number because that number has a hundred- thousand in it, and even only 1 hundred-thousand is bigger than 9 ten-thousands! But 9 is more than 5! Yes, but the 9 is in the ten-thousands place and ten-thousands are less than hundred-thousands! What mistake was I making? You didn’t think about the largest place value!” 
• In Unit 7, Lesson 8, Discussion, teachers are provided with guidance and questions to engage students in critiquing the reasoning of others. The materials state, “Last year I had a scholar who told me that when you compare decimals it's like the opposite of comparing whole numbers. TT: What do you think they meant by that? They probably meant that when you compare whole numbers, tens are smaller than hundreds, but with decimals tenths are bigger than hundredths. They probably meant that when you are comparing whole numbers, the more digits you have the larger the number is, but that isn’t always true with decimals, like with 0.8 and 0.09. Yes! How are comparing whole numbers and comparing decimals similar? For both we have to think about the place value that digits are in.”
• In Unit 7, Math Stories, students are provided with an opportunity to share their math thinking as they solve problems involving fractions. Problem 12 states, “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 ____.”

The instructional materials reviewed for Achievement First Mathematics Grade 4 meet  expectations that materials explicitly attend to the specialized language of mathematics. 

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

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

• In Unit 4, Lesson 15, teachers are provided with instructions to explicitly teach the term remainder. Try One More states, “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. Reveal KP on VA. What was our answer before the remainder and why? 114 because we have 1 hundred + 1 ten + 4 ones. Add to VA. Yes. Now our answer becomes 114 R 1 because our answer is 114 with a remainder of 1.”
• In Unit 6 Lesson 1, teachers are provided with guidance in reviewing vocabulary related to fractions. The introduction states, “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.”
• In Unit 9, Lesson 4, provides teachers with guidance in introducing terminology related to triangles through a series of questions. The introduction states, “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 materials state, “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.” 

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

• In Unit 2, Lesson 1, Exit Ticket, students solve problems requiring them to demonstrate an understanding of the term expanded form. Problem 3 states, “What two hundreds is seven hundred twelve between? Write seven hundred twelve in standard form. Write seven hundred twelve in expanded form.” 
• In Unit 6, Lesson 8, Exit Ticket, accurate terminology is used as students are expected to partition shapes and compare their size. Problem 1 states, “Partition and label the number line below to show the following fractions. Then write TWO inequalities to compare two of the fractions.  $$\frac{2}{3}$$, $$\frac{1}{6}$$, $$\frac{4}{6}$$, Inequality 1: Inequality 2:.”
• In Unit 8, Lesson 2, Independent Practice, accurate terminology is used as students solve problems related to volume and capacity. Problem 1 states, “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?”