The instructional materials reviewed for Achievement First Mathematics Grade 5 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, Common Core, Item 23 states, “What is 43.98 rounded to the nearest tenths place?” (5.NBT.4)
• In Unit 4, Unit Assessment, Item 3 states, “A rectangular garden has an area of 400 square meters. If the garden has a width of 5 meters, how long is the garden?” (5.NBT.2)
• In Unit 5, Unit Assessment, Item 1 states, “At a Sand Castle building contest, the tallest tower was 2 yards tall and the shortest tower was 1 foot and 4 inches tall. How much taller was the tallest tower than the shortest tower?” (5.MD.2)
• In Unit 8, Post-Assessment, Item 4 states, “Anthony has 12 marbles if $$\frac{3}{4}$$ of the marbles are clear, how many clear marbles does Anthony have. Draw a model to show your answer.” (5.NF.4) 

Achievement First Mathematics Grade 5 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 5 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 88 out of 132, which is approximately 67%.
• The number of days devoted to major work (including assessments and supporting work connected to the major work) is 113 out of 143, which is approximately 79%. 
• The instructional minutes were calculated by taking the number of minutes devoted to the major work of the grade (11,365) and dividing it by the total number of instructional minutes (12,870), which is approximately 88%. 

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 88% of the instructional materials focus on major work of the grade.

The instructional materials reviewed for Achievement First Mathematics Grade 5 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 3, Lesson 2, Independent Practice, Ph.D Level Problem 1 states, “Carol sells bracelets and pairs of earrings at a craft fair. Each item sells for $8. Write an expression to show how much money Carol makes if she sells 23 bracelets and 17 pairs of earrings, but pays $25 to rent her booth.” This problem connects the major work of 5.NBT.5, fluently multiply multi-digit whole numbers, to the supporting work of 5.OA.A, writing and interpreting numerical expressions, as students write an expression and solve the problem. 
• In Unit 5, Lesson 5, Exit Ticket, Problem 2 states, “Valerie uses 12 fluid oz of detergent each week for her laundry. If there are 5 cups of detergent in the bottle, in how many weeks will she need to buy a new bottle of detergent. Explain how you know.” This problem connects the major work of 5.NBT.B, perform operations with multi-digit whole numbers and with decimals to the hundreths, to the supporting standard 5.MD.1, convert among different sized standard measurement units within a given measurement system, as students perform a conversion and utilize at least one of the four operations to solve the problem. 
• In Unit 10, Cumulative Review 10.1, Problem of the Day, Day 3 states, “This year, the managers of the farm will change the fraction of the budget for housing to $$\frac{1}{8}$$ but will leave the fraction of the budget for food and medical care the same. Again, the remaining portion of the budget will be for maintenance expenses. What is the difference between the fraction of the budget for maintenance this year and last year?” This problem connects the major work of 5.NF.1 to the supporting cluster 5.MD.B, as students represent and interpret data while solving a multi-step problem involving adding and subtracting fractions with unlike denominators. 

The instructional materials reviewed for Achievement First Mathematics Grade 5 meet expectations that the amount of content designated for one grade-level is viable for one school year. As designed, the instructional materials can be completed in 143 days.

• There are 10 units with 132 lessons total. 
• There are 11 days for Post-Assessments.

According to The Guide to Implementing Achievement First Mathematics Grade 5, each lesson is designed to be completed in 90 minutes. For example:

• The math lessons are divided into three structural lesson types: conjecture-based lesson, exercise-based lesson, and error analysis lesson. The materials state, “On a given day students will be engaging in either a conjecture-based, exercise-based lesson or less often an error analysis lesson.”  
• Four days of the instructional week contain a Math Lesson (55 minutes) and Cumulative Review (35 minutes). The Cumulative Review is broken down into different parts: 
• Three days of Cumulative Review include Fluency (10 minutes), Mixed Practice (15 minutes), and Problem of the Day (10 minutes). 
• One day of Cumulative Review includes Fluency (10 minutes) and Reteach/Quiz (25 Minutes). 
• One day within the instructional week contains a Math Lesson (55 minutes) and Reteach Time Based on Data (35 minutes).

The instructional materials reviewed for Achievement First Mathematics Grade 5 partially 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. However, there is inconsistency across the materials in the identification of content and all students are not given extensive work with grade-level problems. 

Overall, the materials develop according to the grade-by-grade progressions in the Standards. However, content is not consistently connected to future grades 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 2, Overview, Identify the Narrative states, “Following this unit, students study multi-digit whole number computation to develop fluency with standard algorithms for whole number in multiplication and division, before moving into fraction and decimal operations.” The materials further state, “In later grades students continue to leverage this work when forms of rational numbers (grade 6), operating with all forms of rational numbers (grades 6 and 7), understanding ratios and rates of changes (grade 6-8), creating probability models (grade 7), working on coordinate grids (grades 5-8), and creating graphs to represent data (grades 5-8).” 
• In Unit 4, Unit Overview, Identify the Narrative states, “Throughout elementary school students are also writing simple expressions or equations to represent and solve word problems (2.OA.1, 3.OA.3, 4.OA.2). They use bar models to make sense of, think about, and solve simple real-world applications of multiplication. In fifth grade, students will leverage early work in Operations and Algebraic thinking to represent and solve real-world problems, and to write and evaluate mathematical expressions using the order of operations (5.OA.1, 5.OA.2).” 
• In Unit 7, Unit Overview, Identify the Narrative states, “In 5th grade, students will progress to adding fractions and mixed numbers with unlike denominators. In 4th grade and Unit 1 in 5th grade, scholars learned to find equivalent fractions using models and the identity property. This skill will be a crucial prerequisite to this unit. Additionally, scholars also learned how to add and subtract fractions and mixed numbers with like denominators by using fraction tiles, drawing models, and using the standard algorithm. In fourth grade, this included some regrouping, which is typically where scholars struggle the most. It is recommended to assess prior knowledge/skill for adding and subtracting mixed numbers (with like denominators) where regrouping is required to determine how to best target pre-existing gaps while progressing in this unit.” 

The instructional materials in Achievement First Mathematics Grade 5 do not provide opportunities for all students to engage with grade-level problems. Each unit consists of three types of lessons: conjecture based lessons, exercise based lessons, or an error analysis lesson. All three of these lessons types provide dedicated times for Partner Practice and Debrief, Independent Practice and Debrief, and Exit Tickets. The materials include Bachelor Level, Master Level, and Ph.D. Level specific tasks for Partner Practice and Independent Practice. As a result, all students are not provided with the opportunity to engage with grade-level problems to meet the full intent of the standard. Examples include: 

• 5.NBT.2: According to the Implementation Guide, this standard is identified as being taught in two lessons: one in Unit 1, and the other in Unit 4; with 17 independent practice questions for Bachelor Level students, nine for Master Level students, and 11 for Ph.D Level students. While all students are provided with the opportunity to solve problems with exponents, students at the Bachelor Level do not have an opportunity to explain the patterns. For example, Unit 4, Lesson 8, Independent Practice, Bachelor Level, Problem 3 states, “Ms. Jenkins decided it was time to donate and sell all of her old books. She has 1,042 books. She donated 300 and sold the rest. A used book store let her drop off the books she was selling in boxes of 80. How many boxes did she have to drop off?”
• 5.OA.1: There are not opportunities to solve problems with brackets or braces. According to the Implementation Guide, this standard is identified as being taught in two lessons in Unit 3; with 11 independent practice questions for the Bachelor Level students, nine for Master Level students, and nine for Ph.D students. However, students at the Bachelor Level are not provided with the opportunity to solve problems with brackets or braces. For example, Unit 3, Lesson 1, Independent Practice, Bachelor Level, Problem 2 states, “Evaluate $$15 × (7 - 7) + (5 × 2) - 3$$.” 
• 5.OA.2: According to the Implementation Guide, this standard is only identified and present in Unit 3, Lesson 2, with one independent practice question for Bachelor Level students, zero independent practice questions for Master Level students, and two independent practice questions for Ph.D students. For example, Unit 3, Lesson 2, Independent Practice, Ph.D Level states, “Carol sells bracelets and pairs of earrings at a craft fair. Each item sells for $8. Write an expression to show how much money Carol makes if she sells 23 bracelets and 17 pairs of earrings, but pays $25 to rent her booth.”
• 5.MD.1: There are not opportunities for students at all levels to solve multi-step word problems. According to the Implementation Guide, this standard is identified as being present in five lessons in Unit 5 and one lesson in Unit 6, with 24 independent practice questions for the Bachelor Level students, 22 for Master Level students, and nine for Ph.D students. At the Bachelor Level, students are not provided with the opportunity to solve multi-step problems and problems are focused mainly on conversions. For example, Unit 5, Lesson 2, Independent Practice, Bachelor Level, Problem 1 states, “___ pounds = 160 ounces.” 

Achievement First Mathematics Grade 5 relates grade-level concepts explicitly to prior knowledge from earlier grades. Each unit has a Unit Overview and a section labeled “Identify Desired Results” where the standards for the unit are provided as well as a correlating section “Previous Grade Level Standards/Previously Taught & Related Standards” where prior grade-level standards are identified. Examples include:

• In Unit 2, Unit Overview, Identify Desired Results: Identify the Standards lists 5.NF as being addressed in this unit and 4.NF.1, 4.NF.2, and 4.NF.3 as Previous Grade Level Standards/ Previously Taught & Related Standards connections. The materials state, “Starting in 3rd grade, students learn to recognize fractions as numbers (3.NF.A). They learn to represent fractions concretely and pictorially using unit fractions, on a number line and with equivalent fractions. They also learn to reason about relative sizes of fractions that have the same numerator or denominator. In 4th grade, students extend their understanding of fractions to compare and order fractions using equivalent fractions (4.NF.A), add and subtract fractions with like denominators, and multiply fractions and whole numbers (4.NF.B).” 
• In Unit 6, Unit Overview, Identify the Narrative connects the work of this unit to prior work in 3rd and 4th grades. The materials state, “Unit 6 draws heavily from Geometry and Numbers in Base Ten content learned in grades 3 and 4. In grade 3, students develop an understanding of area and relate the concept to both multiplication and addition. They also apply the concept to explore number properties (commutative and distributive) (3.MD.C). In fourth grade, students solidify their understanding of area and learn to apply the area formula fluently when measuring the area of rectangles (4.MD.3). These understandings and skills are useful moving into 5th grade as the concept of volume is developed concretely, pictorially and abstractly by making connections between volume and base-area using unit cubes, pictures and formulas as well as addition and multiplication to calculate the volume of a right rectangular prism. (5.MD.3,4,5).”

The instructional materials reviewed for Achievement First Mathematics Grade 5 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 1, Lesson 8, the Aim states, “SWBAT explain the effect of multiplying or dividing by powers of ten on the location of digits in a number,” which is shaped by 5.NBT.A, “Understand the place value system.”
• In Unit 3, Lesson 2, the Aim states, “SWBAT write simple numerical expressions that record calculations with numbers and interpret numerical expressions without evaluating them,” which is shaped by 5.OA.A, “Write and interpret numerical expressions.” 
• In Unit 6, Lesson 2 the Aim states, “SWBAT explore volume by building 3D figures and counting with unit cubes,” which is shaped by 5.MD.C, “Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.”

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 6, Lesson 5, students connect 5.MD.C, understand concepts of volume and relate volume to multiplication and addition, to 5.NBT.B, perform operations with multi-digit whole numbers and decimals, as they determine unknown values for measurements based on a given volume. In Exit Ticket, Problem 2 states, “Bernard is packing a box with a volume of 96 cubic inches. Enter a possible base area and height for his box below.” 
• In Unit 8, Cumulative Review 8.3, Problem of the Day, Day 3 connects 5.NBT.A, 5.NBT.B, and 5.OA.A, as students use their understanding of the place value system to evaluate a multi-step problems involving decimals, giving the answer in various forms. The materials state, “A.) Evaluate and express your answer in the three given forms: $$[(15×2)+(2×4)]+[12.06-(3×4)]$$; Standard Form, Expanded Form, Word Form.”
• In Unit 9, Cumulative Review 9.3, Problem of the Day, Day 2 connects 5.NBT.B with 5.NF.B, as students perform operations with multi-digit whole numbers and fractions. For example, “A chocolate factory produced 5,301 pounds of chocolate every day for 31 days in the month of January and 4,592 pounds of chocolate every day for 28 days in the month of February. Of their total chocolate produced, $$\frac{5}{8}$$  was milk chocolate. How many ounces of non-milk chocolate did the factory produce?”
• In Unit 11, Lesson 5, students connect 5.MD.B, represent and interpret data to 5.G.A, graph points on the coordinate plane to solve real-world and mathematical problems, as they generate data and develop a coordinate graph. In the Independent Practice, Bachelors Level states, “There is a $25 annual fee for membership at the gym. It also costs $5 per visit to use the gym. Fill in the table to show the total cost of $$\frac{5}{8}$$ visits to the gym. A. Write the ordered pairs, and graph the data on the coordinate graph. B. Write the ordered pair that represents 6 visits to the gym. Explain what the ordered pair means. C. If Amaya can only spend up to $50 in one month, how many times can she visit the gym? Explain.”

The instructional materials for Achievement First Mathematics Grade 5 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 3, Lesson 4, students develop conceptual understanding of 5.NBT.5, as they calculate products of two- and three-digit numbers by one-digit factors using area models. For example, Partner Practice, Problem 1 states, “Taliyah’s brother sells 654 gallons of cookie dough for $7 each. How much money does her brother raise? a) Find the product using the distributive property and an area model.” (a partially filled out area model is provided) “b) Use the standard algorithm to solve the multiplication problem. c) Describe each of the partial products you calculated, in order, when using the standard algorithm.”
• In Unit 6, Lesson 4, students develop conceptual understanding of 5.MD.5, as they use visual models of shapes to write expressions related to volume. In the Independent Practice, Bachelor Level, Problem 1, provides students with a $$4×4×5$$ rectangular prism. The materials state, “The same prism is shown below three times. Each cube represents one cubic meter. On each prism, use the lines to show you how you can deconstruct it into layers in a different way. Then, below each prism, write an expression to find the volume of each prism and solve.”  
• In Unit 8, Lesson 2, students develop conceptual understanding of 5.NF.3, as they use tape diagrams to solve division problems. In Think About It, students are introduced to tape models to solve, “$$8 ÷ 4 =$$ and $$3 ÷ 4 =$$ .” The materials state, “The models below are called tape diagrams. Part A. Use the models provided to determine each quotient. Circle the quotation in your model. Part B. In the space below each model, show a check step to prove that each quotient is correct.”

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

• In Unit 2, Mixed Practice 2.1, students demonstrate conceptual understanding of 5.NBT.A, as they explain patterns in products when multiplying by powers of ten. Problem 2 states, “Matthew multiplied $$1.5×10^3$$ and said that the answer was 1.5000. Which statement, if any, explains Matthew’s error? a. Matthew multiplied 10 by the exponent 3 b. Matthew multiplied 1.5 by the exponent 3 c. Matthew added 3 zeroes to the end of 1.5 d. Matthew’s statement is correct and contains no errors.”
• In Unit 7, Lesson 1, students demonstrate conceptual understanding of 5.NBT.7, as they use a decimal grid to solve a subtraction problem involving decimals. In the Independent Practice, Bachelor Level, Problem 2 states, “Use the decimal grid below to solve: $$0.81-0.16=$$ ?” 
• In Unit 8, Lesson 7, students develop conceptual understanding of 5.NF.4, as they create area models to multiply unit fractions. In the Independent Practice, Bachelor Level, Problem 3 states, “What is the area of a rectangle that is  $$\frac{1}{2}$$ yard long and $$\frac{3}{8}$$ yard wide? A 1 by 1 yard rectangle has been started for you below.” 

The instructional materials for Achievement First Mathematics Grade 5 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 Skill Fluency and Cumulative Review (Mixed Practice) components. 

The publisher states that the Skill Fluency component of the curriculum “addresses the skill, procedures and concepts that students must perform quickly and accurately in order to master a standard or a skill imbedded within a standard. Skill Fluency is delivered during a 10-minutes segment of a 90-minute period.” The Skill Fluency and Cumulative Review (Mixed Practice) components contain resources to support the procedural skill and fluency standard 5.NBT.5: Fluently multiply multi-digit whole numbers using the standard algorithm. 

The instructional materials develop procedural skill and fluency throughout the grade level. For example:

• In Unit 3, Lesson 4, Independent Practice, Bachelor Level, students estimate and connect partial products to the standard algorithm as they multiply a one-digit number by a three-digit number. Problem 3 states, “For each problem, make an estimate first. Then calculate the product using the standard algorithm and show your work. For number 3, list each of the partial products being calculated in order as shown in number 1. Use estimation to check the reasonableness of your product: $$464×5=$$ ____.”  (5.NBT.5) 
• In Unit 3, Lesson 8, students reflect upon and choose an appropriate strategy for multiplication. Think About It states, “We’ve studied several methods for multiplying in this unit and in previous grades, including mental math, the distributive property (with an area model or expression) and the standard algorithm. Look at each problem below and decide which of these strategies makes the most sense to use.” Students solve, “$$7×8$$, $$85×10$$, $$5×17$$, and $$422×329$$” (5.NBT.5) 
• In Unit 5, Mixed Practice 5.1, students develop procedural skill and fluency related to multiplication as they solve a word problem. Problem 3 states, “Over the course of fifteen days, a museum counts the number of guests that enter. They count an average of 2,362 people on each of the days. How many guests visited the museum altogether. Show your work. Answer ________.” (5.NBT.5)

The instructional materials provide opportunities for students to independently demonstrate procedural skill and fluency throughout the grade level. For example: 

• In Unit 3, Mixed Practice, 3.2, Day 2, students demonstrate procedural skill and fluency as they multiply multi-digit factors while solving a problem with a provided chart.  The materials state, “Rory, Elaina and Yashika are all on a reading marathon team. The time each girl reads each day is shown in the chart below. If each girl reads for 36 days, how many total minutes will they have read?” (5.NBT.5)
• In Unit 4, Skill Fluency 4.2, Day 2, students demonstrate fluency in multiplying multi-digit whole numbers using the standard algorithm. Problem 1 states, “Find the product of 736 and 92.” (5.NBT.5)  
• In Unit 7, Skill Fluency 7.1, Day 3, students demonstrate procedural skill and fluency with multiplication. Problem 3 states, “$$62×?=5952$$. Find the value of ?.” (5.NBT.5)

The instructional materials for Achievement First Mathematics Grade 5 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 7, Lesson 12, students engage with 5.NF.2 as they solve a word problem  involving addition and subtraction of fractions. Exit Ticket states, “Sheldon harvests the strawberries and tomatoes in his garden. He picks $$1\frac{2}{5}$$ kg fewer strawberries in the morning than in the afternoon. Sheldon picked $$2\frac{1}{4}$$ kg in the morning. How many kilograms of strawberries did Sheldon pick in the afternoon?”
• In Unit 9, Lesson 3, students engage with 5.NF.7 as they solve a real world problem involving division of unit fractions. In the Independent Practice, Bachelor Level states, “Virgil has $$\frac{1}{6}$$ of a birthday cake left over. He wants to share the leftover cake with 3 friends. What fraction of the original cake will each of the 3 people receive? Draw a picture to support your response.”
• In Unit 9, Lesson 11, students engage with 5.NBT.7 as they solve a word problem involving decimals to the hundreths. In the Independent Practice, Masters Level states, “A group of 14 friends collect 0.7 pounds of blueberries each and decide to make blueberry muffins. They put 0.05 pounds of berries in each muffin. How many muffins can they make if they use all the blueberries they collected?”

Achievement First Mathematics Grade 5 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 in the Interaction with New Material, Test the Conjecture, and Think About it components of the materials. For example: 

• In Unit 7, Lesson 11, students engage with 5.NF.1 as they add and subtract fractions with unlike denominators. Interaction with New Material states, “Victor is making a special enchilada dish for the Latin Heritage festival at his school. To make the dish, he needs a lot of fresh tomatillos. To make enough for 60 servings he needs $$12\frac{1}{2}$$ pounds of tomatillos. He finds $$5\frac{1}{4}$$ pounds at King’s Grocery and $$3\frac{3}{5}$$ pounds at Metropolitan Grocers. He decides to call a third store to see if they’ll have enough in stock. How much should he ask for?”
• In Unit 9, Lesson 3, students engage with 5.NF.7 as they apply and extend previous understanding of division to divide unit fractions by whole numbers and whole numbers by unit fractions. Think About It states, “Carmine and Miguel are working together on the following problem: Mrs. Silverstein is having a college graduation party for her son. She buys enough cake so that each guest at the party can have up to $$\frac{1}{6}$$ of a cake. She buys 3 cakes. How many guests is she expecting? Carmine writes the equation $$\frac{1}{6}÷3=\frac{1}{18}$$.  Miguel writes the equation $$3÷\frac{1}{6}=18$$. Is either student correct? Create a model to prove your thinking. Then explain your reasoning on the lines below.”

The instructional materials for Achievement First Mathematics Grade 5 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 4, Mixed Practice 4.1, students develop procedural skill and fluency as they solve problems involving multi-digit multiplication. Problem 3 states, “Find a 3-digit number and a 1-digit number that when multiplied together will result in a product between 3,000 and 4,000. Show your work.” (5.NBT.5) 
• In Unit 5, Cumulative Review, Problem of the Day, Day 2, students apply skills related to measurement conversions as they solve a routine problem. The materials state, “A city wants to install fencing around two new playgrounds. Playground A is 5 yards long and 25 feet wide. Playground B is 3 yards long and 27 feet wide. A) Which playground will require more fencing, and by how much? B) Fencing costs $15 per two feet. How much will it cost to put up fencing around both playgrounds?” (5.MD.1) 
• In Unit 7, Lesson 2, Independent Practice, Bachelor Level students develop conceptual understanding of adding and subtracting decimals to the hundredths as they use a hundreds grid to solve a problem. In Problem 3, students are shown a 100 grid with two rows of 10 filled in. The materials state, “Jonah added 0.36 to the value below and got 2.36. Is his answer reasonable? Why or why not? (Use the space to the right to explain.)” (5.NBT.7)

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

• In Unit 3, Lesson 4, Partner Practice, students develop conceptual understanding of place value and procedural skills and fluency as they solve a problem involving the standard algorithm, to find a product in a real world context. Problem 1 states, “Taliyah’s brother sells 654 gallons of cookie dough for $7 each How much money does her brother raise? a) Find the product using the distributive property and an area model. b) Use the standard algorithm to solve the multiplication problem.” (5.NBT.5)
• In Unit 7, Lesson 12. Independent Practice, Master Level, students develop conceptual understanding of fractions and apply skills related to addition and subtraction of fractions as they solve a problem and develop a model. Problem 1 states, “Directions: Create a model of both scenarios. Write an equation that could be used to find a solution in each scenario. Explain how the scenarios are similar and how they are different. Problem A: Jennah has one piece of string that is $$3\frac{1}{8}$$ meters long, and another that is $$3\frac{5}{10}$$ meter. How much longer is the longer string? Model: ____, Equation: ____ . Problem B: Jennah had a piece of string that was $$3\frac{5}{10}$$ meters long. She used $$3\frac{1}{8}$$  meters. How much string was left? Model: ____ Equation: ___. How are the problem scenarios mathematically similar? What is one important difference in the problem scenarios?” (5.NF.1, 5.NF.2) 
• In Unit 8, Lesson 18, Independent Practice, Masters Level, students apply their understanding of fractions as they solve problems involving multiplication of fractions and mixed numbers, and demonstrate procedural skill to add and subtract decimals to hundredths. Problem 1 states, “Oliver came home from the store with .250 L of heavy cream only to find that he needed $$1\frac{1}{3}$$ times that much for his recipe. How much more heavy cream does he need when he goes back to the store? Represent the problem with a model and an expression or equation. Then solve.” (5.NF.6, 5.NBT.7)

The instructional materials reviewed for Achievement First Grade 5 partially meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade-level. 

All Standards for Mathematical Practice are identified throughout the materials in the Unit Overviews and Lesson Overviews. However, in the Unit Overviews, all MPs are listed in each Unit Overview and some are bolded. There is not a rationale for why MPs are bolded and the materials do not include a connection as to how the MPs are demonstrated within the unit. In the Lesson Overview, MPs are identified in the Standards section. 

In addition, MP5 is only identified in one unit, Unit 11. There is also only one lesson within Unit 11 that specifically identifies MP5 in the Standards section of the Lesson Overview. 

Examples that the MPs are identified throughout the materials, with few or no exceptions, include:

• MP1: In Unit 5, Lesson 4, Standards, MP1, make sense of problems and persevere in solving them, is identified in this lesson as, “SWBAT solve conversion problems that involve measurements in mixed units.”
• MP2: In Unit 9, Unit Overview, Identify the Desired Results, Identify the Standards, lists all eight MPs in a table, MP2, reason abstractly and quantitatively, is bolded in this Unit Overview. Identify the Narrative states, “In the final lesson, students apply what they have learned during this unit as well as from prior units. In Lesson 12, students add, subtract, multiply, and divide decimals to solve real-world and mathematical problems. These problems may also require converting between measurement units using skills and strategies developed earlier in the year. Problems aligning to 5.NBT.7 often require multiple computations using different operations like those seen in this lesson.”
• MP4: In Unit 10, Unit Overview, Identify the Desired Results, Identify the Standards, lists all eight MPs in a table, MP4, model with mathematics, is bolded in this Unit Overview. Identify the Narrative states, “Students measure angles and create angles to meet given specifications. Simultaneously, students review or acquire important angle vocabulary (acute, right, and obtuse) that they will need to classify triangles.”
• MP5: In Unit 11, Unit Overview, Identify the Desired Results, Identify the Standards, lists all eight MPs in a table, MP5, use appropriate tools strategically, is bolded in this Unit Overview. Identify the Narrative states, “Lesson 2 builds directly on this understanding with applications in which students plot and locate points. When justifying the locations and/or coordinates of points, students explain that the first number in an ordered pair indicates how far to travel horizontally along the x-axis from the origin while the second number indicates how far to travel vertically along the y-axis from the origin. Scholars will apply this skill to represent and solve real world and mathematical problems on where a coordinate system might be useful (5.G.2) throughout early lessons and then more deeply in lesson 5.”
• MP6: In Unit 11, Lesson 3, Standards, MP6, attend to precision, is identified as being embedded in the lesson. The materials state, “SWBAT identify patterns in coordinate pairs that lead to vertical and horizontal lines, and interpret points on the plane as distances from the axes.”
• MP7: In Unit 6, Unit Overview, Identify the Desired Results, Identify the Standards, lists all 8 MPs in a table, MP7, look for and make use of structure, is bolded in this Unit Overview. Identify the Narrative states, “Students examine and articulate patterns that emerge between these properties of the prism and its volume, and look at the effects of changing the base or height on the total volume of the prism.”
• MP8: In Unit 11, Lesson 7, Standards, MP8, look for and express regularity in repeated reasoning, is identified as being embedded in the lesson. The materials state, “SWBAT generate two number patterns from given rules, plot the points, and analyze the patterns.”

The instructional materials reviewed for Achievement First Mathematics Grade 5 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 5, Lesson 5, Independent Practice, Master’s Level, students analyze information and select a strategy to solve a multi-step problem involving perimeter. Problem 4 states, “Mr. Rice needs to replace the 166 feet of fencing on the flower beds in his backyard. The fencing is sold in lengths of 5 yards each. How many lengths of edging will Mr. Rice need to purchase? Will he have any extra?”
• In Unit 6, Lesson 6, Think About It, students recognize what information is known in a problem in order to apply a formula accurately. The materials state, “The inside of Jackson’s desk where he stores his books and supplies has a total volume of 3200 cubic inches. The base area of the inside is 400 square inches. How tall is the inside of the desk?”  
• In Unit 8, Lesson 4, Independent Practice, PhD Level, students make sense of a problem by relating previously learned content about angles and connect this to understanding of fractions. Problem 2 states, “Three angles are labeled below with arcs. They form a complete circle of 360 degrees. The smallest angle is $$\frac{3}{8}$$ as large as the $$160\degree$$ angle. Find the value of angle a. Then explain how you know your solution is correct.”

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

• In Unit 2, Skill Fluency 2.2, students use a shaded decimal grid to identify a decimal and fraction. Problem 4 states, “Write the value as a decimal fraction and a fraction.”
• In Unit 10, Lesson 7, Interaction With New Material, students are provided a protractor to use to solve a problem involving angle measurement. The materials state, “Example 1: Haley runs a roller skating and skateboarding park. She asked her team to construct a new ramp with a 25-30 degree incline. Michael says the sketch to the right will not work, because the incline is roughly 150 degrees. On the lines below, answer both questions: A. Is Michael’s claim reasonable? Why or why not? B. What is the actual measure of the incline?” 
• In Unit 11, Lesson 3, Independent Practice, Masters Level, students use provided grids to solve a problem involving plotting points to construct a parallel line. Problem 3 states, “Write the coordinate pairs of 3 points that can be connected to construct a line that is $$5\frac{1}{2}$$ units to the right of and parallel to the y-axis. a. ____ b. ____ c. ____.”

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

• MP2: In Unit 5, Lesson 2, Independent Practice, Ph.D Level, students reason with provided quantities to solve a problem and explain their answer. Problem 1 states, “Ms. Jackson mixes a solution using 2,200 milliters of saline and 1,500 milliters of another liquid. Can she pour the mixture into a bottle that is 3.5 liters? Explain why or why not.”
• MP4: In Unit 9, Lesson 12, Independent Practice, Bachelor Level students create a model to solve a real life problem involving decimals. Problem 1 states, “Two wires, one 17.4 meters long and one 7.5 meters long, were cut into pieces 0.3 meters long. How many such pieces can be made from both wires? Create a model to represent the problem and solve it.”
• MP6: In Unit 4, Lesson 4, Exit Ticket, students attend to precision as they calculate the width of a space based on the area and provided length. Problem 2 states, “A 90 square foot bathroom has a length of 15 feet. It is rectangular in shape. What is the width of the bathroom?”
• MP7: In Unit 10, Lesson 6, Think About It, students look for structure as they classify quadrilaterals. The materials state, “Below each shape, list as many names as you can for the shape. Then, circle every name that they have in common.”
• MP8: In Unit 7, Lesson 9, Day 2, Independent Practice, Bachelor Level, students find the least common denominator as an efficient shortcut or additional subtraction strategy with fractions. Problem 3 states, “Madame Curie made some radium in her lab. She used $$\frac{15}{36}$$ kg of the radium in an experiment and had $$1\frac{1}{18}$$ kg left. Part A. How much radium did she have at first?”

The instructional materials reviewed for Achievement First Mathematics Grade 5 meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

The materials provide students with the opportunity to critique the work of others and engage them in extending their thinking to justify their responses. Examples of opportunities for students to construct viable arguments and/or critique the reasoning of others include:

• In Unit 4, Lesson 7, Partner Practice, Masters Level, students critique the reasoning of others and construct an argument based on their knowledge of division. Problem 1 states, “Paul divided 8,280 by 36 and got 23. Do you agree or disagree? Prove your thinking and explain in the space below.”
• In Unit 5, Lesson 3, Independent Practice, Bachelor Level, students construct a viable argument and critique the reasoning of others based on their knowledge of multiplication. Problem 3 states, “The Town of Andover has an annual road running race that is 5,000 meters long. Alexis and Keith did mental math to determine the length of the race in kilometers. Which runner’s thinking is correct? Explain. Alexis’s Thinking $$5000 × 1000 = 5,000,000$$ km, Keith’s Thinking $$5000÷1000 = 5$$ km.”
• In Unit 6, Lesson 2, Independent Practice, Bachelor Level, students critique the reasoning of others and construct an argument based on their knowledge of shapes. Problem 7 states, “Tyler builds the shape below and then turns it on its side. He says that the figure takes up less space now because it is shorter. Do you agree or disagree with his claim and why?” 
• In Unit 7, Lesson 11, Day 2, Partner Practice, Bachelor Level, students construct an argument based on their knowledge of fractions. Problem 1 states, “Which of the following differences will require regrouping to solve? $$1\frac{1}{3} -\frac{1}{2}$$  OR $$1\frac{1}{2} -\frac{1}{3}$$  Explain how you know without doing any calculations.” 
• In Unit 8, Lesson 17, Day 2, Exit Ticket, students critique the reasoning of others as they use estimation to assess the reasonableness of an answer. Problem 2 states, “Tyler multiplies 3.1 and 4.2. He gets a product of 130.2. Using estimation as your evidence explain if his product is reasonable or unreasonable and what his mistake might have been.” 
• In Unit 10, Lesson 9, Interaction With New Material, students critique the reasoning of others as they classify triangles. The materials state, “Ms. Cox’s class is analyzing the two figures below. Mya says that they can be given the same name. Justin says the shapes have different names. Ms. Cox says that both students are correct. Part A. How is it possible that both students are correct? Explain your reasoning. Part B. What is the most specific name that can be given to each triangle? Justify your response.”

The instructional materials reviewed for Achievement First Mathematics Grade 5 meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. The materials provide teachers guidance to engage students in both constructing viable arguments and analyzing the arguments of others across the mathematics of the grade-level. For example: 

• In Unit 3, Lesson 3, guidance is provided to teachers in the form of a think-aloud exemplar to help students compare fictional students’ strategies. Think About It, Debrief states, “Which of these strategies was better in this problem? Using compatible numbers was better because they were closer to the actual numbers, so the estimate will be much closer to the real cost.” 
• In Unit 4, Lesson 2, Day 2, Debrief, provides teachers with guidance to support engaging students in analyzing the work of others in an error analysis problem.
  The materials state, “See the error: Which student’s work did you DISAGREE with most? Vote. Turn and tell your partner who you chose and why. Revote. CC. I see you disagree with student A. Why do you disagree with student A? SMS: I disagree with student A’s work because 500 and 70 are not compatible numbers. You can’t divide 500 by 70 mentally, it doesn’t include a basic fact. BPQ: What is missing from Student A’s number sentence that would make it possible to mentally divide? SMS: A basic fact they are not compatible numbers (NOTE: Give students the vocab compatible numbers as values that include a basic fact making them easy to compute mentally, if they do not remember from yesterday.) BPQ: Why is it important to round to compatible numbers when estimating quotients? SMS: Estimation is supposed to be done mentally and if you don’t round to compatible numbers that are a basic fact, it is hard to compute mentally. What error did this student make? CC. SMS: S/he rounded to the highest place value instead of to compatible numbers that included a basic fact.Let’s cross this one off.” 
• In Unit 8, Lesson 5, guidance is provided to teachers in the form of language to support them in helping students compare fictional student’s work and critique. Talk About It states, “Which do you agree with ...Vote...You agree with scholar B, which means you disagree with scholar A. Why do we disagree with Scholar A?” 
• In Unit 11, Lesson 5, Day 2, the materials guide the teacher to support engaging students in analyzing the reasoning of others in a real world word problem. The Debrief states, “See the Error: Which student’s explanation did you agree with? Vote. Turn and tell your partner who you chose and why. (Listening for exemplar understandings during TT). Revote. CC. SMS: Student B is correct. Both students chose the correct ordered pair, but Student B read the label for the x-axis carefully and saw that it did not show hours but instead showed actual times of day, so you had to calculate how many hours there are between 6 and 12.  What error did this student make? CC. SMS: Student A assumed that the coordinate was the number of hours that passed not the time that it was.”

The instructional materials reviewed for Achievement First Mathematics Grade 5 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 2, guidance is available for teachers to provide explicit instruction on the term compatible numbers. The debrief states, “We call numbers like 240 and 20 ‘compatible numbers’ in a division expression because together they contain basic facts that are easy to compute mentally, which is ONE requirement of estimation. But is this a valid method? How do you know? TT. CC. SMS: It is a valid method because it leads to an estimate that is close to the actual answer. The actual quotient is 7.219 when I use a calculator, and our estimate is 8 so that’s definitely in the same ballpark/reasonable.”
• In Unit 8, Lesson 12, provides teachers with guidance to introduce the new vocabulary term scaling. Think About It states, “When we want to increase or decrease an amount by a certain factor, it is called ‘scaling.’ You have probably solved problems before that asked you to find $$1\frac{1}{2}$$ times as much, or $$\frac{3}{4}$$ of an amount. The number we multiply by when doing this is called a ‘scale factor.’”
• In Unit 10, Lesson 8, provides strategies for teachers to use in guiding students to use precise vocabulary when classifying triangles. The debrief states, “Using the precise words for angles less than, equal to, or greater than 90, what name could we give each group, and why? TT. CC. SMS: Acute, Right, and Obtuse, because group 1 has only acute angles, group 2 has a right angle, and group 3 has an obtuse angle.What attribute are we using to sort these? CC. SMS: The measure of the angles.”

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

• In Unit 3, Lesson 2, Independent Practice, Bachelors Level, accurate terminology is used as students identify expressions. Problem 2 states, “Which expression represents twice the product of 15 and 4? Circle all that apply. a. $$2 + (15 × 4)$$ b. $$2 × (15 × 4)$$ c. $$2 × (15 + 4)$$ d. 62 e. 120.”
• In Unit 5, Lesson 4, Independent Practice, Bachelors Level, students are expected to understand and use accurate terminology as they solve a division problem and explain their answer. Problem 5 states, “Myra converted 5,300 feet into miles using the correct expression $$5,300÷5,280$$. She got a correct answer of 1 R20. What does the 1 in her quotient represent? What does the 20 represent? Explain.”
• In Unit 8, Lesson 7, Exit Ticket, accurate terminology is used as students create an area model to solve a problem. Problem 1 states, “What is $$\frac{1}{2}$$ of $$\frac{2}{5}$$? Part A. Use an area model to solve. Part B. Why does it make sense that your product is less than $$\frac{2}{5}$$?”