## Achievement First Mathematics

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### Overall Summary

The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for Alignment to the CCSSM. In Gateway 1, the materials meet expectations for focus and coherence, and in Gateway 2, the materials meet expectations for rigor and practice-content connections.

###### Alignment
Meets Expectations
###### Usability
Does Not Meet Expectations

### Focus & Coherence

The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for focus and coherence. For focus, the materials assess grade-level content and provide all students extensive work with grade-level problems to meet the full intent of grade-level standards. For coherence, the materials are coherent and consistent with the CCSSM.

##### Gateway 1
Meets Expectations

#### Criterion 1.1: Focus

Materials assess grade-level content and give all students extensive work with grade-level problems to meet the full intent of grade-level standards.

The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for focus as they assess grade-level content and provide all students extensive work with grade-level problems to meet the full intent of grade-level standards.

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

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

The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for assessing grade-level content and, if applicable, content from earlier grades. Each unit contains a Post-Assessment which is a summative assessment based on the standards designated in that unit.

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

• Unit 1, Common Core, Item 23, “What is 43.98 rounded to the nearest tenths place?” (5.NBT.4)

• Unit 4, Unit Assessment, Item 3, “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)

• Unit 5, Unit Assessment, Item 1, “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)

• Unit 8, Post-Assessment, Item 4, “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.

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

Materials give all students extensive work with grade-level problems to meet the full intent of grade-level standards.

The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards.

Each lesson provides State Test Alignment practice, Exit Tickets, Think About It, Test the Conjecture or Exercise Based problems, Error Analysis, Partner Practice, and Independent Practice, which all include grade-level practice for all students. The Partner and Independent Practice provide practice at different levels: Bachelor, Masters and PhD. Each unit also provides Mixed Practice, Problem of the Day, and Skill Fluency practice. By the end of the year, the materials address the full intent of the grade-level standards. Examples include:

• Unit 1, Lesson 6, Day 2 Independent Practice Question 2 (Master Level), students explain the pattern in the placement of the decimal point when a decimal is divided by a power of ten. “How many powers of ten would you need to divide 4,700 by to get a result of forty-seven thousandths? Prove your thinking in the space below.” (5.NBT.2)

• Unit 3, Lesson 2, Mixed Practice, Problem 5, students multiply multi-digit whole numbers. “A petroleum company has 14 large barrels full of oil that they sell to local gas stations. All 14 barrels hold approximately 315 gallons of oil, and a gas station will typically buy 500 gallons of oil each time they order. The owner of the company estimates that they have around 4500 gallons of oil, so they allow 9 gas stations to buy oil. a. Explain how the owner likely found the estimate of 4500. b. Does the petroleum company have enough oil to allow 9 gas stations to buy oil? If not, how many more barrels of oil would they need to produce?” (5.NBT.5)

• Unit 7, Lesson 4, Exit Ticket, students estimate sums and differences of fractions with unlike denominators. “Kendra made the following statements while estimating. Determine whether you agree with each, and mark yes or no. 0.33 + \frac{9}{10} is about 1\frac{1}{2}; 1\frac{4}{5} - \frac{3}{7} is approximately 2 - \frac{1}{2}” (5.NF.1, 5.NF.2).

• Unit 8, Lesson 4, Mixed Practice, Day 3, Problem 2, students divide decimals using place value strategies (5.NBT.7). “Alyssa had $0.70. She put $$\frac{1}{10}$$ of it in her penny jar to save. How much money did she save? a) .07 cents b) 7 cents c) 70 cents d) 7 dollars.” #### Criterion 1.2: Coherence Each grade’s materials are coherent and consistent with the Standards. The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for coherence. The materials: address the major clusters of the grade, have supporting content connected to major work, make connections between clusters and domains, and have content from prior and future grades connected to grade-level work. ##### Indicator {{'1c' | indicatorName}} When implemented as designed, the majority of the materials address the major clusters of each grade. The materials reviewed for Achievement First Mathematics Grade 5 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each grade. • The number of lessons devoted to major work of the grade (including assessments and supporting work connected to the major work) is 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 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 materials focus on major work of the grade. ##### Indicator {{'1d' | indicatorName}} Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade. The materials reviewed for Achievement First Mathematics Grade 5 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade. There are opportunities in which supporting standards/clusters are used to support major work of the grade and are connected to the major standards/clusters of the grade. Examples include: • Unit 3, Lesson 2, Independent Practice, Ph.D Level Problem 1, “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. • Unit 5, Lesson 5, Exit Ticket, Problem 2, “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. • Unit 10, Cumulative Review 10.1, Problem of the Day, Day 3, “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 ##### Indicator {{'1e' | indicatorName}} Materials include problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade. The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for including problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade. Examples include: • Unit 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, “Bernard is packing a box with a volume of 96 cubic inches. Enter a possible base area and height for his box below.” • 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.” • 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. “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?” • 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. Independent Practice, Bachelors Level, “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.” ##### Indicator {{'1f' | indicatorName}} Content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades. The materials reviewed for Achievement First Mathematics Grade 5 meet expectations that content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades. Each unit has a Unit Overview and a section labeled “Identify Desired Results” where the standards for the unit are provided as well as a correlating section “Previous Grade Level Standards/Previously Taught & Related Standards” where prior grade-level standards are identified. Examples include: • Unit 2, Unit Overview, Identify Desired Results: Identify the Standards lists 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. “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).” • Unit 6, Unit Overview, Identify the Narrative connects the work of this unit to prior work in 3rd and 4th grades. “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 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: • Unit 2, Overview, Identify the Narrative, “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. 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).” • Unit 4, Unit Overview, Identify the Narrative, “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).” • Unit 7, Unit Overview, Identify the Narrative, “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.” ##### Indicator {{'1g' | indicatorName}} In order to foster coherence between grades, materials can be completed within a regular school year with little to no modification. The instructional materials reviewed for Achievement First Mathematics Grade 5 foster coherence between grades and can be completed within a regular school year with little to no modification. 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). ###### Overview of Gateway 2 ### Rigor & the Mathematical Practices The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for rigor and balance and practice-content connections. The materials reflect the balances in the Standards and help students develop conceptual understanding, procedural skill and fluency, and application. The materials make meaningful connections between the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs). ##### Gateway 2 Meets Expectations #### Criterion 2.1: Rigor and Balance Materials reflect the balances in the Standards and help students meet the Standards’ rigorous expectations, by giving appropriate attention to: developing students’ conceptual understanding; procedural skill and fluency; and engaging applications. The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for rigor. The materials develop conceptual understanding of key mathematical concepts, give attention throughout the year to procedural skill and fluency, spend sufficient time working with engaging applications of mathematics, and do not always treat the three aspects of rigor together or separately. ##### Indicator {{'2a' | indicatorName}} Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. The instructional materials for Achievement First Mathematics Grade 5 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. The materials include problems and questions that develop conceptual understanding throughout the grade level. Examples include: • Unit 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. Partner Practice, Problem 1, “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.”

• 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 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.”

• 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 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. Examples include:

• 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, “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.”

• 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. Independent Practice, Bachelor Level, Problem 2, “Use the decimal grid below to solve: 0.81-0.16=?”

• Unit 8, Lesson 7, students develop conceptual understanding of 5.NF.4, as they create area models to multiply unit fractions. Independent Practice, Bachelor Level, Problem 3, “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.”

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

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

The materials for Achievement First Mathematics Grade 5 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. The materials include opportunities for students to build procedural skill and fluency in both 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 materials develop procedural skill and fluency throughout the grade level. Examples include:

• 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, “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)

• Unit 3, Lesson 8, students reflect upon and choose an appropriate strategy for multiplication. Think About It, “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)

• Unit 5, Mixed Practice 5.1, students develop procedural skill and fluency related to multiplication as they solve a word problem. Problem 3, “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 materials provide opportunities for students to independently demonstrate procedural skill and fluency throughout the grade level. Examples include:

• 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. “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)

• Unit 4, Skill Fluency 4.2, Day 2, students demonstrate fluency in multiplying multi-digit whole numbers using the standard algorithm. Problem 1, “Find the product of 736 and 92.” (5.NBT.5)

• Unit 7, Skill Fluency 7.1, Day 3, students demonstrate procedural skill and fluency with multiplication. Problem 3, “$$62 × ? = 5952$$. Find the value of ?.” (5.NBT.5)

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

Materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Students are given multiple opportunities to engage in real-world applications especially within exercise based lessons as well as the problem of the day in each cumulative review.

Materials include multiple routine and non-routine applications of the mathematics throughout the grade level. Examples include:

• Unit 2, Lesson 1, Mixed Practice Day 2, Problem 5, students solve a real-world, non-routine problem by comparing two decimals to thousandths based on meanings of the digits in each place. (5.NBT.3) "Noah threw a Frisbee 4.89 yards. a) Noah threw the Frisbee farther than Lin. How far could Lin have thrown the Frisbee? b) Andre threw the Frisbee farther than Noah but less than 4.9 yards. How far could Andre have thrown the Frisbee? Explain your reasoning.”

• Unit 3, Lesson 3, Problem of the Day, Day 1, students write and interpret an expression then solve a routine real-world problem involving multiplying multi-digit whole numbers using the standard algorithm (5.NBT.5). "Use the chart to solve. (note: the chart shows minimum and maximum length and width of High School and FIFA Regulation Soccer Field Dimensions) a. Write an expression to find the difference in the maximum area and minimum area of NYS high school soccer fields. Then, evaluate your expression. b. Would a field with a width of 75 yards and an area of 7,500 square yards be within FIFA regulation? Explain why or why not.”

• Unit 7, Lesson 11, Interaction with New Material, students engage in a routine problem with 5.NF.1 as they add and subtract fractions with unlike denominators. "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?”

• Unit 9, Lesson 3, Think About It, 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 in a non-routine problem. Think About It, “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.”

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

• Unit 6, Lesson 10, Independent Practice Question 1 (Bachelor Level), students apply the volume formula (5.MD.5) and convert among different-sized standard measurement units within a given measurement system (5.MD.1) in the context of solving a non-routine real-world problem. "At the flea market, a shopper asks Geoffrey if it is possible to use his 3 foot by 2 foot by 2-foot large planter as a bookcase or storage instead. Geoffrey considers this and estimates that a typical book has a volume of about 40 cubic inches. How many books would a large planter hold if filled with as many books as possible?”

• Unit 9, Lesson 3, Independent Practice Question 2 (Bachelor Level), students engage with 5.NF.7 as they solve a routine real-world problem involving division of unit fractions. "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.”

• Unit 11, Lesson 5, Real World Problems, students represent routine real-world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation (5.G.2). "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 x 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.” ##### Indicator {{'2d' | indicatorName}} The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade. The materials reviewed for Achievement First Mathematics Grade 5 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade. The instructional materials include opportunities for students to independently demonstrate the three aspects of rigor. Examples include: • Unit 4, Mixed Practice 4.1, students develop procedural skill and fluency as they solve problems involving multi-digit multiplication. Problem 3, “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) • Unit 5, Cumulative Review, Problem of the Day, Day 2, students apply skills related to measurement conversions as they solve a routine problem. “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)

• 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. Problem 3, students are shown a 100 grid with two rows of 10 filled in. “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. Examples include:

• 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, “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)

• 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, “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)

• 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, “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)

#### Criterion 2.2: Math Practices

Materials meaningfully connect the Standards for Mathematical Content and Standards for Mathematical Practice (MPs).

The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for practice-content connections. The materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs). However, there is no intentional development of MP5 to meet its full intent in connection to grade-level content.

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

Materials support the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. The Standards for Mathematical Practice are identified and incorporated within mathematics content throughout the grade level. The Mathematical Practices are listed in the Unit Overviews as well at the beginning of each lesson. There are instances where the Unit Overview gives a detailed explanation of the MPs being addressed within the unit, but the lessons do not cite the same MPs.

There is intentional development of MP1 to meet its full intent in connection to grade-level content. Examples include:

• The Unit 6 Overview outlines the intentional development of MP1. “In lesson 6, students work to make sense of problems by identifying unknowns in various volume-related contexts. In lessons 9 and 10, students make sense of complex volume problems in various contexts, persevering to properly formulate solution pathways and solutions.”

• Unit 7, Lesson 12, Partner Practice Question 1 (Bachelor Level), students make sense of how equations connect to verbal descriptions. “Which equation or equations can be used to represent the following: Team A built a tower that was 1\frac{1}{2} feet taller than Team B’s. Team B’s tower was 3\frac{1}{5} tall. How tall is Team A’s tower? Create a model, and then circle all equations that apply. a) 1\frac{1}{2} - \frac{1}{5} =?; b) 1\frac{1}{2} + ? = 3\frac{1}{5}; c) 3\frac{1}{5} - 1\frac{1}{2} =?; d) 3\frac{1}{5} + 1\frac{1}{2} = ?”

• The Unit 9 Overview describes development of MP1. “In both lessons 5 and 12, students extend their understandings of division by making sense of and persevering in solving multi-step problems in real world contexts. In lesson 5 students do this with the division of fractions, in lesson 12 with all operations of decimals.”

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

• Unit 4, Lesson 8, Interaction With New Material, students reason abstractly and quantitatively when working with dimensions of a room. “The owner of the art gallery knows that his rectangular space is 1500 square feet. The width of the space is 60 feet. In order to plan for a new exhibit, he needs to know the full perimeter of the gallery. Help him find it in the space below.”

• Unit 7, Lesson 13 Check for Understanding, students engage with MP2 as they solve a real world problem involving addition and subtraction of fractions with different denominators. “John got directions to his new high school for orientation day. He knows the school is 9 miles away. When he pulls the directions from his pocket, some of the last step was rubbed off. ~Take Atlantic Ave. 3\frac{1}{4} miles. Turn right on 17th. ~Go 2\frac{1}{8} miles. The road becomes Jefferson Ave. ~Take Jefferson Ave .... If the school is on Jefferson Ave., how many miles should John be on Jefferson Ave.? Draw a model and write an equation to represent the scenario.”

• The Unit 8 Overview outlines the intentional development of MP2. “In lesson 1-3, students relate fractions to division in different contexts. By de- and re-contextualizing fractions in these contexts students reason abstractly and quantitatively about the situations. In lessons 16 and 17, students reason quantitatively about the placement of their decimal in a product based on their estimates.”

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

Materials support the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

There is intentional development of MP3 to meet its full intent in connection to grade-level content. Examples include:

• The Guide to Implementing AF Math describes Error Analysis lessons as one way to address MP3. “Purpose: Through the use of error analysis, guided questioning and discussion students will identify and fix a common misconception related to a skill they learned the previous day. These are sequenced so that after a particularly complex conceptual lesson or a lesson involving a skill that surfaces a common misconception, students get another focused at bat to either fix their misunderstanding or deepen their reasoning around key mathematical concepts and viable strategies to guide them away from making the same error. These lessons start with analyzing fictional student work and are structurally based off of the Standards for Mathematical Practice 3.”

• 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, “Paul divided 8,280 by 36 and got 23. Do you agree or disagree? Prove your thinking and explain in the space below.”

• 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, “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?”

• Unit 7, Lesson 11, Day 2, Partner Practice, Bachelor Level, students construct an argument based on their knowledge of fractions. Problem 1, “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.”

• 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, “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.”

• Unit 10, Lesson 9, Interaction With New Material, students critique the reasoning of others as they classify triangles. “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.”

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

Materials support the intentional development of MP4: Model with mathematics; and MP5: Choose tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials reviewed for Achievement First Mathematics Grade 5 partially meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Choose tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. Students are provided with occasions to develop their own task pathways, but have limited opportunities to choose tools.

There is intentional development of MP4 to meet its full intent in connection to grade-level content. Examples include:

• Unit 5, Lesson 1, Test the Conjecture Question 2, students engage in MP4 to solve a real-world problem. “Alex needs 5,100 milliliters of distilled water to create homemade liquid soap. Distilled water is sold by the liter, so he buys 5.5 liters. Does he have enough? Show your work.”

• The Unit 9 Overview identifies MP4 as “a major focus of Unit 9 as students utilize modeling to establish their conceptual understanding of division of fractions and decimals. In lessons 1-4, students build the foundation of their division with fraction understanding with tape diagrams and other models.”

• Unit 9, Lesson 1, Think About It, students engage with MP4 as they use a model or diagram to divide a whole number by a fraction. “Jessiah has two feet of cord for making bracelets. He will cut it into pieces of equal length to make all bracelets the same size. He has two different options for how long to cut each piece. Create a model or diagram to determine the number of bracelets he can make with two feet of cord. Option A. \frac{1}{2} foot of cord per bracelet. 2 ÷ \frac{1}{2} =?; Option B. \frac{1}{3} foot of cord per bracelet. $$2 ÷ \frac{1}{3} =$$?”

• Unit 9, Lesson 12, Independent Practice, Bachelor Level students create a model to solve a real-life problem involving decimals. Problem 1, “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.”

There is no intentional development of MP5 to meet its full intent in connection to grade-level content. Examples include:

• MP5 is identified for Units 10 and 11, so there is very limited exposure to the practice.

• Lack of intentional development of MP5 is seen in misaligned identification in the Unit Overviews and lessons. The Overview for Unit 11 identifies MP5 in all lessons, but within the lessons, MP5 is labeled once in Lesson 4 when students use a coordinate grid.

• Students are rarely given choice in tools to solve problems. Unit 11, Lesson 4, students use ordered pairs to construct and name shapes on the coordinate grid. The materials list has a handout and a protractor and the students are given a pre-numbered and pre-labeled coordinate grid for each problem. There is no opportunity to choose a tool to solve the problems.

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

Materials attend to the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

There is intentional development of MP6 to meet its full intent in connection to grade-level content. Many problems present students with the opportunity to attend to precision within the mathematics and the reasoning of the answer. Examples include:

• The Unit 4 Overview, "In lessons 3-7 students attend to precision when they are transitioning their preferred division strategy to the written methods. This requires students to make connections to more concrete representations and to keep place values of quotients in order. MP 6 is a major focus of unit 4 as students must precisely identify place values when dividing throughout the unit."

• The Unit 6 Overview, “In lesson 1, students attend to precision by intentionally learning the proper expression of cubic units in volume and the why behind these three dimensions. This understanding is expanded upon in lesson 4 when students learn a different way of breaking down a 3D structure into layers. Finally in lesson 7, students learn of the additive nature of volume and how to define their units when adding multiple structures together. Students often attend to precision in lessons, however, in Unit 6, MP6 is specifically emphasized with a focus on the identification and tracking of proper units in volume contexts.”

• Unit 10, Lesson 8 provides strategies for teachers to use in guiding students to use precise vocabulary when classifying triangles. The debrief, “Using the precise words for angles less than, equal to, or greater than 90, what name could we give each group, and why?” Students might say, “Acute, Right, and Obtuse, because group 1 has only acute angles, group 2 has a right angle, and group 3 has an obtuse angle.”

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

• At the beginning of each lesson plan, there is a section labeled “Key Vocabulary” for the teacher. Unit 2, Lesson 2, Key Vocabulary,

• “Denominator – The bottom number in a fraction; shows the number of parts in the whole

• Numerator – The top number in a fraction; shows how many parts of the whole are being described

• Equivalent fraction – A fraction with the same value as another fraction but with different numerators and denominators.”

• Unit 3, Lesson 2, Independent Practice, Question 2 (Bachelors Level), accurate terminology is used as students identify expressions. “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.”

Unit 5, Lesson 4, Independent Practice, Question 5 (Bachelors Level), students are expected to understand and use accurate terminology as they solve a division problem and explain their answer. “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.”

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

Materials support the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards.

There is intentional development of MP7 to meet its full intent in connection to grade-level content. Examples Include:

• Unit 2, Lesson 4, Independent Practice, Question 1 (Bachelor Level), students engage with MP7 as they compare and order fractions. “Review: Compare the pairs of fractions by reasoning about the size of the units. Use >, <, or =. a) 1 fourth _____ 1 fifth; b) 3 fourths _____ 3 fifths; c) 1 tenth _____ 1 twelfth; d) 7 tenths _____ 7 twelfths.”

• Unit 10, Lesson 6, Think About It, students look for structure as they classify quadrilaterals. “Below each shape, list as many names as you can for the shape. Then, circle every name that they have in common.”

• Unit 11, Lesson 1, Exit Ticket, students interpret the structure of the coordinate plane as they construct a coordinate plane and use it to name the location of points. “Use a ruler on the grid below to construct the axes for a coordinate plane. The x-axis should intersect points L and M. Construct the y-axis so that it contains points K and L. Label each axis. a) Place a hash mark on each grid line on the x- and y-axis. b) Label each hash mark so that A is located at (1, 1). c) What are the coordinates of point M? d) What is point L called?”

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

• Unit 2, Lesson 1, Independent Practice, Question 2 (Bachelor Level), students repeatedly create equivalent fractions and make connections to multiplying and dividing by fractions equal to one. “Generate four fractions that have the same value as the fraction \frac{2}{5} . Show your work and record your answers on the line below.”

• Unit 7, Lesson 9, Day 2, Independent Practice, Question 3 (Bachelor Level), students find the least common denominator as an efficient shortcut or additional subtraction strategy with fractions. “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?”

Unit 10, Lesson 3, Test the Conjecture, Question 2, students use repeated reasoning to make sense of polygons by classifying quadrilaterals based on the presence of parallel sides. “True or false, a quadrilateral is always a trapezoid.”

### Usability

The materials reviewed for Achievement First Mathematics Grade 5 do not meet expectations for Usability. The materials partially meet expectations for Criterion 1, Teacher Supports, partially meet expectations for Criterion 2, Assessment, and do not meet expectations for Criterion 3, Student Supports.

##### Gateway 3
Does Not Meet Expectations

#### Criterion 3.1: Teacher Supports

The program includes opportunities for teachers to effectively plan and utilize materials with integrity and to further develop their own understanding of the content.

The materials reviewed for Achievement First Mathematics Grade 5 partially meet expectations for Teacher Supports. The materials: provide teacher guidance with useful annotations and suggestions for enacting the materials, include standards correlation information that explains the role of the standards in the context of the overall series. The materials do not provide a comprehensive list of supplies needed to support instructional activities. The materials contain adult-level explanations and examples of the more complex grade-level concepts, but do not contain adult-level explanations and examples and concepts beyond the current grade so that teachers can improve their own knowledge of the subject. The materials provide explanations of the instructional approaches of the program but do not contain identification of the research-based strategies.

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

Materials provide teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development. Teacher guidance is found throughout the materials in the Implementations Guides, Unit Overviews, and individual lessons.

Materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials. Examples include:

• The Guide to Implementing AF Math provides a Program Overview for the teacher with information on the program components and scope and sequence. This includes descriptions of the types of lessons, Skill Fluency, Mixed Practice, and Problem of the Day.

• The Teacher’s Guide supports whole group/partner discussion, ask/listen fors, common misconceptions and errors, etc.

• Each lesson includes a table identifying the steps and actions for the teacher which helps in planning the lesson and is intended to be reviewed with a coach.

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. Each lesson includes anticipated challenges, misconceptions, key points, sample dialogue, and exemplar student responses. Examples from Unit 6, Measurement and Data-Volume, Lesson 5 include:

• “What do we want every student to take away or do as a result of this lesson? How will a teacher know if students have met this goal? As a result of this lesson, every student can determine the volume of a prism using one of two formulas and given dimensions.”

• “Anticipated Misconceptions: Scholars may struggle to identify the given dimensions as a base area, length, width, or height. This will be particularly true if the area of the “base” is not given as the bottom or top layer, but rather the side. Students will need to rotate their perspective (but practiced this in L4). If students do not have a solid understanding of how the base area is derived, they may struggle to make connections between the formulas and understand why B and lw are interchangeable.”

• “Debrief: Show student work that created 3 layers of 8 cubes. How are the students' methods similar? Student might say, We think Jade’s method is correct because she found that there were 8 cubes in one layer and multiplied by 3 layers. Michael did the same thing, but he multiplied the length and width to find the number of cubes in one layer and then multiplied by 3 layers [Planner’s note: Scholars must be able to articulate a clear connection between the strategies].”

• Test the Conjecture provides multiple prompts to help teachers guide students through the problem. “What is the question asking us to do? How can we apply our conjecture to solve this problem? Is the height given, and if so, what is it, how do you know? We can use your conjecture to write a formula and evaluate. [Write v = Bh]. What do we substitute for B and h? Talk to your partner, and solve. How can we prove that our conjecture worked?”

Each lesson includes a “How” section that lists the key strategies of the lesson and delineates what “top quality” work should include. Examples from Unit 6, Measurement and Data-Volume, Lesson 5 include:

• “Key Strategy: To apply the volume formulas: Identify whether the base area or the length and width are given. Determine the height. Select a formula and substitute known dimensions in.”

• “CFS (Criteria for Success) for top quality work (generating equivalent fractions): Annotate and label given dimensions using their variables. The formula is written before substitutions are made. Work is shown for substitution and computation. The answer is recorded with the correct units.”

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

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

The materials reviewed for Achievement First Mathematics Grade 5 partially meet expectations for  containing adult-level explanations and examples of the more complex grade/course-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject. There is very little reference or support for content in future courses.

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

• Unit Overviews provide thorough information about the content of the unit which often includes definitions of terminology, explanations of strategies, and the rationale about incorporating a process. Unit 8 Overview, “In lessons 12 and 13 students explain multiplication as scaling (5.NF.5). They rationalize: 1) Why multiplying by a second factor greater than 1 results in a product that is greater than the first factor; and 2) why multiplying a second factor less than 1 results in a product that is less than the first factor. This is the middle school mathematician’s first introduction to the concept of a fractional scale factor, which is an important concept throughout middle school math, and reappears later in this unit as “scaling factors” are used to convert between units of measure. In lesson 12, students develop this concept with fractions. They see leverage patterns in products as well as number sense from earlier in the unit to explain that multiplication by a number less than one can never result in a number bigger than the second factor (i.e. \frac{3}{4} of 6 can never be more than 6, because it is just 3 of the 4 parts of the whole). At the same time, multiplying by a number greater than 1 will always result in a product bigger than the second factor. In lesson 13 students explain and prove that the same is true with decimal fractions. For now, the concept of scaling will be key to developing number sense for multiplying and dividing by rational numbers, estimating products containing fractions and decimal fractions, and solving problems involving “x times as many.”

• Some Unit Overviews include background knowledge for the teacher at the end of the file. The sources include Envision by Pearson, EngageNY, the progression documents, and several listed as “author and source unknown.” The Unit 4 Overview includes, “Multi-digit division requires working with remainders. In preparation for working with remainders, students can compute sums of a product and a number, such as 4\times8+3. In multi-digit division students will need to find the greatest multiple less than a given number.  For example, when dividing by 6, the greatest multiple of 6 less than 50 is 6\times8=48.  Students can think of these ‘greatest multiples’ in terms of putting objects into groups. For example, when 50 objects are shared among 6 groups, the largest whole number of objects that can be put into each group is 8 and there are 2 objects left over.”

• The Unit Overview includes an Appendix titled “Teacher Background Knowledge” which includes a copy of the relevant pages from the Common Core Math Progression documents which includes on grade-level information.

Materials rarely contain adult-level explanations and examples of concepts beyond the current course so that teachers can improve their own knowledge of the subject. Examples include:

• The Common Core Math Progression documents in the Appendix are generally truncated to the current grade level and do not go beyond the current course. At times, they may reference how the content connects to the next grade.

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

Materials include standards correlation information that explains the role of the standards in the context of the overall series.

The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.

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

• Guide to Implementing AF Grade 5, Program Overview, “Scope and Sequence Detail is designed to help teachers identify the standards on which each lesson within a unit is focused, whether on grade level or not. You will find the daily lesson aims within each unit and the content standards addressed within that lesson. A list of the focus MPs for each lesson and unit and details about how they connect to the content standards can be found in the Unit Overviews and daily lesson plans.”

• The Program Overview informs teachers “about how to ensure scholars have sufficient practice with all of the Common Core State Standards. Standards or parts thereof that are bolded are addressed within a lesson but with limited exposure. It is recommended that teachers supplement the lessons addressing these standards by using the AF Practice Workbooks to ensure mastery for all students. Recommendations for when to revisit these standards during Math Practice and Friday Cumulative Review are noted in the Practice section of each unit.”

• The Unit Overview includes a section called Identify Desired Results: Identify the Standards which lists the standards addressed within the unit and previously addressed standards that relate to the content of the unit.

• In the Unit Overview, the Identify The Narrative provides rationale about the unit connections to previous standards for each of the lessons. Future grade-level content is also identified.

• The Unit Overview provides a table listing Mathematical Practices connected to the lessons and identifies whether the MP is a major focus of the unit.

• At the beginning of each lesson, each standard is identified.

• In the lesson overview, prior knowledge is identified, so teachers know what standards are linked to prior work.

Explanations of the role of the specific grade-level/course-level mathematics are present in the context of the series.

In the Unit Overview, the Identify the Narrative section provides the teacher with information to unpack the learning progressions and make connections between key concepts. Lesson Support includes information about connections to previous lessons and identifies the important concepts within those lessons. Examples include:

• Unit 4 Overview, “Unit 4 content is integral to students’ success in the rest of 5th grade as well as future grade levels. Later in 5th grade, students apply strategies for division of whole numbers to decimals (5.NBT.7) and general principals of division to division of fractions (5.NF.7). In 6th grade, students solidify their division of whole number skills as they learn to fluently apply the division algorithm (6.NS.2). Later, they develop fluency with decimal division as well (6.NS.3). They are expected to use this fluency in applications of division across a number of topics in 6th grade and in 7th grade when they learn to operate with rational numbers (7.NS. A). Students continue to calculate quotients throughout the rest of middle school. Therefore, it’s imperative that the foundation built in 5th grade is strong in order to set them up for success in future grades.”

• Unit 8 Overview, “Immediately following Unit 6, scholars move into Unit 7 on dividing fractions, decimals, and mixed numbers. By 6th grade, students are expected to have a deep conceptual understanding of multiplying fractions and decimals as well as fluency with multiplying fractions. Once in 6th grade, students extend their understanding of multiplying fractions to working with division of fractions and mixed numbers, and they extend their understanding of decimals to fluently multiply decimal numbers. Additionally, scholars apply their knowledge of fractions to ratios and rates, fluently compute multi-digit multiplication and division with decimals, and begin working with percents. In 7th grade, students’ understanding of and ability to perform calculations culminates with performing all operations with rational numbers. This unit is pivotal in students’ continued understanding of and ability to fluently calculate with rational numbers. For High School, fluency with rational numbers sets students up to focus on learning new algebraic material in high school that incorporates the use of these numbers and assumes knowledge of them. An understanding of rational number operations also facilitates the understanding of rational functions and how to work with them appropriately.”

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

Materials provide strategies for informing all stakeholders, including students, parents, or caregivers about the program and suggestions for how they can help support student progress and achievement.

The materials reviewed for Achievement First Mathematics Grade 5 do not provide strategies for informing all stakeholders, including students, parents, or caregivers about the program and suggestions for how they can help support student progress and achievement. No evidence could be found related to informing stakeholders about the materials.

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

Materials provide explanations of the instructional approaches of the program and identification of the research-based strategies.

The materials reviewed for Achievement First Mathematics Grade 5 partially meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies.

Materials explain the instructional approaches of the program.

• The Implementation Guide states, "Our program aims to see the mathematical practices come to life through the shifts (focus, coherence, rigor) called for by the standards. For students to engage daily with all 3 tenets, we structure our program into two main daily components: math lesson and math cumulative review. The math lessons are divided into three structural lesson types: conjecture-based lesson, exercise-based lesson, and error analysis lesson. On a given day students will be engaging in EITHER a conjecture-based, exercise-based lesson or less often an error analysis lesson. The math cumulative review component has three sub-components: skill fluency, mixed practice, and problem of the day. Three of the five school days students engage with all three sub-components of the math cumulative review. The last two days of the week have time reserved for lessons, reteach lessons, and assessments. See the diagram below followed by each category overview for more information.”

Materials do not include and reference research-based strategies.

• The materials do not explicitly name any strategies as research-based strategies.

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

Materials provide a comprehensive list of supplies needed to support instructional activities.

The materials reviewed for Achievement First Mathematics Grade 5 do not meet expectations for providing a comprehensive list of supplies needed to support instructional activities.

• Each lesson includes a list of materials, but it often does not support the teacher in preparing the lesson. For example, “Handout” is commonly named on the materials list, but there is no link provided to the document and the title of the handout is not provided. For example, in Unit 10 Lesson 7, the Lesson Overview includes, “Materials: Handout, Protractors.”

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

This is not an assessed indicator in Mathematics.

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

This is not an assessed indicator in Mathematics.

#### Criterion 3.2: Assessment

The program includes a system of assessments identifying how materials provide tools, guidance, and support for teachers to collect, interpret, and act on data about student progress towards the standards.

The materials reviewed for Achievement First Mathematics Grade 5 partially meet expectations for Assessment. The materials identify the standards, but do not identify the practices assessed for the formal assessments. The materials provide multiple opportunities to determine students' learning and sufficient guidance to teachers for interpreting student performance but do not provide suggestions for following-up with students. The materials include opportunities for students to demonstrate the full intent of grade-level/course-level standards and practices across the series.

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

Assessment information is included in the materials to indicate which standards are assessed.

The materials reviewed for Achievement First Mathematics Grade 5 partially meet expectations for having assessment information included in the materials to indicate which standards are assessed. There are connections identified for standards, but not the mathematical practices.

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

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

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

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

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

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

The materials reviewed for Achievement First Mathematics Grade 5 partially meet expectations for including an assessment system that provides multiple opportunities throughout the grade, course, and/or series to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

The assessment system provides multiple opportunities to determine students' learning and sufficient guidance to teachers for interpreting student performance but does not provide suggestions for following-up with students. Examples include:

• Assessments include an informal Exit Ticket in each lesson and a formal Unit Assessment for every unit.

• There is guidance, or “look-fors,” to teachers about what the student should be able to do on the assessments.

• All Unit Assessments include an answer key with exemplar student responses.

• The is a rubric for exit tickets that indicates, “You mastered the learning objective today; You are almost there; You need more practice and feedback.”

• There are no strategies or suggestions if students do not demonstrate understanding of the concept, and no next steps based on the results of the assessment.

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

Assessments include opportunities for students to demonstrate the full intent of grade-level/course-level standards and practices across the series.

The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series. There are a variety of question types including multiple choice, short answer, and constructed response. Mathematical practices are embedded within the problems.

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

• The Unit 4 Assessment contributes to the full intent of 5.NBT.6 (Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors). Item 5, “Tom’s teacher wrote the following word problem on the board. Elvis wrote 155 songs during his career and shared them on 9 albums without repeating any songs on multiple albums. About how many tracks were on each of his albums? What is a reasonable estimate for the number of tracks on each album? a) 6 b) 12 c) 16 d) 1,600.”

• Unit 6, Lesson 4, Exit Ticket, Problem 2 contributes to the full intent of 5.MD.5 (relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume). “The prism to the right measures 18 cubic units. Which of the following statements may correctly describe the prism? Select 2:  a) 3 layers, 2 cubes in each layer. b) 6 layers, 3 cubes in each layer. c) 3 cubes wide, 2 cubes high, 6 cubes long. d) 2 cubes wide, 1 cube high, and 9 cubes long.”

• The Unit 8 Assessment contributes to the full intent of 5.NF.4 (apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction). “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. ( 2 points)”

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

• Unit 3 Assessment, Item 10, supports the full development of MP1 as students make sense of a complex problem. “Layla starts her own slime business. She orders 63 boxes of glue and 45 boxes of baking soda and 10 bottles of different colors of dye. Every box of glue has 24 bottles of glue in it. Each bottle contains 16 ounces of glue. Every bottle of dye has 36 ounces of dye in it. How many more ounces of glue does Layla have than ounces of dye?”

• Unit 10 Assessment, Item 8, supports the full development of MP3 as students construct a viable argument and explain their reasoning. “What is the best, or most specific name of the quadrilateral shown below? (3 points) Can you use any other classifications to describe the quadrilateral above? If so, what are they, and how do you know? If not, why not?”

• Unit 5 Assessment, Item 10, supports the full development of MP6 as students attend to precision in place value as they convert units. “Brandon and Kwame both buy sodas at the Nets concession stand during the game. Brandon looks at the label for his soda and sees that his soda contains 1,260 mL. Kwame looks at his bottle and sees that his soda contains 1.4 liters. Kwame insists that he has more soda than Brandon. Is he correct? Explain your thinking on the lines below. ”

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

Assessments offer accommodations that allow students to demonstrate their knowledge and skills without changing the content of the assessment.

The materials reviewed for Achievement First Mathematics Grade 5 do not provide assessments which offer accommodations that allow students to demonstrate their knowledge and skills without changing the content of the assessment. This is true for both formal unit assessments and informal exit tickets.

#### Criterion 3.3: Student Supports

The program includes materials designed for each child’s regular and active participation in grade-level/grade-band/series content.

The materials reviewed for Achievement First Mathematics Grade 5 do not meet expectations for Student Supports. The materials do not provide strategies and supports for students in special populations or for students who read, write, and/or speak in a language other than English to support their regular and active participation in learning grade-level mathematics.

The materials provide multiple extensions and/or opportunities for students to engage with grade-level/course-level mathematics at higher levels of complexity, and manipulatives, both virtual and physical, are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

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

Materials provide strategies and supports for students in special populations to support their regular and active participation in learning grade-level/series mathematics.

The materials reviewed for Achievement First Mathematics Grade 5 do not meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level/series mathematics.

The materials do not provide specific strategies and supports for differentiating instruction to meet the needs of students in special populations.

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

Materials provide extensions and/or opportunities for students to engage with grade-level/course-level mathematics at higher levels of complexity.

The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for providing extensions and/or opportunities for students to engage with grade-level/course-level mathematics at higher levels of complexity.

Materials provide opportunities for students to investigate the grade-level content at a higher level of complexity. Examples include:

• Implementation Guide, the philosophy of the Problem of the Day, “The typical question types selected for this component of the program are of the highest level of rigor in the program. They often cross standards, are multi-step and require a full problem-solving process in order to solve.”

• In the Implementation Guide, philosophy of the Mixed Practice Overview states, “[problems are] presented in mixed problem types and normally at a middle or high level of rigor. These questions are often in the form of word problems, multi-step problems, or a novel context.”

• Independent Practice work in each lesson provides three levels of rigor in the lesson for student work: Bachelor, Master, and PhD work, with the PhD including the most rigorous problems. In Unit 8, Lesson 18, Independent Practice PhD Level states, “Jacob buys a project board that is 2.5 feet long and 1.75 feet wide. Before he gets started, he needs to get the length and width down to \frac{3}{4} of its original size. Part A. What will the new dimensions of his project board be, in inches? Show your work. Part B. What is the area of his new project board, in square feet? Show your work.”

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

Materials provide varied approaches to learning tasks over time and variety in how students are expected to demonstrate their learning with opportunities for students to monitor their learning.

The materials reviewed for Achievement First Mathematics Grade 5 provide varied approaches to learning tasks over time and variety in how students are expected to demonstrate their learning; however, there are few opportunities for students to monitor their learning.

The program uses a variety of formats and methods to deepen student understanding and ability to explain and apply mathematics ideas. These include: Conjecture Based Lessons, Exercise Based Lessons, Error Analysis Lessons, and Math Cumulative Review. The Math Cumulative Review includes Skill Fluency, Mixed Practice, and Problem of the Day.

In the lesson introduction, the teacher states the aim and connects it to prior knowledge. In Pose the Problem, the students work with a partner to represent and solve the problem. Then the class discusses student work. The teacher highlights correct work and common misconceptions. Then students work on the Workshop problems, Independent Practice, and the Exit Ticket. Students have opportunities to share their thinking as they work with their partner and as the teacher prompts student responses during Pose the Problem and Workshop discussions. For each Exit Ticket, students have the opportunity to evaluate their work as well as get teacher feedback.

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

Materials provide opportunities for teachers to use a variety of grouping strategies.

The materials reviewed for Achievement First Mathematics Grade 5 provide some opportunities for teachers to use a variety of grouping strategies. Grouping strategies within lessons are not consistently present or specific to the needs of particular students. There is no specific guidance to teachers on grouping students.

The majority of lessons are whole group and independent practice; however, the structure of some lessons include grouping strategies, such as working in a pair for games, turn-and-talk, and partner practice. Examples include:

• Unit 2, Lesson 1, Test the Conjecture, students work in pairs to “come up with a conjecture about how to create equivalent fractions efficiently.” The lesson includes a ‘Bachelor Level’ handout to be solved with a partner.

• Unit 11, Lesson 5, Debrief, “Which student’s explanation did you agree with? Vote. Turn and tell your partner who you chose and why.”

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

Materials provide strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

The materials reviewed for Achievement First Mathematics Grade 5 do not meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

Materials do not provide any resources for students who read, write, and/or speak in a language other than English to meet or exceed grade-level standards through regular and active participation in grade-level mathematics.

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

Materials provide a balance of images or information about people, representing various demographic and physical characteristics.

The materials reviewed for Achievement First Mathematics Grade 5 provide a balance of images or information about people, representing various demographic and physical characteristics. Examples include:

• Lessons portray people from many ethnicities in a positive, respectful manner.

• There is no demographic bias seen in various problems.

• Names in the problems include multi-cultural references such as Mario, Tanya, Kemoni, Jiang, Paige, and Tomi.

• The materials are text based and do not contain images of people. Therefore, there are no visual depiction of demographics or physical characteristics.

• The materials avoid language that might be offensive to particular groups.

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

Materials provide guidance to encourage teachers to draw upon student home language to facilitate learning.

The materials reviewed for Achievement First Mathematics Grade 5 do not provide guidance to encourage teachers to draw upon student home language to facilitate learning.

The materials do not provide suggestions or strategies to use the home language to support students in learning mathematics. There are no suggestions for teachers to facilitate daily learning that builds on a student’s multilingualism as an asset nor are students explicitly encouraged to develop home language literacy. Teacher materials do not provide guidance on how to garner information that will aid in learning, including the family’s preferred language of communication, schooling experiences in other languages, literacy abilities in other languages, and previous exposure to academic everyday English.

##### Indicator {{'3t' | indicatorName}}

Materials provide guidance to encourage teachers to draw upon student cultural and social backgrounds to facilitate learning.

The materials reviewed for Achievement First Mathematics Grade 5 do not provide guidance to encourage teachers to draw upon student cultural and social backgrounds to facilitate learning.

The materials do not make connections to linguistic and cultural diversity to facilitate learning. There is no teacher guidance on equity or how to engage culturally diverse students in the learning of mathematics.

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

Materials provide supports for different reading levels to ensure accessibility for students.

The materials reviewed for Achievement First Mathematics Grade 5 do not provide supports for different reading levels to ensure accessibility for students.

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

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

Manipulatives, both virtual and physical, are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

The materials reviewed for Achievement First Mathematics Grade 5 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

Manipulatives are described as accurate representations of mathematical objects in the narrative of the Unit Overviews, and although there is little guidance for teachers or students about the use of manipulatives in the lessons, the use of manipulatives can be connected to written methods. Examples include:

For example, in Unit 6 Overview, “In lesson 2, students use cubic units (unit cube manipulatives) to build figures to a specified volume. Through this, they see that the units may be rearranged to create a different looking figure which has the same volume. More importantly, in this lesson students get an introduction to the process of finding volume as they count the number of cubic units making up a solid figure to determine its volume.”

#### Criterion 3.4: Intentional Design

The program includes a visual design that is engaging and references or integrates digital technology, when applicable, with guidance for teachers.

The materials reviewed for Achievement First Mathematics Grade 4 do not integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the grade-level standards, include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other, or provide teacher guidance for the use of embedded technology to support and enhance student learning. The materials have a visual design that supports students in engaging thoughtfully with the subject, and is neither distracting nor chaotic.

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

Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the grade-level/series standards, when applicable.

The materials reviewed for Achievement First Mathematics Grade 5 do not integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the grade-level/series standards, when applicable.

The materials do not contain digital technology or interactive tools such as data collection tools, simulations, virtual manipulatives, and/or modeling tools. There is no technology utilized in this program.

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

Materials include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other, when applicable.

The materials reviewed for Achievement First Mathematics Grade 5 do not include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other, when applicable.

The materials do not provide any online or digital opportunities for students to collaborate with the teacher and/or with other students. There is no technology utilized in this program.

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

The visual design (whether in print or digital) supports students in engaging thoughtfully with the subject, and is neither distracting nor chaotic.

The materials reviewed for Achievement First Mathematics Grade 5 have a visual design that supports students in engaging thoughtfully with the subject, and is neither distracting nor chaotic.

The student-facing printable materials follow a consistent format. The lesson materials are printed in black and white without any distracting visuals or an overabundance of graphic features. In fact, images, graphics, and models are limited within the materials, but they do support student learning when present. The materials are primarily text with white space for students to answer by hand to demonstrate their learning. Student materials are clearly labeled and provide consistent numbering for problem sets. There are several spelling and/or grammatical errors within the materials.

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

Materials provide teacher guidance for the use of embedded technology to support and enhance student learning, when applicable.

The materials reviewed for Achievement First Mathematics Grade 5 do not provide teacher guidance for the use of embedded technology to support and enhance student learning, when applicable.

There is no technology utilized in this program.

## Report Overview

### Summary of Alignment & Usability for Achievement First Mathematics | Math

#### Math K-2

The materials reviewed for Achievement First Mathematics Grades K-2 meet expectations for Alignment to the CCSSM. In Gateway 1, the materials meet expectations for focus and coherence. In Gateway 2, the materials meet expectations for rigor and practice-content connections. The materials reviewed for Achievement First Mathematics Grades K-2 do not meet expectations for Usability, Gateway 3.

##### Kindergarten
###### Alignment
Meets Expectations
###### Usability
Does Not Meet Expectations
###### Alignment
Meets Expectations
###### Usability
Does Not Meet Expectations
###### Alignment
Meets Expectations
###### Usability
Does Not Meet Expectations

#### Math 3-5

The materials reviewed for Achievement First Mathematics Grades 3-5 meet expectations for Alignment to the CCSSM. In Gateway 1, the materials meet expectations for focus and coherence. In Gateway 2, the materials meet expectations for rigor and practice-content connections. The materials reviewed for Achievement First Mathematics Grades 3-5 do not meet expectations for Usability, Gateway 3.

###### Alignment
Meets Expectations
###### Usability
Does Not Meet Expectations
###### Alignment
Meets Expectations
###### Usability
Does Not Meet Expectations
###### Alignment
Meets Expectations
###### Usability
Does Not Meet Expectations

#### Math 6-8

The materials reviewed for Achievement First Mathematics Grades 6-8 meet expectations for Alignment to the CCSSM. In Gateway 1, the materials meet expectations for focus and coherence. In Gateway 2, the materials meet expectations for rigor and practice-content connections. The materials reviewed for Achievement First Mathematics Grades 6-8 do not meet expectations for Usability, Gateway 3.

###### Alignment
Meets Expectations
###### Usability
Does Not Meet Expectations
###### Alignment
Meets Expectations
###### Usability
Does Not Meet Expectations
###### Alignment
Meets Expectations
###### Usability
Does Not Meet Expectations

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### Overall Summary

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###### Usability
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