## Achievement First Mathematics

##### v1.5
###### Usability
Our Review Process

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## Report for 6th Grade

### Overall Summary

The materials reviewed for Achievement First Mathematics Grade 6 meet expectations for Usability. The materials meet expectations for Criterion 1, Teacher Supports, Criterion 2, Assessment, and Criterion 3, Student Supports.

###### Alignment
Meets Expectations
###### Usability
Meets Expectations

### Focus & Coherence

The materials reviewed for Achievement First Mathematics Grade 6 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 6 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.

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

The materials reviewed for Achievement First Mathematics Grade 6 meet expectations for assessing grade-level content and, if applicable, content from earlier grades. Assessment questions are aligned to grade-level standards.

No examples of above grade-level assessment items were noted. Each unit contains a Post-Assessment which is a summative assessment based on the standards designated in that unit. The assessments contain grammar and/or printing errors which could, at times, interfere with the ability to make sense of the materials.

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

• Unit 1 Assessment, Question 11, “Write an equivalent expression to 40 + 32 using the distributive property. The sum of two whole numbers within the parentheses should have no common factor greater than 1.” (6.NS.4)

• Unit 5 Assessment, Question 6, “Manufacturer A requires 70 buttons for the manufacture of 10 shirts. Manufacturer B requires 68 buttons for the manufacture of 17 shirts. How many buttons will each require to manufacture 15 shirts?” (6.RP.3b)

• Unit 7 Assessment, Question 6, “Given the set {4.2, 4.7, 5}, determine which of the values are solutions to the inequality, 4.3 + x < 9. Defend your answer by explaining why or why not for each of the values.” (6.EE.5)

• Unit 9 Assessment,  Question 7, “Julia is repainting her jewelry box with a length of 8 inches, a height of 5.5 inches, and a width of 3\frac{1}{2} inches. She is painting all 4 sides blue, the top purple, and she is not painting the bottom. How many square inches of each paint color does Julia need?” (6.G.4)

• Unit 10 Assessment, Question 1, “For each of the following, identify whether or not it would be a valid statistical question you could ask about people at your school. Explain for each why it is, or is not, a statistical question. a) What was the mean number of hours of television watched by students at your school last night? (Yes, this is a valid statistical question because there is more than one answer and you can collect information from multiple sources.) b) What is the school principal’s favorite television program? c) Do most students at your school tend to watch at least one hour of television on the weekend?” (6.SP.1)

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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 6 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, Mixed Practice 1.3, Day 1, Question 4, students fluently subtract and multiply multi-digit decimals using the standard algorithm. “Rashad drove at an average speed of 50.55 miles per hour for 1.75 hours. He stopped at a rest stop and then drove at an average speed of 45.2 miles per hour for 2.25 hours. Did Rashad drive more miles before or after the rest stop? How many more miles?” (6.NS.3)

• Unit 2 Lesson 7, Independent Practice Question 3 (Masters level), students use ratio and rate reasoning to solve percent problems. “Steven orders 8 pizzas for his birthday party. He expects that each person will eat 25% of a pizza. How many people can attend the party based on his prediction? Use a model to prove your answer.” (6.RP.3c)

• Unit 6, Lesson 7, Independent Practice Question 3 (Master level), students understand that a variable can represent an unknown number. “Jeff planned to run a few miles over a certain number of weeks. He planned to run half a mile each weeknight and 4 miles on Saturday. Which expression(s) shows how many miles Jeff planned to run over a certain number of weeks. Circle all that apply. a) \frac{5}{2}+4; b) w(\frac{5}{2}+4); c) \frac{1}{2}w+4; d) \frac{5}{2}w+10; e) \frac{5}{2}+4w; f) 5w+(2)(4).” (6.EE.6)

• Unit 7, Lesson 3, Independent Practice, Question 9 (PhD level), students write and solve real-world problems. “Michael bought 8 Granny Smith apples, 7 Macintosh apples, and p Red Delicious apples. She bought a total of 27 apples. Write and solve an equation that represents this problem.” (6.EE.7)

#### Criterion 1.2: Coherence

Each grade’s materials are coherent and consistent with the Standards.

The materials reviewed for Achievement First Mathematics Grade 6 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.

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When implemented as designed, the majority of the materials address the major clusters of each grade.

The materials reviewed for Achievement First Mathematics Grade 6 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each grade.

• The approximate number of units devoted to major work of the grade (including assessments and supporting work connected to the major work) is 6 out of 10, which is approximately 60%.

• The number of days devoted to major work of the grade (including assessments and supporting work connected to the major work) is 92 out of 140, which is approximately 66%.

• The number of minutes devoted to major work (including assessments and supporting work connected to the major work) is 8316 out of 12,600 (90 minutes per lesson for 140 days), which is approximately 66%.

A minute-level analysis is most representative of the materials because of the way lessons are designed, where 55 minutes are designated for the lesson and 35 minutes are designated for cumulative review each day, so it was important to consider all aspects of the lesson. As a result, approximately 66% of the materials focus on major work of the grade.

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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 6 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.

Although connections are rarely explicitly stated, problems clearly connect supporting and major work throughout the curriculum. Examples where supporting work enhances major work include:

• Unit 1, Lesson 16, supporting standard 6.NS.4 enhances the major work of 6.EE.3. Students use the distributive property to create an expression that expresses the sum of two whole numbers between 1-100 and explains how to apply the concept of factoring to the distributive property. Independent Practice Question 7 (Master level), “What values of a and b make the two expressions below equivalent? 36 + 54 = a(2 + b).”

• Unit 3, Lesson 12, supporting standard 6.G.3 enhances major work standards 6.NS.6 and 6.NS.8. Students solve real-world and mathematical problems that involve points, lines, and polygons on the coordinate plane. In Independent Practice Question 4 (PhD level), students are given a coordinate graph and instructed, “Edwina is creating a diagram of her bedroom so that she can plan how to rearrange her furniture before moving anything. Use the following information below to help Edwina rearrange her room. Her room is in the shape of a rectangle. The area of her room is 86 square feet. Her bed covers 48 square feet of the floor. Her dresser covers 3 square feet of the floor. Her two night stands each have the dimensions \frac{3}{4} ft. by 1\frac{1}{4} ft. Use the information provided above and draw a plan for arranging Edwina’s furniture. Start by drawing the floor of her room. Be sure to label all vertices with coordinate pairs and label the dimensions of each figure you create.”

• Unit 4, Lesson 11, supporting standard 6.NS.3 enhances major work standards 6.RP.2 and 6.RP.3. Given a unit rate, students find ratios associated with the unit rate and recognize that all ratios associated to a given unit rate are equivalent. Independent Practice #3 (Master level), “Aubrey has to type a 5-page article but only has 18 minutes until she reaches the deadline. If Aubrey is able to type at a constant rate of 0.25 page every 1 minute, will she meet her deadline? Show your work to defend your answer.”

• Unit 9, Lesson 4, supporting standard 6.G.2. enhances the major standard 6.EE.7. Students apply the formulas V = lwh and V = bh to find the volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. Independent Practice Question 5 (Master level), “A rectangular box is going to be filled with sand. The length of the box is 412 feet. The width 2\frac{1}{4} feet and the height is 9 feet. If sand is sold in bags of 12 cubic feet, how many bags of sand will be needed to fill the box?”

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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 6 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 4, Lesson 7 connects the work with decimal operations (6.NS.A) with the work of one variable equations (6.EE.B) as students draw a model and use substitution to check their work. Independent Practice, Question 4 (Master level), ”$$\frac{1.05}{k}=3.5$$.”

• Unit 7 Curriculum Review, Problem of the Day 7.1 connects interpret and compute quotients of fractions (6.NS.A) with ratio concepts and reasoning (6.RP.A) as students create and compare equivalent ratios in a table using fractions and decimals. Problem of the Day 7.1, Day 2, “Edgar and Teri are each driving from their house to Disneyland for a vacation. Edgar is driving at a rate of 75.5 miles every 2.5 hours and Teri is driving at a speed of 41\frac{2}{5} miles every 1\frac{1}{5} hours. Fill out both ratio tables and graph the equivalent ratios representing each drivers’ speed to determine how much farther has Teri driven than Edgar after 5 hours and 30 minutes?”

• Unit 10, Lesson 2 connects 6.SP.A and 6.SP.B as students create dot plots and describe the distribution of data in the dot plot in terms of center and variability. In the Partner Practice, Question 2 (Master Level), “At a local hospital, babies in the intensive care unit are given 24-hour attention from doctors and nurses. The staff records the amount of milk babies drink each hour in order to ensure that they are eating properly. The amount of milk consumed by the babies is recorded: 3\frac{1}{2} oz; 3\frac{3}{4} oz; 2\frac{3}{4} oz; 22 oz  3.5 oz; 3 oz; 4\frac{1}{4} oz; 3.75 oz; 3\frac{1}{2} oz; 2\frac{3}{4} oz. a) Create the dot plot below. b) Are there any clusters in the data set? c) Are there any peaks in the data set? d) Is the data symmetrical? How do you know? e) Are there any gaps in the data set? f) What is the spread of the values in the data set? g) What value best represents the center? What does the value mean in the context of the problem?”

• Students’ work with ratios and proportional relationships (6.RP.A) is combined with their work in representing quantitative relationships between dependent and independent variables (6.EE.C). In Unit 4, Lesson 10 students develop 6.RP.A by using information from a ratio table and placing the ratios on the coordinate plane. In Unit 7, Lessons 9-12 students develop 6.EE.C when working with dependent and independent variables and graphing them. Students work in both domains independently though there is an implied connection between Unit 7 and the previous learning with proportional reasoning in Unit 4 when student graph relationships.

Examples where the materials miss the opportunity to connect two or more clusters in a domain or two or more domains in a grade:

• Plotting rational numbers in the coordinate plane (6.NS.C) is part of analyzing proportional relationships (6.RP.A). Students work on these independently and no connections are made between the domains. In Unit 3, Lesson 13, students learn about graphing rational numbers but there is no connection to RP work. In Unit 4, Lesson 9, students graph proportional relationships on the coordinate plane but there is no connection with rational numbers.

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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 6 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.

The materials relate grade-level concepts explicitly to prior knowledge from earlier grades. This can be found in the progressions descriptions listed above, but also often focuses explicitly on connecting prior understanding. Examples include:

• The Unit Overview includes “Previous Grade Level Standards and Previously Taught and Related Standards” which describes in detail the progression of the standards within each unit. In Unit 3, “Prior to 6th grade, students learned to represent positive rational numbers on a number line (3.NF.2 and 4.NF.6) and to plot points in the first quadrant of the coordinate grid (5.G.A). Students have also compared and ordered positive rational numbers, including decimal fractions (5.NBT.3b) and fractions (4.NF.2). This prior knowledge of location on a number line, coordinate geometry, and ordering positive numbers is fundamental for students being introduced to negative rational numbers for the first time in this unit.”

• Each Unit Overview provides a narrative for the teacher that introduces the student learning of the unit and the background students should have. In Unit 10, Overview, Statistics and Probability – Representing and Analyzing Data, “Prior to this unit, students have had little exposure to statistics. Throughout their elementary schooling, however, they do talk about data analysis in each grade. While their study of measuring, representing, and interpreting data starts in 1^{st} grade, I will quickly review the previous three years. In 3^{rd} grade (3MDB), students generate, represent, and interpret data in bar graphs. In 4^{th} (4MD4) and 5^{th} (5MD2) grades, students represent and interpret data using line/dot. Unit 10 is the first time students learn about statistical questions and measures of center and variability. They also learn about new graphical representations – the box plot, frequency table, and histogram. On account of this, the unit focuses the majority of the time on these topics in order to develop students’ understanding of statistical representations and analysis.”

• Each lesson includes a “Connection to Learning and Conceptual Understanding” section that relates to prior knowledge. In Unit 1, Lesson 11 Connection to Learning, “More directly, students learn to find factor pairs of numbers between 1 and 100, and identify numbers as prime or composite in 4th grade (4.OA.4). Ss (students) build directly off of this foundational knowledge in this lesson. FENCEPOST #1: Factors are numbers that you multiply together to get a product. FENCEPOST #2: Common factors are factors shared by two numbers.”

• In the Scope and Sequence Detail from the Implementation Guide, the Notes + Resources column for some lessons includes a lesson explanation that makes connections to prior learning. In Unit 6, “Students have been taught the order of operations in 5th grade. The inclusion of exponents into the process is new.”

• In the Scope and Sequence Detail from the Implementation Guide, there are additional progression connections made. The Cumulative Review column for each unit provides a list of lesson components and the standards addressed. Prior (Remedial) standards are referenced with an “R” and grade level standards are referenced with an “O.” In Unit 2, The Number System- Dividing Fractions, “Skill Fluency (4 days a week): 6.NS.3 (O)* Division, 6.NS.2 (O), 6.NS.4 (O)* GCF,LCM, Distributive Prop. Mixed Practice (3 days a week): 5.NBT.3 (R), 5.NBT.4 (R), 6.NS.3 (O), 6.NS.1 (O).”

The materials clearly identify content from future grade levels and use it to support the progressions of the grade-level standards. These connections are made throughout the materials including the Implementation Guide, the Unit Overviews, and the lessons. For example:

• The end of the Unit Overview previews, “Mastery of this unit is essential to 6th grade and in future grades. In later units in 6th grade, students apply coordinate geometry when working with areas of a variety of polygons. In the 7th grade, students rely heavily on the number line to make sense of and form generalizations about rational number operations. They apply rational numbers to represent and solve real world and mathematical problems as well as to evaluate expressions and solve equations and inequalities. Additionally, students graph proportional relationships on the coordinate plane. In 8th grade, students work heavily on the coordinate plane as they learn about transformations, functions, linear equations, systems of equations, and bivariate data. Students also work with rational number operations when solving equations and systems of equations. Rational numbers and coordinate geometry continue to be integral throughout High School mathematics as well.”

• Throughout the narrative for the teacher in the Unit Overview, there are descriptions of how the lessons will be used as the grade level work progresses. In the Unit 8 Overview, “In lesson 6, students use what they learned about calculating the area of a triangle to derive the formula for the area of trapezoids.”

• The last paragraph of each narrative for the teacher in the Unit Overview describes the importance of the unit in the progressions. In Unit 7, “This unit is essential for 6th grade and future grades. Students will work with equations and inequalities throughout all future math courses. It is imperative in 6th grade that the conceptual foundations for equations and inequalities are deeply understood in order to set students up for more abstract manipulation and application of equations and inequalities in future grades. Students learn to represent problems with and solve multi-step equations and inequalities using inverse operations and number properties in 7th grade and continue to apply their understanding of and skills with equations and inequalities throughout the remainder of their math career.”

• The narrative for the teacher in the Unit Overview makes connections to current work. In Unit 8, “Additionally, in 5^{th} grade, students used area models as a way to understand multiplication and division of whole numbers as well as multiplication of fractions. All of this work with area provides students with (a) strong base understanding of the concept from which to build as students dive deeper into their study of area in Unit 8.”

• For units that correlate with the progressions document, the materials attach the relevant text so that connections are made. In Unit 7, Appendix A: Teacher Background Knowledge (after the assessment), the “6-8 Expression and Equations” progression document is included with the footnote, “‘Common Core Expressions and Equations Progressions 6-8’ by Common Core Tools. Achievement First does not own the copyright in ‘CC Expressions and Equations Progression’ and claims no copyright in this material.”

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In order to foster coherence between grades, materials can be completed within a regular school year with little to no modification.

The instructional materials for Achievement First Mathematics Grade 6 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 140 days.

• There are 10 units with 130 lessons total; each lesson is 1 day.

• There are 10 days for summative Post-Assessments.

• There is an Optional Cumulative Project at the end of Unit 10 on Statistics. The amount of time is not designated. Since it is optional, it is not included in the total count.

According to The Guide to Implementing Achievement First Mathematics Grade 6, each lesson is completed in one day, which is designed for 90 minutes.

• Each day includes a Math Lesson (55 minutes) and Cumulative Review (35 minutes).

• The Implementation Guide states, “If a school has less than 90 minutes of math, then the fluency work and/or mixed practice can be used as homework or otherwise reduced or extended.”

### Rigor & the Mathematical Practices

The materials reviewed for Achievement First Mathematics Grade 6 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 6 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.

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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 6 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

The instructional materials develop conceptual understanding throughout the grade level. Materials include problems and questions that promote conceptual learning. Examples include:

• Unit 2, Lesson 1, Exit Ticket #1, students develop conceptual understanding of fraction division by modeling problems using a tape diagram. “Draw a model and evaluate the expression \frac{9}{10}\div\frac{3}{10}.” (6.NS.1)

• Unit 4, Lesson 9, Partner Practice, Question 2 (Master), students develop conceptual understanding of equivalent ratios by interpreting data points in a table and a graph. “The table below shows the relationship between the number of ounces in various sized boxes of Cheerios and the number of Cheerios in the box. (Table provides 4 data points.) Using the template below (Quadrant I of a coordinate plane provided), make a graph showing the relationship between the number of ounces in a box of Cheerios and the actual number of Cheerios in the box. a) What does the point (14, 4,500) represent? How do you know? b) Are the ratios in the table equivalent? Provide two reasons for how you know.” (6.RP.3)

• Unit 6, Lesson 10, Test the Conjecture #1, students develop conceptual understanding of equivalence by analyzing an equation. Teacher prompts include, “Is the following equation true? 4m + 12 = 2(m + 6). What is the question asking us to do? How do you know? How can we apply our conjecture to this problem?” (6.EE.3)

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Examples include:

• Unit 4, Lesson 2, Independent Practice, Question 3 (Bachelor level), students demonstrate conceptual understanding of ratio relationships by creating equivalence models. “Write two ratios that are equivalent to 3:5. Use a model to prove that each ratio is equivalent.” (6.RP.3a)

• Unit 6, Lesson 11, Exit Ticket, Problem 1, students demonstrate conceptual understanding of generating equivalent expressions by using properties of operations to rewrite expressions. “Without substituting a value in for x, prove that 3x+9x-2x is equivalent to 10x .” (6.EE.3)

• Unit 8, Lesson 2, Independent Practice, Question 5 (Bachelor Level), students demonstrate conceptual understanding of finding area by decomposing parallelograms into triangles. “Brittany and Sid were both asked to draw the height of a parallelogram. Their answers are below. Who is correct? Explain your answer. Is there another way they could have drawn in the height? If so, draw the different way to identify the height on one of their parallelograms and explain.” (6.G.1)

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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 6 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.Although there are not many examples to practice within a lesson, students are provided opportunities to practice fluency both with a partner and individual practice, especially within exercise based lessons and the skill fluency of the cumulative review section.

The materials develop procedural skill and fluency throughout the grade level. Examples include:

• Unit 1, Lesson 11, Interaction with New Material, students develop procedural skill and fluency by finding common factors and multiples. “Ex. 1) The Ski Club members are preparing identical welcome kits for the new skiers. The Ski Club has 72 hand warmer packets and 48 foot warmer packets. What is the greatest number of identical kits they can prepare using all of the hand warmer and foot warmer packets? How many hand warmer packets and foot warmer packets will there be in each kit? ...Based on our understanding of the problem, what is our plan for solving this problem? ...Note to teacher: Ss will likely struggle to make the connection to the GCF. Push hard on Ss understanding that you are dividing each total up and make sure that students truly understand that the number of groups will be the same for both types of warmer and the size of the group will be different.” (6.NS.4)

• Unit 7, Skill Fluency 7.3, Day 1, Question 5, students develop procedural skill and fluency by using substitution to make equations true. “Which equation is true if x = 5? a) 3x = 8; b) 2x = 10; c) x + 5 = 5; d) 25 - 5 = x.” (6.EE.5)

• Unit 8, Skill Fluency 8.2, Day 1, Question 6, students develop procedural skill and fluency by using properties of operations to generate equivalent expressions. “What is the correct first step to take in order to simplify the expression below? [3.5 × (5 - 4.3)] + 2.7: a) Subtract 4.3 from 5; b) Multiply 3.5 by 2.7; c) Multiply 3.5 by 5.” (6.EE.3)

The materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade-level. Examples include:

• Unit 1, Lesson 1, Independent Practice, Question 1 (Bachelor Level), students demonstrate procedural skill and fluency by using operations on decimals. “Evaluate each expression: a) 23 – 0.324; b) 9.3 + 19.59.” (6.NS.3)

• Unit 2, Skill Fluency 2.2, Day 1, Problems 1-4, students demonstrate procedural skill and fluency by dividing multi-digit numbers. "1) 1,986 ÷ 60 = ?; 2) Solve: 80.25\div20= ?; 3) Find the quotient: 540\div0.60= ?; 4) 35.2\div5.5= ?” (6.NS.2)

• Unit 6, Lesson 2, Independent Practice, Question 3 (Master Level), students demonstrate procedural skill and fluency by evaluating expressions. “Evaluate each expression: a) 24\frac{3}{5}+(4^3\times(8.2-2)); b) 6^2 + (13.5 - 5 + 2) × 2^3 + 3\frac{8}{10}” (6.EE.2c)

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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 6 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 6, Mixed Practice 6.3, Day 1, Question 4, students apply skills related to solving routine problems using division of fractions. “Carla wants to know how many batches of birdseed she can make with 3\frac{1}{2} cups of  sunflower seeds. She puts \frac{1}{6} cup of sunflower seeds in every batch. Carla divides 3\frac{1}{2}  by \frac{1}{6}  to find the answer. She says this is the same as multiplying \frac{1}{6}  by \frac{7){2}. a) Is Carla correct? Why or why not? b) Using a correct method, find the solution. Show your work.” (6.NS.1)

• Unit 7, Lesson 12, Partner Practice, Question 2 (Master Level), students represent and analyze routine quantitative relationships between dependent and independent variables. "Sam drove his car at a constant speed for t minutes and traveled a total of m miles. This relationship is represented in the table below. (3 data points provided, leading to 1.5t = m) If Sam drove 14.25 miles in all, how many minutes had he been traveling?” (6.EE.9)

• Unit 10, Problem of the Day 10.1, Question 2 - students apply skills related to solving problems utilizing decimals in a non-routine application. "Four 6th graders are working on a project. They are going to paint a large banner and need to protect the floor. They measured the floor, which is 3.05 meters by 3.68 meters. Plastic is sold in rolls of  0.5 square meters each. How many rolls of plastic will they need to buy in order to cover the floor?” (6.NS.3)

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

• Unit 3, Lesson 12, Independent Practice, Question 3 (Master Level), students solve real life, non-routine problems such as graphing points on the coordinate plane and using them to find area. "Mason Rice Elementary School is creating a new playground in the park behind the school. The playground’s perimeter is rectangular and is 60 feet long with a width of 15\frac{1}{2} feet. The planning committee is drafting the design of the new playground on a coordinate grid. They started by placing one corner of the park at (-8, 8). Each unit on the coordinate plane represents 1 foot. a) Plot the other three corners of the playground, label the coordinates of each corner, and connect the corners to create a rectangle. b) The committee is planning on splitting the playground diagonally in order to make two separate spaces for younger kids and older kids. Draw a line that divides the playground diagonally. How many square feet of space is the committee allocating for each part of the playground?” (6.NS.6c, 6.NS.8, 6.G.1)

• Unit 4, Lesson 5 Exit Ticket, Question 2, students apply skills related to solving problems using ratio reasoning in non-routine ways. “Josh was solving the following problem: The Superintendent of Highways is interested in  the numbers of commercial vehicles that frequently use the county’s highways. He obtains information from the Department of Motor Vehicles for the month of  September and finds that for every 14 non-commercial vehicles, there were 5 commercial vehicles. If there were 108 more noncommercial vehicles than  commercial vehicles, how many of each type of vehicle frequently use the county’s highways during the month of September? He says that he cannot solve it because \frac{108}{14} = a number with a decimal remainder. What is the mistake that Josh is making? Find the mistake, and solve correctly.” (6.RP.3a)

• Unit 5, Day 2, Problem of the Day, students apply skills related to solving problems using ratio reasoning in a non-routine problem. "Gylissa and Alicia are developing a business of making and selling slime. The table below shows corresponding amounts of all ingredients they use to make their slime. Part A: If Gylissa and Alicia always use the same recipe when making slime, what are the values of x and y? Part B: Gylissa and Alicia receive a huge order of 6 cups of slime for each student in their class of 24 students. How many cups of each ingredient will they need to fill the order? Part C: Jadine and Tiarah also decide to make a slime business, but their recipe uses 6 cups of water, 8 cups of glue, and 3 cups of borax. Whose recipe will make a stickier slime? Show your work below to prove your answer.” (6.RP.3a)

• Unit 7, Lesson 7, Independent Practice, Question 7 (PhD Level), students apply skills related to writing and solving routine one-step equations. "Nadia bought five food tickets that each cost x dollars and three drink tickets that each cost $2 to attend a spaghetti fundraiser at her school. She spent a total of$33.50. Write an equation that represents the cost of each food ticket.” (6.EE.7)

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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 6 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. Overall, there is an emphasis on the application aspect with the conceptual component of rigor being slightly less represented; however, each aspect of rigor is demonstrated throughout the curriculum. The materials often demonstrate a combination of aspects of rigor within single lessons and even single problems.

All three aspects of rigor are present independently throughout the program materials. Examples include:

Conceptual Understanding:

• Unit 1, Lesson 5, Independent Practice, Question 4b (Master Level), students use models and equations to conceptualize division with the standard algorithm. "For each word problem, draw a model, write an equation, and solve the problem. b. Thomas has 575 pennies that he wants to exchange for quarters. How many quarters will he receive in exchange for his 575 pennies?" (6.NS.2)

Fluency and Procedural Skill:

• Unit 6, Lesson 1, Independent Practice, Question 6 (Master Level), students develop fluency with evaluating numerical expressions that include exponents. “Evaluate the expressions: a) 90 - 5^2 × 3.5; b) 6.4 - 2^2\div2 + 0.3^2.” (6.EE.1)

Application:

##### 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 6 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.

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

The materials reviewed for Achievement First Mathematics Grade 6 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 3, Lesson 11, Key Learning Synthesis, “Let’s form our KP for today. With your partner, TT and come up with our key point about comparing rational numbers.”

• Unit 8, Lesson 3, Test the Conjecture,  students work in pairs to test the conjecture the area formula of a right triangle applies to acute triangles. “For the next 5 minutes, you’ll be working with your partner applying the conjecture that we just stamped.”

##### 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 6 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.

Guidance is consistently provided for teachers to support students who read, write, and/or speak in a language other than English, providing scaffolds for them to meet or exceed grade-level standards. According to the Program Overview, Guide to Implementing AF Math: Grade 6, Differentiation, Supporting Multilingual and English Language Learners, “The Achievement First Mathematics Program appreciates the importance of creating a classroom environment in which Multilingual and English language learners (MLLs/ ELLs) can thrive socially, emotionally, and academically. MLLs/ ELLs have the double-task of learning mathematics while continuing to build their language mastery. Therefore, additional support and thoughtful curriculum is often needed to ensure their mastery and support in learning. Our materials are designed to help teachers recognize and serve the unique educational needs of MLLs/ ELLs while also celebrating the assets they bring to the learning environment, both culturally and linguistically. Our three structural lesson types: conjecture-based lesson, exercise-based lesson, and error analysis lesson all build on the four design principles for promoting mathematical language use and development in curriculum and instructions outlined by Stanford’s Graduate School of Education, Understanding Language/SCALE.” The series provides the following principles that promote mathematical language use and development:

• Design Principle 1: Support sense-making - Daily lesson resources (DLRs) are designed to promote student sense-making with an initial ‘Think About It’ task that engages students with a meaningful task upon which they can build connections. Students have time to read and understand the problem individually and the debrief of these tasks include clear definitions of new terminology and/or key ideas or conjectures…Additionally, teachers are provided with student-friendly vocabulary definitions for all new vocabulary terms in the unit plan that can support MLLs/ELLs further.”

• “Design Principle 2: Optimize output - Lessons are strategically built to focus on student thinking. Students engage in each new task individually, have opportunities to discuss with partners, and then analyze student work samples during a whole class debrief…All students benefit from the focus on the mathematical discourse and revising their own thinking, but this is especially true of MLLs/ELLs who will benefit from hearing other students thinking and reasoning on the concepts and/or different methods of solving.”

• “Design Principle 3: Cultivate conversation - A key element of all lesson types is student discussion. Daily lesson resources (DLRs) rely heavily on the use of individual think/write time, turn-and-talks with partners, and whole class discussion to answer key questions throughout the lesson script. The rationale for this is that all learners, but especially MLLs/ELLs benefit from multiple opportunities to engage with the content. Students that are building their mastery of the language may struggle more with following a whole-class discussion; however, having an opportunity to ask questions and discuss with a strategic partner beforehand can help deepen their understanding and empower them to engage further in the class discussion….”

• “Design Principle 4: Maximize linguistic and cognitive meta-awareness - The curriculum is strategically designed to build on previous lesson mastery. Students are given opportunities to discuss different methods to solve similar problems and/or how these concepts build on each other. The focus of the ‘Think About It’ portion of the Exercise-Based lesson is to help students build on their current understanding of mathematics in order to make a new key point for the day’s lesson. The entire focus of the Test the Conjecture lesson is for students to create their own conjecture about the new learning and then to test this by applying it to an additional problem(s). Students focus on building their own mathematical claims and conjectures and see mathematics as a subject that involves active participation of all learners. By ending each lesson type with this meta-awareness, all learners, but especially MLLs/ELLs benefit by building deeper connections.”

The series also provides Mathematical Language Routines in each unit. According to the Program Overview, Guide to Implementing AF Math: Grade 6, Differentiation, Supporting Multilingual and English Language Learners, “Beyond these design principles, our program outlines for teachers in every unit plan the most appropriate mathematical language routines (MLRs) to support language and content development of MLLs/ELLs with their learning within the specific unit.” Examples include:

• Unit 8 Overview, Geometry:  Area, Differentiating for Learning Needs, Supporting MLLs/ELLs,

• Vocabulary: “MLLs/ELLs should be provided with a student-friendly vocabulary handout throughout the unit that is either completed for them and/or that they add to each day. All terms included in the ‘Vocabulary’ section below should be included. This scaffold can be incredibly helpful for other learners to help them see a verbal and visual definition for each term. Each of the terms, definitions, and examples should be translated into the students preferred language using Google Translate or a translator (Spanish in the example provided).”

• Sentence Frames: “MLLs/ELLs and all students can greatly benefit from specific guidance around sentence frames for standard justifications or explanation within the unit. In lessons 1 and 5, students learn the formula for the area of a triangle and trapezoid, and construct arguments to explain why the formula works. Teachers can provide students with the following sentence frames to use throughout these problems and class discussions: ‘These figures do/do not have the same area. I know this because _____.’ ‘I agree/ disagree with _____’s claim because _____.’  ‘We can use triangles to derive the area formula for trapezoids by _____.’”

• Language Development Routines: “Throughout the unit, teachers should focus on student discussion and use of critical thinking when analyzing student work samples. See the ‘Implementing Language Routines’ of the Implementation Guide for the course for further detail of how these routines live within all lessons. Within this unit, students should specifically focus on the following Mathematical Routines.

• MLR 1: Stronger and Clearer Each Time: Students will focus in ALL lessons on analyzing student work and revising their thinking either during the Think About It, Interaction with New Material, or Test the Conjecture portion of each lesson.

• MLR 2: Collect and Display: In the Think About it, Interaction with New Material sections, or Test the Conjecture portion of ALL lessons, students will be justifying their response. Collect student Responses that are partially correct, leading a discussion and allowing students to build, revising as needed. Note any specific vocabulary or specific evidence that strengthens the response.

• MLR 7: Compare and Connect: In lessons 8 – 10, students will be finding the area of compound shapes by decomposing the shape into regular figures. In this, students have numerous ways in which they can decompose their figures, some being more efficient than others. Leverage student work to have students compare approaches, analyzing what makes a particular decomposition more efficient (less missing dimensions, more whole numbers, etc.), and stamping that both are viable.

• MLR 8: Discussion Supports: Lessons 1 and 5 asks students to describe how they were able to derive the formulas for a triangle and trapezoid, respectively, from different shapes. Students may be able to illustrate the connection, but my struggle to articulate themselves in a written response. Here, the teacher may provide sentence stems and key terms to include in the discussion.”

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

The materials reviewed for Achievement First Mathematics Grade 6 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.

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

The materials reviewed for Achievement First Mathematics Grade 6 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 6 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 6 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 6 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:

• In Unit 9 Overview, “The learning in this unit relies heavily on concrete and pictorial representations to solidify students’ conceptual understanding of volume and surface area. For example, when packing the solid below with cubes that have an edge length of 1⁄2 inch, students first find the number of cubes that fit in the cube. They do this by determining how many cubes fit across the length (3), how many cubes fit across the width (3), and how many cubes fit across the height (3). Students multiply the number of cubes that fit across the length, width, and height to determine that a total of 27 one-half inch cubes fit in the cube below.”

• In Unit 10, Lesson 4, cubes are listed as an option for a representation of people when graphing a frequency table.

#### 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 6 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 6 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 6 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 6 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 6 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. In Gateway 3, the materials meet expectations for Usability.

##### Kindergarten
###### Alignment
Meets Expectations
###### Usability
Meets Expectations
###### Alignment
Meets Expectations
###### Usability
Meets Expectations
###### Alignment
Meets Expectations
###### Usability
Meets 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. In Gateway 3, the materials meet expectations for Usability.

###### Alignment
Meets Expectations
###### Usability
Meets Expectations
###### Alignment
Meets Expectations
###### Usability
Meets Expectations
###### Alignment
Meets Expectations
###### Usability
Meets 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. In Gateway 3, the materials meet expectations for Usability.

###### Alignment
Meets Expectations
###### Usability
Meets Expectations
###### Alignment
Meets Expectations
###### Usability
Meets Expectations
###### Alignment
Meets Expectations
###### Usability
Meets Expectations

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### Overall Summary

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