The instructional materials reviewed for Achievement First Mathematics Grade 6 meet expectations that they assess grade-level content. 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: 

• In Unit 1 Assessment, Question 11 states, “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)
• In Unit 5 Assessment, Question 6 states, “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)
• In Unit 7 Assessment, Question 6 states, “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)
• In Unit 9 Assessment,  Question 7 states, “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) 
• In Unit 10 Assessment, Question 1 states, “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)

The instructional materials reviewed for Achievement First Mathematics Grade 6 do not meet expectations for spending a majority of instructional time on major work of the grade. For example:

• 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 87 out of 140, which is approximately 62%.
• The number of minutes devoted to major work (including assessments and supporting work connected to the major work) is 7285 out of 12,600 (90 minutes per lesson for 140 days), which is approximately 58%. 

A minute level analysis is most representative of the instructional 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 58% of the instructional materials focus on major work of the grade.

The instructional materials reviewed for Achievement First Mathematics Grade 6 meet expectations that supporting work 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:

• In 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. For example, Independent Practice Question 7 (Master level) states, “What values of a and b make the two expressions below equivalent? $$36 + 54 = a(2 + b)$$.” 
• In 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.”
• In 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. For example, Independent Practice #3 (Master level) states, “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.”
• In 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. For example, Independent Practice Question 5 (Master level) states, “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?”

Instructional materials for Achievement First Mathematics Grade 6 meet expectations that the amount of content designated for one grade-level is viable for one year. As designed, the instructional materials can be completed in 140 days. For example:

• 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.”

The instructional materials for Achievement First Mathematics Grade 6 meet expectations for the materials being consistent with the progressions in the Standards. 

The materials clearly identify content from prior and 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 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. For example, Unit 3 states, “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.” The end of this section adds, “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. For example, Unit 8 Overview states, “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. For example, Unit 7 states, “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.”
• For units that correlate with the progressions document, the materials attach the relevant text so that connections are made. For example, 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.”
• Each lesson includes a “Connection to Learning and Conceptual Understanding” section that describes the progression of the standards within the unit. In Unit 5, Lesson 11 states, “In the previous four lessons, students learned to use a DNL” (double number line) “to find an unknown percent, part, and total given two other pieces of known information. In this lesson, students make a connection between using an equation and a DNL in order to make sense of and use equations to solve ‘percent of’ problems. This is the first day of two days working with equations to solve percent problems.”
• 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 states, “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 attend to the full intent of the grade-level standards by giving all students extensive work with grade-level problems. 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:

• In Unit 1, Mixed Practice 1.3, Day 1, Question 4, students fluently subtract and multiply multi-digit decimals using the standard algorithm. The materials state, “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)
• In Unit 2 Lesson 7, Independent Practice Question 3 (Masters level), students use ratio and rate reasoning to solve percent problems. The materials state, “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)
• In Unit 6, Lesson 7, Independent Practice Question 3 (Master level), students understand that a variable can represent an unknown number. The materials state, “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)
• In Unit 7, Lesson 3, Independent Practice, Question 9 (PhD level), students write and solve real-world problems. The materials state, “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)

The instructional 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. For example:

• 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 states, “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.”
• The narrative for the teacher in the Unit Overview makes connections to current work. Unit 8 states, “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.”
• Each lesson includes a “Connection to Learning and Conceptual Understanding” section that relates to prior knowledge. In Unit 1, Lesson 11 Connection to Learning explains, “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. Unit 6 states, “Students have been taught the order of operations in 5th grade. The inclusion of exponents into the process is new.”

The instructional materials for Achievement First Mathematics Grade 6 partially meet expectations that materials foster coherence through connections at a single grade, where appropriate and required by the Standards.

The materials include learning objectives, identified as AIMs, that are visibly shaped by the CCSM cluster headings. The Guide to Implementation, as well as individual lessons display each learning objective along with the intended standard. The instructional materials utilize the acronym SWBAT to stand for “Students will be able to” when identifying the lesson objectives. Examples include:

• The AIM for Unit 4, Lesson 1: “SWBAT define ratio and use ratio language to describe a ratio relationship between two quantities,” is shaped by 6.RP.A: Understand ratio concepts and use ratio reasoning to solve problems.
• The AIM for Unit 6, Lesson 10: “SWBAT identify and explain when two expressions are equivalent regardless of which value is substituted in for the variable” is shaped by 6.EE.A: Apply and extend previous understandings of arithmetic to algebraic expressions. 
• The AIM for Unit 8, Lesson 7: “SWBAT apply the area formulas for rectangles, squares, triangles, parallelograms and trapezoids in the context of solving real-world problems,” is shaped by 6.G.A: Solve real-world and mathematical problems involving area, surface area, and volume.

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

• In Unit 4, Lesson 7 connects the work with decimal operations (6.NS.A) with the work of one variable equations (6.EE.B) when students draw a model and use substitution to check their work. For example, Independent Practice Question 4 (Master level) states, ”$$\frac{1.05}{k}=3.5$$”
• In Unit 7 Curriculum Review, Problem of the Day 7.1 connects the ability to interpret and compute quotients of fractions (6.NS.A) with ratio concepts and reasoning (6.RP.A) when students create and compare equivalent ratios in a table using fractions and decimals. In Problem of the Day 7.1, Day 2 states, “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?”
• In 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; $$2$$ 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?”

Examples where the materials miss the opportunity to make natural connections among clusters or domains that are expected and important:

• Students’ work with ratios and proportional relationships (6.RP.A) can be combined with their work in representing quantitative relationships between dependent and independent variables (6.EE.C). In Unit 4, Lesson 9 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. However, students work in both domains independently; in Unit 7, no connection is made to the previous learning with proportional reasoning in Unit 4. 
• 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, Lessons 11 and 12, 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.

The instructional materials for Achievement First Mathematics Grade 6 meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific 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:

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

• In Unit 4, Lesson 2, Independent Practice, Question 3 (Bachelor level), students demonstrate conceptual understanding of ratio relationships by creating equivalence models. The materials state, “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. The materials state, “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. The materials state, “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)

The instructional materials for Achievement First Mathematics Grade 6 meet expectations that they attend to those 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 instructional materials develop procedural skill and fluency throughout the grade level. Examples include:

• In Unit 1, Lesson 11, Interaction with New Material, students develop procedural skill and fluency by finding common factors and multiples. The materials state, “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)
• In Unit 7, Skill Fluency 7.3, Day 1, Question 5, students develop procedural skill and fluency by using substitution to make equations true. The materials state, “Which equation is true if $$x = 5$$? a) $$3x = 8$$; b) $$2x = 10$$; c) $$x + 5 = 5$$; d) $$25 - 5 = x$$.” (6.EE.5)
• In 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. The materials state, “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 instructional materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade-level. Examples include:

• In Unit 1, Lesson 1, Independent Practice, Question 1 (Bachelor Level), students demonstrate procedural skill and fluency by using operations on decimals. The materials state, “Evaluate each expression: a) 23 – 0.324; b) 9.3 + 19.59.” (6.NS.3)
• In Unit 2, Skill Fluency 2.2, Day 1, Problems 1-4, students demonstrate procedural skill and fluency by dividing multi-digit numbers. The materials state, "1) $$1,986 ÷ 60 =$$ ?; 2) Solve: $$80.25\div20=$$ ?; 3) Find the quotient: $$540\div0.60=$$ ?; 4) $$35.2\div5.5=$$ ?” (6.NS.2)
• In Unit 6, Lesson 2, Independent Practice, Question 3 (Master Level), students demonstrate procedural skill and fluency by evaluating expressions. The materials state, “Evaluate each expression: a) $$24\frac{3}{5}+(4^3x(8.2-2))$$; b) $$6^2+(13.5-5+2)×2^3+3\frac{8}{10}$$” (6.EE.2c)

The instructional materials for Achievement First Mathematics Grade 6 partially meet expectations that the materials are 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. However, students do not have consistent opportunities to explore non-routine problems.

The instructional materials include multiple opportunities for students to engage in routine application of mathematical skills and knowledge of the grade level. Students are rarely presented with problems that involve a context that they have not already practiced. Examples include:

• In Unit 2, Mixed Practice 2.3, Day 3, Question 3, students apply skills related to fraction division. The materials state, “Ellie ran a $$1\frac{3}{4}$$ mile special race course. Every $$\frac{1}{8}$$ mile, there was an obstacle, with a final obstacle at the finish line. How many obstacles did Ellie encounter in the race? Use a model and/or words to explain your thinking.” (6.NS.1)
• In Unit 3, Lesson 12, Independent Practice, Question 3 (Master), students solve real life problems such as graphing points on the coordinate plane and using them to find area. The materials state, “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)
• In Unit 5, Problem of the Day 5.2, Day 2, students apply skills related to solving problems using ratio reasoning. The materials state, “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)
• In Unit 7, Lesson 7, Independent Practice, Question 7 (PhD level), students apply skills related to writing and solving one-step equations. The materials state, “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)
• In Unit 7, Lesson 12, Partner Practice, Question 2 (Master Level), students represent and analyze quantitative relationships between dependent and independent variables. The materials state, “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)
• In Unit 9, Problem of the Day 9.2, Day 3, Question 1, students apply skills related to solving rate problems using decimals. The materials state, “Alan and his family are on their way to visit a national park that is 780 miles from their house. They choose a route so that it will take them 3 days to get to their destination. The first day they drive for 4 hours at an average speed of 60 miles per hour. The second day they drive for 6 hours at an average speed of 65 miles per hour. If the average speed on the third day is 60 miles per hour, how many more hours will it take them to reach the national park? Show your work.” (6.RP.3b, 6.NS.3)

The instructional materials for Achievement First Mathematics Grade 6 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. 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:

• In Unit 1, Lesson 4, Independent Practice, Question 2 (Bachelor Level), students use models and equations to conceptualize division with the standard algorithm. The materials state, "Cheyenne collects movies. She has 435 movies that are either comedy, scary, or sci-fi. She has the same number of videos for each genre. How many movies does she have in each genre? Draw a model and write an equation that represents the problem. Solve." (6.NS.2)

Fluency and Procedural Skill:

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

Application:

• In Unit 7, Problem of the Day, 7.3, Day 2, students apply their knowledge about using a variable to represent an unknown in an equation to find out about fast food profits. The materials state, “Wendy’s has a number of franchises, f, in Brooklyn. Each franchise makes $223 every hour. a) Write an expression to represent, m, the total amount of money all Wendy’s franchises make per hour. b) If m = 7,136 how many Wendy’s franchises are there in Brooklyn?” (6.EE.6) 

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

• In Unit 1, Lesson 11, Independent Practice, Question 3 (Master Level), students demonstrate both procedural skill and application as they use ratio reasoning to determine final floral bouquets. The materials state, “Felix owns a flower shop and is preparing to put together bouquets to sell. He has 66 lilies and 88 roses that he wants to distribute among the bouquets. He wants there to be the same number of lilies and the same number of roses in each bouquet. What is the largest number of bouquets Felix can make? How many roses and lilies will be in each bouquet?” (6.RP.A)
• In Unit 2, Lesson 4, Partner Practice, Question 1 (Bachelor level), students demonstrate both conceptual understanding and procedural skill as they create a model and use the standard algorithm to divide fractions. The materials state, “Evaluate each expression using a model and using the algorithm: a) $$2\frac{4}{5}\div\frac{2}{5}$$; b) $$3\frac{5}{6}\div\frac{2}{3}$$”  (6.NS.1)
• In Unit 3, Lesson 12, Independent Practice #2 (Master Level), students demonstrate procedural skill and application as they meet the requirements to construct a rectangle. The materials state, “Construct a rectangle on the coordinate plane that satisfies each of the criteria listed below. Identify the coordinates of each of its vertices. 1) Each of its vertices lies in a different quadrant; 2) Its sides are either vertical or horizontal; 3) The perimeter of the rectangle is 28 units; 4) Using absolute value, show how the lengths of the sides of your rectangle provide a perimeter of 28 units.” (6.G.3)

The instructional materials reviewed for Achievement First Mathematics Grade 6 partially meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade level. The Mathematical Practices are listed in the Unit Overviews as well at the beginning of each lesson, however there is little direction provided about how the MPs enrich the content or make connections to enhance student learning.

All eight MPs are clearly identified throughout the materials, with few or no exceptions, though they are not always accurate. For example:

• In the Guide to Implementing AF Math, “Math Lesson Types” explains how different types of lessons engage students with the Mathematical Practice Standards. For example, Conjecture Based Lessons states, “Purpose: Through the use of investigation and guided inquiry, students develop conceptual understanding of math topics and strategies. They persevere by developing and proving mathematical conjectures. Structurally based off of the Standards for Mathematical Practice 3, these lessons push students to make viable arguments and critique the thinking of others to generate a conjecture that will then be tested. They must make connections to previously learned content, apply sound mathematical practices, and think flexibly.”
• The MPs are listed at the beginning of each lesson in the Standards section. 
• All MPs are represented throughout the materials, though lacking balance. For example, MP5 is emphasized in two units, while MP6 is emphasized in ten.
• The Mathematical Practices are not always identified accurately. For example:
• At the unit level for Unit 2, MP3 is not identified as an emphasized practice. However, at the lesson level, 11 out of the 12 lessons identify it as connected. MP6 and MP7 are bold (indicating emphasis) at the unit level, but aren't connected in any of the lessons.
• The Unit 5 Overview bolds MPs 1, 5, 6 and 7 as the emphasis of the unit. However, eight of the lessons bold MP2, and 10 lessons bold MP4, while MP7 is only bolded in Lesson 6. 
• In Unit 7, MPs 2, 4, 6, 7, and 8 are bold; however, MP7 does not appear in any of the lessons in this unit. 
• There is no stated connection to the MPs within the skill fluency, mixed review, problem of the day, or assessments. In the Guide to Implementation in Problem of the Day Overview it explains, “The problem of the day provides students with practice applying mathematical practices and multiple standards to a rigorous problem.” While the learning standards are listed for these problems, the relevant MPs are not identified.

There are a few instances where the MPs are addressed, but are not clear in the content. For example:

• It is generally left to the teacher to determine where and how to connect the emphasized mathematical practices within each lesson.
• There are connections to the content described in the Overview, though not specifically linked to an MP. If a teacher was not familiar with the MPs, the connection may be overlooked. Examples include:
• Unit 4 Overview states, “In lesson 2, students use tape diagrams to model ratios and solve problems given a part-to-part ratio and one quantity from the second ratio. Students understand that a tape diagram is used to model relationships that involve ‘like units’ because the size of each part is the same." 
• Unit 5 Overview states, ”In lessons 6-12, students are introduced to percentages for the first time and apply their understanding of ratios and rates to make sense of and solve problems involving percentages.”
• Unit 7 Overview states, “In lesson 6, students understand that mathematical situations and structures can be translated and represented abstractly using one-step equations that are made up of variables, symbols, and numbers.”

Unit 1 is the exception in the materials. The Unit 1 Overview makes connections to help teachers understand how to emphasize the content to incorporate the MPs. None of the other units make these connections. For example:

• MP1 - Unit 1 Overview states, “In lesson 10, students solve problems that incorporate all four operations with decimals and multiple steps. The lesson is intended to push students to make decisions regarding the correct operation to utilize given known and unknown information as well as to help them make sense of multi-step problems and apply their learning flexibly (SMP.1; 6.NS.3).” 
• MP2 - Unit 1 Overview states, “By the end of the lesson, students are fluent in creating and recognizing equivalent division expressions to find quotients when divisors are decimal numbers. Again, throughout the division lessons, teachers consistently reinforce the use of estimation and reasoning about the magnitude of a quotient relative to the dividend to gauge reasonableness (SMP.2).”
• MP7 - Unit 1 Overview states,  “...Once they have realized that, they can factor out the GCF from the addends and rewrite the expression as the product of the GCF and the sum of the two other factors (SMP.7).”
• MP8 - Unit 1 Overview states, “Given a pair of addends made up of whole numbers from 1-100, students recognize that they can find factors of the addends as a strategy to apply the distributive property to rewrite the expression. Students recognize that both addends are multiples with common factors (SMP.8).”

The instructional materials reviewed for Achievement First Mathematics Grade 6 partially meet expectations that the instructional materials carefully attend to the full meaning of each practice standard. 

The materials do not attend to the full meaning of two MPs. For example:

• MP4: Model with mathematics - Students have limited opportunities to develop their own solution pathways that would best support mathematical tasks and are often directed to represent the problem in a certain way. 
• In Unit 3, Lesson 1, Think About It!, students find the absolute value of numbers. The materials state, “Use the number line below to answer the following prompts. a) Labeling the number line has been started for you. Label all the hash marks to the left of 0; b) What is the relationship between 2 and -2?” Students complete a number line without developing any strategies or applying any real-world context.
• In Unit 5, Lesson 6, Independent Practice, Question 1, (Bachelor Level), students use a five-column chart to represent a value as a percentage - decimal - fraction - ratio - model (given a 10x10 grid). In this example, MP4 is a visual representation but students do not choose a strategy, nor is there any real-world connection.
• In Unit 7, Lesson 5, Independent Practice, Question 5, (Master Level) states, “Directions: Draw a model to solve each equation and check your answer using substitution. $$\frac{3.15}{k}=0.5$$.” Again, students create a visual representation without any real-world context. 

• MP5: Use appropriate tools strategically - Students have limited opportunities to choose tools that would best support mathematical tasks and are often provided with only one type of tool to solve a problem. MP8 is only identified in Unit 8 of the materials. 
• In Unit 8, Lesson 8, students calculate the area of figures given a coordinate plane. There is no opportunity to choose a tool to help solve the problems, all problems include a coordinate grid that is pre-numbered and labeled. 
• In Unit 8, Lesson 9, Partner Practice, Question 1, (Bachelor Level) states, “Determine the area of the figure drawn on the grid below.” The grid provided is pre-numbered and labeled; students do not have a chance to choose tools. 
• In Unit 8, Lesson 11, students decompose compound figures and calculate area. Students are not given a choice of tools, as calculators are simply listed in the materials.

The following practices are connected to grade-level content and are developed to their full intent over the course of the materials. 

• MP1: Make sense of problems and persevere in solving them  
• In Unit 2, Lesson 10, Independent Practice Question 6 (PhD level), students find an entry point and persevere to solve a multi-step problem involving rational numbers. The materials state, “Amare paid $16.50 to buy a book. The cost of the book is $$2\frac{1}{2}$$ the cost of a magazine. He bought 3 books and 2 magazines with a $50 bill. How much change should he receive?”
• In Unit 7, Lesson 5, Independent Practice, Question 8 (PhD Level), students make sense of an unusual situation in order to find a solution. The materials state, “A toy-maker is attempting to make a stackable set of discs to help recreate the famous Tower of Hanoi problem. To do this, the toy-maker wants to create discs whose diameters decrease by 1 inch for each new level of the stacked discs. The toy-maker wants to eventually have 5 discs stacked on top of each other and the largest disc will be 10 inches in diameter. What will be the diameter of the smallest disc?”
• In Unit 8, Lesson 7, Interaction with New Material, students are provided with an unusual shape and the additional challenge of determined cost. The materials state, “Bob the builder is constructing a lot with the shape and dimensions below. (Given a trapezoid with all 4 sides and the height labeled.) Part A: Before starting to build, he ropes off the entire lot to ensure no one walks on it during construction. How many feet of rope does he need? Part B: After roping off the lot, he covers the entire lot with a layer of cement. Cement costs $4 per bag and each bag covers 20 sq. ft. How much does he need to spend on cement?”

• MP2: Reason abstractly and quantitatively 
• In Unit 2, Lesson 3, Independent Practice Question 2 (Master Level), students have to use ratio reasoning with fractions and then put their answer back into context in order to answer the question. The materials state, “Melanie is planning a hiking trip. She knows that she estimates that she will finish a liter of water every $$\frac{3}{4}$$ mile that she hikes. Using the table below (given 3 trails and their distance), answer each of the following questions. a) How many liters of water will Melanie drink if she does the Lily Pond loop? b) How many liters of water will Melanie drink if she hikes to Starlight point and back?”
• In Unit 5, Lesson 9, Independent Practice, Question8 (PhD level), students use percent reasoning to quantitatively derive a final solution. The materials state, “Peter and his friends are all driving from Boston to New York. Peter is driving 120 miles per hour. Lauren is driving 80% as fast as Peter. Gabe is driving 75% as fast as Lauren. And, Deshawn is driving 25% as fast as Lauren. What percent of Peter’s speed is Deshawn driving?” 
• In Unit 7, Lesson 14, Exit Ticket, students reason abstractly and quantitatively as they solve and graph the solution for one-step inequalities. The materials state, “For each inequality below, solve, graph the solution, and check your answer. a) $$20≥m+7$$; b) $$\frac{1}{3}<4$$”

• MP6: Attend to precision  

• In Unit 3, Lesson 11, Independent Practice, Question 1, (Bachelor Level), students must accurately create an absolute value expression to calculate the distance of line segments. The materials state, “Find the lengths of the line segments whose endpoints are given below. Write an expression using absolute value to represent the length of each line segment. a) (-3, 4) and (-3, 9); b) (2, -2) and (-8, -2); c) (0, -11) and (0, 8).”
• In Unit 6, Lesson 3, Exit Ticket Question 3, students must clearly communicate using accurate vocabulary. The materials state, “When you take a taxi, the driver charges an initial fee when you start a ride and then an additional charge for every mile driven in the cab. If the total fare for a cab ride is $$3.5m + 2.5$$ after riding for m miles, what does the 2.5 represent? Explain (use appropriate math vocabulary in your explanation).”
• In Unit 7, Lesson 6, Independent Practice, Question 4, (Master Level), students must understand mathematical symbols in order to demonstrate understanding of a description. The materials state, “Which equation(s) represent the following description? Select all that apply. One-fourth of a number is $$3\frac{1}{4}$$. a) $$\frac{1}{4}x=3\frac{1}{4}$$; b) $$x-\frac{1}{4}=3\frac{1}{4}$$; c) $$x\div\frac{1}{4}=3\frac{1}{4}$$; d) $$x\div4=3\frac{1}{4}$$; e) $$(4)(3\frac{1}{4})=x$$; f) $$x=3\frac{1}{4}+3\frac{1}{4}+3\frac{1}{4}+3\frac{1}{4}$$.”

• MP7: Look for and make use of structure
• In Unit 8, Lesson 1, Opening, THINK ABOUT IT!, students understand that rectangles with the same area can have different dimensions and perimeters. The materials state, “Mr. Boyer is creating a rectangular garden outside of his house that has an area of 18 square units. Using the grid below, draw all the possible gardens that he can create with whole number dimensions. Fill out the table below for each of the figures you drew. Mr. Boyer believes that no matter which garden he chooses, the perimeter will be the same. Do you agree or disagree? Explain.”
• In Unit 6, Lesson 12, Think About It! states, “You can represent a number next to parentheses as repeated addition.” During instruction, the teacher states, “The expression $$2(a + 4)$$ means, ‘$$2 × (a + 4)$$.’ Using this understanding, write two equivalent expressions to $$2(a + 4)$$ and explain how you came up with the two expressions.” Students create various examples, and provided with the conjecture, “Applying the distributive property creates equivalent expressions.” Then students are guided through identifying a “quicker route” to use the distributive property. Students are guided through two expressions in Test the Conjecture, followed by independent practice. 
• In Unit 6, Lesson 2, Think About It!, students analyze the structure of expressions with order of operations. The materials state, “Antoine, Rosa, and Michelle are having a debate about how to evaluate an expression. Analyze their work and settle the debate by explaining: 1) Who solved correctly. 2) How you know that they solved correctly.”

• MP8: Look for and express regularity in repeated reasoning 

• In Unit 1, Lesson 8, THINK ABOUT IT!, Debrief, students have the opportunity to generalize understanding about multiplying by form of 1 using powers of 10. The teacher prompts, “Let’s zoom in on A and C. Aside from multiplying each expression by a form of one to create an equivalent expression, what else do you notice is similar about how equivalent expressions were created? Why do you think that similarly is important?  Discuss. SMS: I notice that in both cases the first expression is being multiplied by a form of one made up of two powers of ten $$-\frac{10}{10}$$ in the first and $$\frac{100}{100}$$ in the second. This is necessary because multiplying by a power of ten shifts the digits and in both cases we are now dividing by a whole number divisor. We don’t know how to divide by a decimal divisor, so this step helps us create an expression that we can evaluate. BPQ: What is the impact of multiplying by $$\frac{10}{10}$$ on the dividend and divisor in part A? What about the impact of multiplying by $$\frac{100}{100}$$ in part C? BPQ: What is different about the divisors in the equivalent expressions from the original expressions? How do you think that would be helpful for division? BPQ: Why did the S multiply by a $$\frac{10}{10}$$ and not another form of 1, like $$\frac{6}{6}$$?”
• In Unit 4, Lesson 11, Independent Practice, Question 6 (Master Level), students develop an understanding of repeated addition to create equivalent rates and ratios, and determine that all ratios with the same unit rate are equivalent. The materials state, “At Fun Burger, the burger master can make 25 hamburgers every 5 minutes. The master’s apprentice can make 35 burgers every 7 minutes. The master says that the apprentice still has much to learn because it takes the apprentice more time to make burgers. The apprentice says that they make the same number of burgers in the same amount of time. With whom do you agree? Explain why.”
• In Unit 6, Lesson 1, Think About It, students have the opportunity to make sense of repeated reasoning as they evaluate exponents. The materials state, “Complete the table below. Explain how you came up with the exponential expressions.” The table includes the multiplication expression, product, and exponential expression.

The instructional materials reviewed for Achievement First Mathematics Grade 6 meet expectations that the instructional materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

Student materials consistently prompt students to both construct viable arguments and analyze the arguments of others. Examples include:

• The Guide to Implementing AF Math describes Error Analysis lessons as one way to address MP3. The materials state, “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.”
• In Unit 1, Lesson 7, Independent Practice, Question 2 (Master level), students perform operations with decimals. The materials state, “A plane travels 3,625.2 miles in 18 hours. The pilot told the passengers that the plane is traveling 20.14 miles per hour. Is he correct? How do you know? If he is incorrect, what is the actual speed per hour?” (6.NS.3)
• In Unit 6, Lesson 14, THINK ABOUT IT!, students identify equivalent expressions. The materials state, “Angela drew a regular octagon (meaning all the sides are the same length) with a side length of $$3p+2$$. Write two equivalent expressions that represent the perimeter of the octagon. Explain how you know that the expressions that you wrote are equivalent.” (6.EE.4)
• In Unit 7, Lesson 5, Error Analysis Lesson, Think About It, students investigate 1-step equations. The materials state, “Compare and contrast Scholar A’s work and Scholar B’s work on yesterday’s exit ticket question. Is either scholar correct? Use numbers and/or words to justify your answer on the lines below.” (6.EE.7)
• In Unit 8, Lesson 6, Independent Practice, Question 4 (Master Level), students decompose shapes into triangles to find area. The materials state, “Explain how you can use triangles to derive the area formula for trapezoids. Use an example to help illustrate the explanation.” (6.G.1)
• In Unit 10, Lesson 1, Test the Conjecture, Question 1, students work with measures of center to understand what a single value represents. The materials state, “Over the last ten days, the temperatures in Miami, Florida have been $$80\degree$$, $$78\degree$$, $$80\degree$$, $$76\degree$$, $$85\degree$$, $$84\degree$$, $$82\degree$$, $$79\degree$$, $$76\degree$$, and $$40\degree$$. The weatherman made an error and forgot to include the 40 degrees when finding the mean and median of the data. Should the weatherman report out the incorrect mean or the median in order to try to hide his mistake and report accurately about the weather? Prove your answer mathematically.” (6.SP.3)

The instructional materials reviewed for Achievement First Mathematics Grade 6 meet expectations that the instructional materials assist teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

Teacher materials assist teachers in engaging students in both constructing viable arguments and analyzing the arguments of others, primarily during the initial instruction when students are exploring a concept and in Back Pocket Questions (BPQs). Examples include:

• In Unit 2, Lesson 9, Error Analysis Lesson, THINK ABOUT IT!, students compare exit ticket responses about the magnitude of the fraction quotient in relationship to 1. Teacher prompts include, “Which scholar’s work did you agree with? Turn and tell your partner who you chose and why. Why does this relationship between the dividend and divisor make sense? What error did this scholar make? What did this scholar do to get this correct, and why was that helpful?” (6.NS.1)
• In Unit 5, Lesson 3, THINK ABOUT IT!, students explore unit rate. Teacher prompts include, “Analyze this work. Which S (student) solved correctly? How do you know? Why did we have to apply the unit rate to this problem? What unit rate did each student find? What does it represent? Does $17.50 make sense? Why or why not? Did both unit rates represent the relationship between bananas and cost?” (6.RP.3)
• In Unit 6, Lesson 7, Error Analysis Lesson, THINK ABOUT IT!, students explore multi-step expressions. Teacher prompts include, “Which scholar’s work did you agree with? Turn and tell your partner who you chose and why. What error did this scholar make? What was the impact of this error? What did this scholar do to get this correct, and why was that helpful? How could we generalize this idea to other problems?” (6.EE.6)

The instructional materials reviewed for Achievement First Mathematics Grade 6 meet expectations that materials explicitly attend to the specialized language of mathematics.

The materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams, and symbols. The materials also use precise and accurate terminology and definitions when describing mathematics, and support students in using them. Examples of explicit instruction on the use of mathematical language include:

• In Unit 1, Lesson 11, Interaction with New Material, Debrief, FENCEPOST #1, students use factors to determine the dimensions of two rooms, one with the area of $$24ft^2$$ and the other with an area of $$36ft^2$$. The materials state, “Factors are numbers that you multiply together to get a product. Show call two pieces of S work side by side and just reveal the work for Part A. One piece of work should have all factor pairs listed and the other should be incomplete. What did both Ss do that is similar and what is different? TT. CC. SMS: In both pieces of work, the Ss listed possible dimensions of the rooms. In the second piece of work, the student listed all of the dimensions while the first piece is missing a few. What is the mathematical name for the numbers listed as dimensions? How do you know? Think time. Hands. SMS: The mathematical name is factors. Factors are numbers that you multiply together to get a product. Name the fencepost: Factors are numbers that you multiply together to get a product.” (6.NS.4)
• In Unit 4, Lesson 1, Opening, Debrief, FENCEPOST #1, students find different ways to describe groups made up of 3 fifth graders for every 2 sixth graders. The materials state, “Ratios are pairs of numbers (that are not zero) that represent a part to part relationship or part to total relationship. CC students to share out pre-identified relationships that describe pairs of numbers. As Ss share out, write the relationships on the board. CC students to share out pre-identified relationships that describe triads of numbers. As Ss share out, write the relationships on the board in a different place. The relationships described on the left represent ratios and the relationships described on the right do not. What is different about these relationships? CC. SMS: The ratios only have two numbers. For the relationship between the fifth graders and sixth graders, we call this a part to part relationship because both the 3 and the 2 represent parts, not totals. In the relationships you all described as 2 fifth graders for the total of five students or 3 sixth graders for the total of five students, we call these part to total relationships. Based on what we noticed about ratios, how would you use your initial understanding to describe a ratio? TT. CC. SMS: Ratios are pairs of numbers that compare parts to parts or parts to totals. Name the fencepost: Ratios are pairs of numbers (that are not zero) that represent a part to part relationship or part to total relationship.” (6,RP.1)
• In Unit 9, Lesson 1, Opening, Debrief, FENCEPOST #1, students use fold outs into nets. The materials state, “The shapes that you cut out of the paper are called ‘nets.’ For the first net, what solid were you able to form? How do you know? CC. SMS: I was able to form a rectangular prism. I know this is a rectangular prism because it has opposite parallel bases that are rectangles. For the second net, what solid were you able to form? How do you know? CC. SMS: I was able to form a triangular prism. I know this is a triangular prism because it has opposite parallel bases that are triangles. If S are not able to name the “why” name it for them. In both cases, what did you have to do to figure out what the solid was? TT. CC. SMS: In both cases we had to fold up the net to make the solid. Based on this, how would you describe a net to someone? Name the fencepost: A net is a flat shape that can be folded up into a solid.” (6.G.4)

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

• At the beginning of each lesson plan, there is a section labeled “Key Vocabulary” for the teacher. For example: Unit 4, Lesson 4 “Key Vocabulary:
• Ratio - A comparison of a pair of non-negative numbers, A:B, which are not both 0. Units can be alike or different.
• Term - 1 part of a ratio (i.e. in ratio A:B, A is a term and B is a term). Terms can represent like or unlike quantities.  
• Like Quantities - Two quantities with the same unit (e.g. girls and boys- both people).
• Unlike Quantities - Two quantities with different units (e.g. dollars per gallon, laps per minute).
• Equivalent ratio – Two ratios that express the same relationship between two terms.
• Tape diagram - A diagram used to represent equivalent ratios that have the same units.”  
• Teachers are routinely prompted to use precise vocabulary such as Unit 6, Lesson 4, Debrief states, “Planner’s note: Students absolutely must understand where the constant, operation, variable, and order come from before moving on.”
• There is very little vocabulary emphasis in student-facing materials. For example, there is not a glossary for student reference.