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)

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)

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 . 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)

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:

• 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. “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:

• 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. “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)

• Unit 3, Lesson 16, Independent Practice #2 (Master Level), students demonstrate procedural skill and application as they meet the requirements to construct a rectangle. “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)

• Unit 4, Lesson 5, Exit Ticket, students demonstrate both procedural skill and application as they use ratio reasoning to solve real-world and mathematical problems. “For every six hot dogs that are shipped to a store, two hamburgers are shipped. Yesterday, 12 hamburgers were added making the amount of hot dogs and hamburgers shipped equal. How many hot dogs and hamburgers were shipped originally?” (6.RP.3)

The materials reviewed for Achievement First Mathematics Grade 6 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. The Standards for Mathematical Practice are identified and incorporated within mathematics content throughout the grade level. The Mathematical Practices are listed in the Unit Overviews as well at the beginning of each lesson. There are instances where the Unit Overview gives a detailed explanation of the MPs being addressed within the unit but the lessons do not cite the same MPs.

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

• The Unit 3 Overview outlines the intentional development of MP1. “In lessons 4 and 12, students make sense of integers in real-world contexts by translating them to a number line. In lesson 16, students solve complex, multi-step problems involving integers on the coordinate plane by making sense of real-world situations and persevering to solve them. MP 1 is a major focus of Unit 3 as students make sense of rational numbers in various contexts and persevere in solving problems and situations involving them.”

• Unit 2, Lesson 10, Independent Practice Question 7 (PhD level), students find an entry point and persevere to solve a multi-step problem involving rational numbers. “Amare paid $16.50 to buy a book. The cost of the book is the cost of a magazine. He bought 3 books and 2 magazines with a $50 bill. How much change should he receive?”

• Unit 5, lesson 12, Independent Practice Question 7 (PhD Level), students make sense of a familiar situation in order to find a solution. “Tricia had a birthday party. During the party, she opened 36 gifts, which was 60% of all of her gifts. After the party, she opened the rest of the gifts and found that 25% of them were the same present, so she returned all but one of the duplicate gifts. How many gifts did she return? Show your work.” 

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

• The Unit 2 Overview outlines the intentional development of MP2. “In lessons 3-5, students reason abstractly and quantitatively about situations involving the division of fractions. By utilizing the context of the problem, students reason to determine the proper division expression for the situation. In lessons 10 and 11, students reason to decontextualize word problems and translate them into expressions and equations for solving. MP 2 is a major focus of Unit 2 as students reason quantitatively about complex situations in order to translate them into mathematical situations they can solve.”

• 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. “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?”

• Unit 7, Lesson 14, Exit Ticket, students reason abstractly and quantitatively as they solve and graph the solution for one-step inequalities. “Exit Ticket: Blayton is at most 2 meters above sea level. Part A: Select all of the statements below that are true about the scenario: a) The phrase at most 2 means values less than two are included; b) The phrase at most 2 means values more than two are included; c) The value 2 would be included in the solution; d) An inequality that represents this scenario is x ≥ 2; e) 2 meters below sea level would be included in the solution set for the inequality representing this scenario. Part B: Represent Blayton’s possible elevation on a graph.”

The materials reviewed for Achievement First Mathematics Grade 6 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

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

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

• Unit 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?” 

• Unit 6, Lesson 14, THINK ABOUT IT!, students identify equivalent expressions. “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.” 

• Unit 7, Lesson 5, Error Analysis Lesson, Think About It, students investigate 1-step equations. “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.” 

• Unit 8, Lesson 6, Independent Practice, Question 4 (Master Level), students decompose shapes into triangles to find area. “Explain how you can use triangles to derive the area formula for trapezoids. Use an example to help illustrate the explanation.” 

• Unit 10, Lesson 5, Test the Conjecture, Question 1, students work with measures of center to understand what a single value represents. “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 materials reviewed for Achievement First Mathematics Grade 6 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

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

• Unit 1, Lesson 5, Question 4 (Master Level), students attend to precision when they apply the division algorithm to divide a three and four-digit dividend by a two- and three-digit divisor. “For each word problem, draw a model, write an equation, and solve the problem: a) Elise has 572 quarters that she has collected over the last several years. She wants to exchange all of her quarters for pennies. How many pennies will she receive in exchange for 572 quarters? b) b. Thomas has 575 pennies that he wants to exchange for quarters. How many quarters will he receive in exchange for his 575 pennies?”

• Unit 4, Lesson 5, Independent Practice, Question 3 (Master Level), students attend to precision as they solve ratio problems involving comparisons using a tape diagram. “Review the student work below and determine whether or not it is correct. If it is correct, explain why. If it is incorrect, explain why and find the correct answer. For every 2 earrings sold at a store, 6 bracelets are sold. Today they had an earring sale and sold 12 extra earrings, making the total number of earrings and bracelets sold the same. How many bracelets did they sell in total?”

• Unit 7, Lesson 5, Independent Practice, Question 1 (Bachelor Level), students attend to precision as they solve and check the solution of one-step equations using substitution. “Directions: Draw a model to solve each equation and check your answer using substitution: 14 = m - 37.” 

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

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

• Unit 6, Lesson 3, Exit Ticket Question 3, students are directed to use appropriate mathematical language. “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).”

• Unit 9, Lesson 1, Opening, Debrief, FENCEPOST #1, students utilize precise vocabulary about 3D shapes. “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?” Students might say, “I was able to form a rectangular prism. I know this is a rectangular prism because it has opposite parallel bases that are rectangles.” The next teacher prompt, “For the second net, what solid were you able to form? How do you know?” Students might say, “I was able to form a triangular prism. I know this is a triangular prism because it has opposite parallel bases that are triangles.”

The materials reviewed for Achievement First Mathematics Grade 6 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards.

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

• Unit 6, Lesson 2, Think About It!, students analyze the structure of expressions with order of operations. “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.” Students look for structure in expressions to manipulate them for efficient simplification. 

• Unit 6, Lesson 12, Think About It!, students learn about the structure of distributive property as a quicker route of creating equivalent expressions. “You can represent a number next to parentheses as repeated addition.” During instruction, the teacher, “The expression $2(a + 4)$ means, ‘.’ Using this understanding, write two equivalent expressions to $2(a + 4)$ and explain how you came up with the two expressions.” Students might say, “I agree with the expression because we know that multiplication is the same as repeated addition. If we think of 2 as the number of groups and $(a + 4)$ as the group size, then we have two groups of $(a + 4)$, which can be written as $(a + 4) + (a + 4)$.” 

• Unit 8, Lesson 1, Opening, THINK ABOUT IT!, students look for and make use of structure as they analyze the structure of each shape they are working with and reason about this structure to derive the area of it. “Louis’ teacher asked him to find the area of the parallelogram pictured below. He wasn’t sure what the area formula was for parallelograms, so he cut a right triangle off of the left side of the parallelogram and moved the triangle to the other side of the parallelogram, forming a rectangle. You can see his steps below. Assuming that what Louis did is okay, how should he find the area of the parallelogram? Show and explain in the space below.” 

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

• Unit 1, Lesson 7, THINK ABOUT IT!, Debrief, students have the opportunity to generalize understanding about “multiplying a division expression by a fraction in the form of 1 does not change the quotient.” The teacher prompts, “Based on your calculations, we know that the expressions in set A are equivalent and the expressions in set C are equivalent because you can multiply the divisor and dividend each by 10 and by 100 respectively. Do you agree or disagree with this student’s explanation and work?” Students might say, “I agree with this work because the student showed that the expressions are equivalent by multiplying by a form of 1. For example, in set A, you can multiply the first expression by $\frac{10}{10}$, which is equal to 1, to get the second expression. For set C, you can multiply the first expression by $\frac{100}{100}$, which is also equal to 1 and therefore the resulting expression is equivalent to the first. For set B, the student did not multiply by a form of 1. Instead, s/he just multiplied the denominator by 10 and the quotients were not equivalent.” Students express regularity in repeated reasoning when manipulating division expressions involving decimals. They shift place value relationships to make solving these expressions easier for them. 

• Unit 4, Lesson 12, Independent Practice, Question 2 (Bachelor 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. “A publishing company is looking for new employees to type novels that will soon be published. The publishing company wants to find someone who can type at least 45 words per minute. Dominique discovered she can type at a constant rate of 704 words in 16 minutes. Does Dominique type at a fast enough rate to qualify for the job? Explain why or why not.”

• Unit 6, Lesson 1, Think About It, students have the opportunity to make sense of repeated reasoning as they evaluate exponents. “Complete the table below. Explain how you came up with the exponential expressions.” The table includes the multiplication expression, product, and exponential expression.

The materials reviewed for Achievement First Mathematics Grade 6 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development. Teacher guidance is found throughout the materials in the Implementations Guides, Unit Overviews, and individual lessons.

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

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

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

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

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

• “What do we want every student to take away or do as a result of this lesson? How will a teacher know if students have met this goal? Understand: As a result of this lesson, every student understands that they can write an expression to represent a real world context. They understand that the real world context and the algebraic expression represent the same thing. Additionally, students understand that they have to very specifically define the meaning of a variable used in an expression to ensure that the expression does in fact represent the real world context accurately and is interpreted correctly. Do: As a result of this lesson, every student is able to define a variable and write an expression to represent a real world context.”

• “ANTICIPATED MISCONCEPTIONS AND ERRORS: Students may not define the variable clearly or may define it inaccurately. Students may switch the order of the number and variable in the expression.”

• “Planner’s note: Students absolutely must understand that the expression does not correctly represent the context if the variable is not defined correctly.”

• Teacher prompts: “Without using numbers, what is this problem asking us to do? How did you annotate the problem and what additional meaning did the annotations give the problem? How should we apply our KP (Key Point) to this problem? Teacher writes on the board and students write in notes. On your own, write an expression to represent this context. Be prepared to defend the expression you wrote. Do we agree with this expression? Why?”

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

• “Key Strategy/ies: Annotate the problem with margin notes (part, total, group size, and number of groups as well as operations when appropriate). Define the variable clearly. Write an expression to represent the context.”

• “CFS (Criterion for Success) for top quality work: Problem is annotated with margin notes. Variable is defined clearly. Expression is written.”

The materials reviewed for Achievement First Mathematics Grade 6 meet expectations for containing adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current grade so that teachers can improve their own knowledge of the subject. 

Materials contain adult-level explanations and examples of the more complex grade/course-level concepts so that teachers can improve their own knowledge of the subject. The Unit Overview includes Appendix: Teacher Background Knowledge which provides “clear links, excerpts, and specific pages from the Common Core the Number System, 6-8 Progression related to the unit content.” Examples include:

• The Unit Overview Appendix also often includes an excerpt from an unknown source which provides a teacher with an understanding of grade-level standards progression. Unit 9, Appendix A: Teacher Background Knowledge 6.NS.1, Dividing Whole Numbers by Fractions, “To divide whole numbers by fractions it is important to first remember that the whole number is also expressed as a fraction with the whole number as the numerator and 1 as the denominator. Often students bring with them misconceptions based on what the problem may look like. It is important to help guide students when they divide a whole number by a fraction. For example, when presented with an exercise in which students are to find the value of $3\div\frac{1}{8}$, they may be tempted to think the solution to the problem is $\frac{3}{8}$, or 0.375. What is really being expressed is the number of eights there are in 3. Following an algorithm, the exercise can be rewritten as $\frac{3}{1}\times\frac{8}{1}=\frac{24}{1}$ or 24. Therefore there are 24 eighths in 3. Think of $5\div\frac{1}{6}$ as the number of sixths that fit into 5. There are 6 sixths in 1, so in 5 there are $5\times6$ or 30 sixths in 5.”

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

• Unit 3 Overview, The Number System – Understanding and Representing Rational Numbers, Identify the Narrative, “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.”

• Unit 5 Overview, Ratios and Proportional Relationships – Application of Ratios and Rates, Identify the Narrative, “While this unit moves students’ understanding and application of ratios to a more abstract level, it is important to continue to ground their thinking in the tables and double number line diagrams used during Uunit 4 to solidify their conceptual understanding of ratio and rate applications. In 7th grade, students will build off of the understanding and skills developed in 6th grade and operate at a more abstract level with expressions, equations, and rational numbers. Students develop an understanding of proportionality and look for proportionality in relationships represented as graphs, tables, and equations. Also, students will solve multi-step real-life and mathematical problems involving scale drawings and percent problems involving real-world scenarios like taxes and tips. In 8th grade, students will apply their understanding of proportional relationships as they work with lines and linear equations, slope and rate of change. Throughout high school, students work extensively with functions.  This work depends on a strong conceptual understanding of ratios and proportional relationships.”

• Unit 10 Overview, Statistics and Probability – Representing and Analyzing Data, Identify the Narrative, “Looking ahead to future grades, students continue their study of statistics in 7th grade with a focus on random sampling, drawing inferences about populations based on random samples, and drawing comparative inferences about two populations by analyzing data represented in graphs and measuring center, spread, and shape. Later, students learn about bivariate data in 8th grade in the context of scatter plots and two-way tables. Then, in high school, students formalize their understanding of variability by measuring the standard deviation to interpret and compare data sets in 9th grade. With standard deviation, students will ultimately be able to discuss the likelihood that an event occurred based on its distance from the mean and the event’s standard deviation. In 12th grade, students will also use more formal rules to determine significance, building on their basic understanding of the concepts from 7th grade.”

The materials reviewed for Achievement First Mathematics Grade 6 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.

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

• Guide to Implementing AF Grade 6, Program Overview, “Scope and Sequence Detail is designed to help teachers identify the standards on which each lesson within a unit is focused, whether on grade level or not. You will find the daily lesson aims within each unit and the content standards addressed within that lesson. A list of the focus MPs for each lesson and unit and details about how they connect to the content standards can be found in the Unit Overviews and daily lesson plans.”

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

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

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

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

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

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

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

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

• Unit 6, Lesson 2, Connection To Learning and Conceptual Understanding identifies previous skills for grade level related standards, “How does this lesson connect to previous lessons? In the previous lesson, students learned how to evaluate exponential expressions. They used the order of operations to evaluate expressions with exponents, multiplication and division, and addition and subtraction. In this lesson, they incorporate grouping symbols. This directly builds off of their work with grouping symbols in 5th grade while adding exponents into the mix.” 

• In the Unit 8 Overview, the narrative provides skills from previous years and connects the units’ lessons. “Starting in elementary school and 5th grade students develop area concepts. Students move from building arrays to understand multiplication and division to using arrays to find the area of rectangles and squares. From there, they use their understanding of area to develop an understanding of volume by decomposing three-dimensional shapes into rectangular layers (5.MD.C.3, 5.MD.C.5). Additionally, in 5th 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 strong base understanding of the concept from which to build as students dive deeper into their study of area in Unit 8.”

The materials reviewed for Achievement First Mathematics Grade 6 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies.

Materials explain the instructional approaches of the program. Examples include:

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

Research-based strategies are cited and described within the Program Overview, Guide to Implementing AF Math: Grade 5-8, Instructional Approach and Research Background and References. Examples of research-based strategies include:

• Concrete-Representational-Abstract Instructional Approach, Access Center: Improving Outcomes for All Students K-8, OESP, “Research-based studies show that students who use concrete materials develop more precise and more comprehensive mental representations, often show more motivation and on-task behavior, understand mathematical ideas, and better apply these ideas to life situations.”

• Introduction to the Math Shifts, by Achieve the Core, 2013, “According to the National Council of Teachers of Mathematics, Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies.”

• Orchestrating Productive Mathematical Discussions: Five Practices for Helping Teachers Move Beyond Show and Tell by Common Core Tools, “According to the National Mathematics Advisory Panel (2008), explicit instruction includes ‘teachers providing clear models for solving a particular problem type using an array of examples,’ students receiving extensive practice, including many opportunities to think aloud or verbalize their strategies as they work, and students being provided with extensive feedback.”

The materials reviewed for Achievement First Mathematics Grade 6 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

The Program Overview, Guide to Implementing AF Math: Grade 6, Scope and Sequence Detail, Supplies List includes a breakdown of materials needed for each Achievement First Mathematics Program. Examples include:

• Grades 5-8 Math Instructional Materials Purchase List, Math Supplies, “ETA Hand2mind Unit Cubes, Set of 1000.”

• Grades 5-8 Math Instructional Materials Purchase List, Math Supplies, “ETA Hand2mind Classroom Number Line (-20 to 100), 1 for each math classroom.”

• Grades 5-8 Math Instructional Materials Purchase List, Math Supplies, “ETA Hand2Mind TI-30X IIS Calculator (Set of 30), Need enough for State Testing.”

The materials reviewed for Achievement First Mathematics Grade 6 meet expectations for having assessment information included in the materials to indicate which standards are assessed. 

Unit Assessments consistently and accurately identify grade-level content standards along with the mathematical practices within each unit. Examples from unit assessments include:

• Unit 2 Overview, Unit 2 Assessment: Division of Fractions, denotes the aligned grade-level standards and mathematical practices. Question 6, “The area of a rectangular poster is $\frac{7}{8}$ square feet. The length of the poster is $\frac{1}{4}$ foot. What is the width of the poster?” (6.NS.1, MP2, MP4, MP8)

• Unit 6 Overview, Unit 6 Assessment: Algebraic Expressions, denotes the aligned grade-level standards and mathematical practices. Question 7, “Write an expression that is equivalent to 64 using each of the following numbers and symbols once in the expression. 7, 7, 7, 2 (exponent of 2),  +, , (  ).” (6.EE.1, MP7, MP8)

• Unit 9 Overview, Unit 9 Assessment: Volume & Surface Area, denotes the aligned grade-level standards and mathematical practices. Question 4, “The volume of a rectangular prism is 758.16 cubic mm. The length is 13mm and the height is 7.2 mm. What is the width of the prism?” (6.G.2, MP1, MP2, MP4)

The materials reviewed for Achievement First Mathematics Grade 6 meet expectations for including an assessment system that provides multiple opportunities throughout the grade, course, and/or series to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

The assessment system provides multiple opportunities to determine students' learning and sufficient guidance to teachers for interpreting student performance. Examples include:

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

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

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

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

Program Overview, Guide to Implementing AF Math: Grade 6, Differentiation, Unit-Level Errors, Misconceptions, and Response, “Every unit plan includes an ‘Evaluating and Responding to Student Learning Outcomes’ section after the post-unit assessment. The purpose of this section is to provide teachers with the most common 1-2 errors as observed on the questions related to each standard, the anticipated misconceptions associated with those errors, and a variety of possible responses that could be taken to address those misconceptions as outlined with possible critical thinking, strategic practice problems, or additional resources.” Examples include: 

• Unit 1 Overview, Unit 1 Assessment: Whole Number and Decimal Operations, Evaluating and Responding to Student Learning Outcomes, Suggestions for How to Respond, 6.NS.2, “Suggested re-teach activities by question group: Q3 – 6: Scholars can make numerous errors during long division including incorrect digit placement, not using a zero place holder, multiplication errors, and subtraction errors. Analyze the most common error and derive a key point for students to focus on during word problem practice. Below are suggested questions to elicit student understanding: ‘Can we make any groups of our divisor? How many?’ ‘If we cannot make any groups, how can we represent that in our quotient?’ ‘What place value should we record our quotient in? How do you know?’ Lessons for possible re-teach focus: Lesson 5: This lesson focuses on dividing three and four digit dividends by two and three digit divisors. The re-teach should allow students to get multiple at-bats with feedback on the accuracy of their division as well as the accuracy of their expression.”

• Unit 4 Overview, Unit 4 Assessment: Ratios and Proportional Relationships – Understanding Ratios and Rates, Evaluating and Responding to Student Learning Outcomes, Suggestions for How to Respond, 6.RP.1, “Suggested re-teach activities by question group: Q1, 2, 5: Allow scholars to compare an exemplar and non-exemplar of a problem where students are tasked with creating or finding an equivalent ratio. The non-exemplar should have mislabeled units or no units labeled at all in their ratio table. Allow scholars to identify the error and derive criteria for success to implement during practice. Below is a suggested CFS: Draw and label ratio table w/ units. Extend ratio, showing calculations. Check units for accuracy. Lessons for possible re-teach focus: Lessons 1 & 2: These lessons focus on students using ratio language to describe a ratio relationship between two quantities, expressing ratios in simplest form, and representing ratio relationships with diagrams. The varied practice allows scholars to apply their understanding of ratios to contextualize the relationship between two quantities.”

• Unit 8 Overview, Unit 8 Assessment: Geometry: Area, Evaluating and Responding to Student Learning Outcomes, Suggestions for How to Respond, 6.G.3, “Suggested re-teach activities by question group: Q7, 8: Scholars who begin counting squares have a strong understanding of area as the space inside a figure. Leverage this understanding during a guided model, creating criteria for success for practice. Prompt students to decompose the figure on their own, leveraging the most common decomposition while acknowledging that there are multiple pathways to solutions. Below are some suggested criteria for success: Plot and connect points. Decompose into regular figures. Label new dimensions. Find area of each figure. Find composite area. Lessons for possible re-teach focus: Lesson 7: This lesson focuses on the skill of calculating area of figures on a coordinate grid. Being that counting the number of squares inside a figure is a valid way to determine area, allow scholars to begin by practicing with figures that have whole number dimensions, prompting them to use a numerical method. As students’ progress, incorporate decimal side lengths, and fractional side lengths for scholars who are demonstrating mastery.”

The materials reviewed for Achievement First Mathematics Grade 6 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series. There are a variety of question types including multiple choice, short answer, and constructed response. Mathematical practices are embedded within the problems. 

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

• The Unit 3 assessment contributes to the full intent of 6.NS.7 (understand ordering and absolute value of rational numbers). Item 1, “1) Lucy's bank statement for last week shows $300 for a deposit into her checking account and –$300 for a withdrawal. (2pts) Select all statements that are true. A) The withdrawal amount is more than the deposit amount. B) The sum of the deposit and withdrawal equals $0. C) The absolute value of both the deposit and withdrawal is $0. D) The amounts of the deposit and withdrawal would be the same distance from 0 on the number line.”

• The Unit 6 assessment contributes to the full intent of 6.RP.3c (use ratio and rate reasoning to solve real-world and mathematical problems). Item 12, “A clothing store offers a 30% discount on all items in the store. a) If the original price of a sweater is $40, how much money will a customer save with the discount? b) A shopper bought a shirt and saved $24 with the discount. What was the original cost of the shirt?”

• Unit 9, Lesson 4, Exit Ticket, Problem 1 contributes to the full intent of 6.G.2 (find the volume of a right rectangular prism with fractional edge lengths). “Jake is filling up the fish tank below completely with water. How many cubic feet of water will he need?” The dimensions of the tank are $2\frac{1}{3}$ feet by $\frac{2}{4}$ foot by $1\frac{1}{2}$ feet.

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

• Unit 2 Assessment, Item 5, supports the full development of MP4 by modeling to represent the problem. "Barry is making pillows. He needs $\frac{3}{4}$ of a yard of fabric to make each pillow. If Barry has 6 yards of fabric, how many pillows will he be able to make? (2 pts) a. Use a model to represent the problem and solution.”

• Unit 5 Assessment, Item 10, supports the full development of MP3 (constructing arguments and critiquing the reasoning of others). "Tyler drew a double number line diagram and stated that 75% of 24 is 15. Is he correct?  Explain why or why not and include your own double number line diagram as part of your justification.”

• Unit 7 Assessment, Item 4, supports the full development of MP2 (reasoning abstractly and quantitatively). “Antoine ate 3 times as many apples as Jose ate on Monday and Tuesday. In total, Antoine ate 12 apples. Jose ate a certain number of apples on Monday and 4 apples on  Tuesday. Write an equation that you could solve to find out how many apples Jose ate on Monday.“

The materials reviewed for Achievement First Mathematics Grade 6 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level mathematics.

Materials regularly provide strategies, supports, and resources for students in special populations to help them access grade-level mathematics as suggestions are outlined within each unit overview. According to the Program Overview, Guide to Implementing AF Math: Grade 6, Differentiation, Supporting Students with Disabilities, “Without strong support, students with disabilities can easily struggle with learning and not feel successful. Therefore, it is critical that strong curricular materials provide support for all student learners, but are created in a way to support students who have diagnosed disabilities. Our program has been designed to align to the elements identified by the Council for Learning Disabilities that should be used in successful curriculum and instruction: Specific and clear teacher models, Examples that are sequenced in level of difficulty, Scaffolding, Consistent Feedback, and Frequent opportunity for cumulative review. Unit Overviews and lesson level materials include guidance around working with students with disabilities, including daily suggested interventions in the Workshop Section of the daily lesson plan.” 

Examples of supports for special populations include: 

• Unit 2 Overview, The Number System – Dividing Fractions, Differentiating for Learning Needs, “Previous Grade Content: Students have worked with fraction division in 5th grade Unit 9: Dividing Fractions. Scholars begin dividing whole numbers by fractions and fractions by whole numbers. Lessons 1 – 4 may be useful in differentiating pre-skill content.” Error Analysis, “The unit includes Error Analysis Lesson 9.2. This lesson focuses on the specificity of student justifications and logic of the relationship of a quotient to the overall magnitude of 1. This lesson has students compare justifications and identify the key components of a strong explanation. This lesson should be included after Lesson 9, Day 1 as an extension.” Responding to Student Learning Outcomes, “See the Unit Assessment ‘Evaluating and Responding to Student Learning Outcomes’ at the end of the Overview for suggestions on unit-level common errors, misconceptions, and suggestions on how to respond. These can be useful for supporting struggling learners proactively throughout the unit.” Student Grouping Suggestions, “Pre-Test: Use the 6th Grade, Unit 2 Pre-test and Key to identify student strengths and weaknesses when it comes to understanding how to multiply fractions, divide a whole or mixed number by a unit fraction, and divide a unit fraction by a whole or mixed number. Identify specific problems to sequence through cumulative review and create groupings of students for small group instruction during that period. Consider changing student seating so that students who struggled with the pre-test are seated next to students who had higher mastery for support throughout the unit or strategically group students who struggle together for teacher support or small group instruction. Exit Tickets: Closely analyze student mastery of the Exit Tickets for Lessons 1, 2, 3, 5 and 8 as these lessons are the foundation for students to understand how to represent fraction division. Students who have struggled with these should be prioritized for small group instruction and/or more support during independent practice prior to the end of the unit where these skills will all need to have reached full mastery to solve complex word problems involving fraction division using a model and computational procedures.”

• Unit 6 Overview, Expressions and Equations –Algebraic Expressions, Differentiating for Learning Needs, “Visual Anchors: Throughout this unit, students will need to define variables and write expressions and equations. Teachers and students may find it useful to use clear visual anchors throughout the unit that show an exemplary problem completed along with student-friendly criteria for success.” Differentiated Problems, “To ensure that all students, regardless of previous mastery level, can engage in regular and active participation in grade-level mathematics, teachers should prepare each lesson with a differentiated set of problems for students to complete based on their mastery either from previous, related content or based upon informal assessment of mastery from the Think About It and Test the Conjecture portion of the lesson. For example, in Lesson 7, to ensure all students are prepared to show mastery on the Exit Ticket, students that are showing lower mastery could be assigned the following problems: PP Bachelor 1 and IP Master 2 & 3. These problems would ensure students have had practice writing an expression, identifying equivalent expressions, and justifying a claim.”

• Unit 9 Overview, Geometry:  Volume and Surface Area, Differentiating for Learning Needs, “Visual Anchors: Throughout this unit, students will need to label key information, apply formulas, and draw nets. Teachers and students may find it useful to use clear visual anchors throughout the unit that show an exemplary problem completed along with student-friendly criteria for success.” Differentiated Problems, “To ensure that all students, regardless of previous mastery level, can engage in regular and active participation in grade-level mathematics, teachers should prepare each lesson with a differentiated set of problems for students to complete based on their mastery either from previous, related content or based upon informal assessment of mastery from the Think About It and Test the Conjecture portion of the lesson. For example, in Lesson 5, to ensure all students are prepared to show mastery on the Exit Ticket, students that are showing lower mastery could be assigned the following problems: PP Bachelor 1, IP Bachelor 1, and PP Master 2. These problems would ensure students have had practice finding the area by counting the amount of space that the net will cover and creating a net from a three-dimensional figure.”

The materials reviewed for Achievement First Mathematics Grade 6 meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.

While there are no instances where advanced students do more assignments than classmates, materials do provide multiple opportunities for students to investigate grade-level content at a higher level of complexity. According to the Program Overview, Guide to Implementing AF Math: Grade 6, Differentiation, Supporting Advanced Students, “Part of supporting all learners is ensuring that advanced students also have opportunities to learn and grow by engaging with the grade level content at higher levels of complexity. A problem-based approach is naturally differentiated as students choose the strategies they use to model and solve the problem. Teachers highlight particular strategies for the class, but they always affirm any strategy that works, regardless of its level of complexity. In a classroom implementing the Achievement First Mathematics program, students are expected to work with a variety of tools and strategies even as they work through the same set of problems; this allows advanced students to engage with the content at higher levels of complexity. Daily lessons resources (DLRs) also provide differentiated problems labeled by difficulty. Teachers should differentiate for student needs by assigning the most challenging problems to advanced students while allowing them to skip some of the simpler ones, so that they can engage with the same number of problems, but at the appropriate difficulty level.” Independent Practice in each lesson provides three levels of rigor in the lesson for student work: Bachelor, Master, and PhD work. Examples include:

• Unit 4, Lesson 5, Independent Practice Bachelor Level, “Dwhite can complete 5 laps around the track for every 2 laps that Jaden completes. Dwhite had an extra energy bar and competed 9 more laps than Jaden today. How many laps did Dwhite run? How many laps did Jaden run?” 

• Unit 4, Lesson 5, Independent Practice Master Level, “At a country concert, the ratio of the number of boys to the number of girls is 2:7. If there are 250 more girls than boys, how many boys are at the concert?”

• Unit 4, Lesson 5, Independent Practice PhD Level, “Luis and Olivia are saving money to buy their favorite candy at the store. For every $5 Luis saves, Olivia saves $7. After a week, Luis saved $42 less than Olivia. If their favorite candy costs $2.50 per pack, how many packs of candy can each of them buy?”

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.”

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.