## Alignment: Overall Summary

The instructional materials reviewed for Achievement First Mathematics Grade 7 partially meet expectations for alignment to the CCSSM. ​The instructional materials meet expectations for Gateway 1, focus and coherence, by assessing grade-level content, focusing on the major work of the grade, and being coherent and consistent with the Standards. The instructional materials partially meet expectations for Gateway 2, rigor and balance and practice-content connections. The materials meet the expectations for rigor and balance and partially meet the expectations for practice-content connections.

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## Gateway 1:

### Focus & Coherence

0
7
12
14
14
12-14
Meets Expectations
8-11
Partially Meets Expectations
0-7
Does Not Meet Expectations

## Gateway 2:

### Rigor & Mathematical Practices

0
10
16
18
15
16-18
Meets Expectations
11-15
Partially Meets Expectations
0-10
Does Not Meet Expectations

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## Gateway 3:

### Usability

0
22
31
38
N/A
31-38
Meets Expectations
23-30
Partially Meets Expectations
0-22
Does Not Meet Expectations

## The Report

- Collapsed Version + Full Length Version

## Focus & Coherence

#### Meets Expectations

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Gateway One Details

The instructional materials reviewed for Achievement First Mathematics Grade 7 meet expectations for Gateway 1, focus and coherence. The instructional materials meet the expectations for focus by assessing grade-level content and spending 62% of instructional time on the major work of the grade, and they also meet expectations for being coherent and consistent with the standards.

### Criterion 1a

Materials do not assess topics before the grade level in which the topic should be introduced.
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Criterion Rating Details

The instructional materials reviewed for Achievement First Mathematics Grade 7 meet expectations for not assessing topics before the grade level in which the topic should be introduced.

### Indicator 1a

The instructional material assesses the grade-level content and, if applicable, content from earlier grades. Content from future grades may be introduced but students should not be held accountable on assessments for future expectations.
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Indicator Rating Details

The instructional materials reviewed for Achievement First Mathematics Grade 7 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 13 states, “Shonte’s bank statement shows that her balance is $25.40. She has an outstanding bill of$65.99 that she has to pay immediately in order to avoid paying a fine. What will her account balance show after she pays her bill?” (7.NS.3)
• In Unit 5 Assessment, Question 1 states, “Talik walked $$\frac{1}{2}$$ of a mile in $$\frac{1}{4}$$ of an hour. Cedric walked $$\frac{3}{4}$$ of a mile in $$\frac{3}{4}$$ of an hour. If these rates remain constant, which of the following statements is true? a) The two boys would walk the same distance in the same amount of time. b) Cedric would walk $$\frac{1}{2}$$ mile in less time tha(n) Talik. c) Talik would walk $$\frac{1}{2}$$ mile in less time than Cedric. d) Who walks faster depends on how far they walk.” (7.RP.1)
• In Unit 6 Assessment, Question 12 states, “There were 48 cookies and 40 brownies in a jar on Monday. The next day, the number of cookies in the jar increased by 25%, and the number of brownies in the jar decreased by 10%. Find the overall percent change in goodies in the jar to the nearest whole number.” (7.EE.3)
• In Unit 8 Assessment, Question 2 states, “Eight of the 32 students in your seventh-grade math class have a cold. The student population is 450. Your classmate estimates that 112 students in the school have a cold. a) Is this a reasonable conclusion to draw from the data? Explain why or why not. b) Describe a survey plan you could use to better estimate the number of students who have a cold. Include all necessary parts of the plan for creating a fair sample and collecting data.” (7.SP.1)
• In Unit 9 Assessment, Question 14 states, “A 3D figure was sliced perpendicular to its base and the plane section that resulted was a triangle. The figure was then sliced horizontally and the plane section that resulted was a square. What is the name of one 3D shape from which the plane section could have come from? Explain on the lines below.” (7.G.3)

### Criterion 1b

Students and teachers using the materials as designed devote the large majority of class time in each grade K-8 to the major work of the grade.
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Criterion Rating Details

The instructional materials reviewed for Achievement First Mathematics Grade 7, when used as designed, spend approximately 62% of instructional time on the major work of the grade, or supporting work connected to major work of the grade.

### Indicator 1b

Instructional material spends the majority of class time on the major cluster of each grade.
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Indicator Rating Details

The instructional materials reviewed for Achievement First Mathematics Grade 7 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 6.5 out of 10, which is approximately 65%.
• The number of days devoted to major work of the grade (including assessments and supporting work connected to the major work) is 95 out of 140, which is approximately 68%.
• The number of minutes devoted to major work (including assessments and supporting work connected to the major work) is 7825 out of 12,600 (90 minutes per lesson for 140 days), which is approximately 62%.

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 62% of the instructional materials focus on major work of the grade. However, because 62% is close to the benchmark and both other measures met or exceeded the benchmark, Grade 7 meets the requirements for spending the majority of class time on major clusters.

### Criterion 1c - 1f

Coherence: Each grade's instructional materials are coherent and consistent with the Standards.
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Criterion Rating Details

The instructional materials reviewed for Achievement First Mathematics Grade 7 meet expectations for being coherent and consistent with the standards. The instructional materials have supporting content that engages students in the major work of the grade and content designated for one grade level that is viable for one school year. The materials also foster coherence through connections at a single grade.

### Indicator 1c

Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.
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Indicator Rating Details

The instructional materials reviewed for Achievement First Mathematics Grade 7 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 4, Lesson 9, supporting standard 7.G.4 enhances the major work of 7.EE.4. Students represent and solve multi-step geometric problems using a complex equation. Independent Practice Question 5 (Master level) states, “A hexagon has six congruent sides and each side length is $$\frac{1}{2}n+2$$. What is the measure of one of the side lengths if the perimeter is 25?”
• In Unit 5, Lesson 12, supporting standard 7.G.1 enhances the major work of 7.RP.2. Students understand a special kind of proportional relationship in scale drawings as either a reduction or the enlargement of a two-dimensional picture and determine the constant of proportionality that relates scale drawings as the scale factor. For example, Independent Practice Question 6 (PhD level) states, “On a blueprint for an apartment building, the height of the door is 4 inches tall. The actual door is 84 inches high. If the rest of the blueprint follows this exact same scale, what would be the actual dimensions of a room that is 10 inches long and 18 inches wide on the blueprint? Express your answer in terms of feet (12 inches = 1 foot).”
• In Unit 7, Lesson 6, supporting standard 7.SP.6 enhances the major work of 7.RP.2. Students develop uniform and non-uniform probability models and use proportional reasoning to predict the approximate relative frequency of outcomes (based on theoretical probability). For example, Independent Practice Question 6 (Master level) states, “Yasmine has a bag of snacks that contains 40% Cheetos, 25% Doritos, 10% Fritos, and 25% pretzels. a) If she reaches into the bag and grabs one snack, and does so 15 times, how many Cheetos do you expect her to get?  b) Yasmine likes all the types of snacks except for Doritos. If she grabs a total of 40 snacks, about how many times will she get a type of snack that she likes?”
• In Unit 10, Lesson 1, supporting standard 7.G.5 enhances the major work of 7.EE.4. Students define complementary and supplementary angles and determine the measurement of a missing angle by writing a simple equation. For example, in Partner Practice Question 5 (Master level), students are given a diagram of two intersecting lines and a ray coming out at $$90\degree$$ and asked, “In the diagram below, angle ABE is $$90\degree$$. Angle EBD measures $$3x$$ and angle DBC measures $$2x-10$$. What are the measures of angles EBD and DBC?”

### Indicator 1d

The amount of content designated for one grade level is viable for one school year in order to foster coherence between grades.
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Indicator Rating Details

Instructional materials for Achievement First Mathematics Grade 7 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 lesson in Unit 8 between Lessons 4 and 5. Since it is optional, it is not included in the total count.

According to The Guide to Implementing Achievement First Mathematics Grade 7, 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.”

### Indicator 1e

Materials are consistent with the progressions in the Standards i. Materials develop according to the grade-by-grade progressions in the Standards. If there is content from prior or future grades, that content is clearly identified and related to grade-level work ii. Materials give all students extensive work with grade-level problems iii. Materials relate grade level concepts explicitly to prior knowledge from earlier grades.
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Indicator Rating Details

The instructional materials for Achievement First Mathematics Grade 7 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. In Unit 1, “In 6th grade, students first developed a conceptual understanding of rational numbers “through the use of a number line, absolute value, and opposites, and extended their understanding to include ordering and comparing rational numbers (6NS5, 6NS6, 6NS7). They further extended their understanding of rational numbers within the context of the coordinate plane (6NS8). $$6^{th}$$ grade also marked the year when students were expected to fluently work with whole number, fraction and decimal operations (6NS1, 6NS2, 6NS3). Early in the unit, students leverage their knowledge of number properties and relationships between operations to understand addition and subtraction of rational numbers.” The end of the Overview previews, “Later, in $$8^{th}$$ grade, students continue to understand rational numbers as they learn about numbers that are not rational, called irrational numbers (8NSA). They also apply their understanding when working with integer exponents (8EEA), graphing and solving (pairs of) linear equations (8EEC), performing translations and dilations (8GA), and using functions to model and compare relationships between quantities (8FA, 8FB). For High School, fluency with rational numbers sets students up to focus on learning new algebraic material in High School that incorporates the use of these numbers and assumes knowledge of them. An understanding of rational number operations also facilitates the understanding of rational functions and how to work with them appropriately.”
• 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. Unit 4 states, “In lessons 4 and 5, students continue to solve two step linear equations using inverse operations and number properties but add on the increased complication of integers (lesson 4) and rational numbers (lesson 5). Students will rely heavily on their learning in the first two units to complete these lessons and future lessons.”
• The last paragraph of each narrative for the teacher in the Unit Overview describes the importance of the unit in the progressions. Unit 6 states, “While percents are not a focus of 8th grade, they may still be applied in a variety of contexts throughout the year, i.e. volume or statistics. In High School, students extend their understanding of percents when applying percents to exponential growth and decay modeling. They need a strong understanding of percentages and their decimal equivalence to be able to conceptually understand the key characteristics of the functions. There is also a connection to proportional reasoning in Geometry, but exponential functions in Algebra are where the connection is most clear.”
• For units that correlate with the progressions document, the materials attach the relevant text so that connections are made. In Unit 8, Appendix A, Teacher Background Knowledge (after the assessment), the “6-8 Statistics and Probability” progression document is included with the footnote, “From the Common Core Progression on Statistics and Probability.”
• Each lesson includes a “Connection to Learning and Conceptual Understanding” section that describes the progression of the standards within the unit. In Unit 4, Lesson 1 states, “In 6th grade, students solved one-step equations through logical reasoning. For example, $$6n = 42$$, students would reason that 6 times some number equals 42 so that number has to be 7. Students will draw from this experience to do the final step but will first have to manipulate the balance/equation in such a way that makes the equation into a one-step equation. Students will determine that they must first remove the additional units (constant) from both sides to keep the equation balanced.”
• 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.” Unit 2 states,  “Unit 2, The Number System- Multiplying and Dividing Rational Numbers states, “Skill Fluency (4 days a week): 7.NS.1 (O), 7.NS.2 (O)” “Mixed Practice (3 days a week): 7.NS.1 (O), 7.NS.3 (O), 7.NS.2 (O), 6.NS.2 (R), 6.EE.2 (R), 7.EE.3.(R).”

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, Lesson 11, Exit Ticket, students solve real-world problems by adding and subtracting rational numbers. The materials state, “Death Valley sits at an elevation of $$212\frac{3}{4}$$ feet below sea level and the temperature at noon is 119.5 degrees.  Mt. Humphrey’s (the tallest point in Arizona) has an elevation of $$13,918\frac{9}{10}$$ feet and the temperature at the top is -19.07 degrees.  What is the change in elevation and temperature between the two locations?” (7.NS.1d)
• In Unit 3, Lesson 3, Independent Practice, Question 9 (PhD level), students understand how quantities are related by rewriting an expression in different forms,. The materials state, “Pretend that you are a test maker. Create four multiple choice answers (one has to be correct) for the problem below. Explain the error that each answer choice addresses. “Write an equivalent expression for $$-4n-3(-2n+3)$$.” (7.EE.2)
• In Unit 4, Lesson 20, Exit Ticket Question 1, students use the formula for the area of a circle to solve problems. The materials state, “The base of John’s coffee cup has a circumference of $$12\pi cm$$. Exactly how much space does the base of the coffee cup take up?” (7.G.4)
• In Unit 6, Lesson 13, Interaction with New Material, Question 1, students use proportional relationships to solve percent problems. The materials state, “Magdalena works at a clothing store and makes both an hourly wage of $8.00 and gets paid a commission rate of 5% on the total cost of all the sales she makes. During her 4-hour shift, only one customer purchased clothing. They bought 2 scarves that normally cost$15 each but were on sale for ‘Buy 1, get 1 50% off’.  How much money did she earn during her shift?” (7.RP.3)

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. Unit 3 states, “The content draws heavily from the work students did in the first two units with rational number operations as well as from the work they did with expressions in $$6^{th}$$ grade. $$6^{th}$$ grade marked the foundation for students beginning to apply algebraic princip to writing expressions that represent real world and mathematical problems. Specifically, students focused on evaluating expressions inclusive of all operations (6.EE.1) and variables and writing and reading expressions with variables (6.EE.2). They also wrote and identified equivalent expressions using their knowledge of properties of operations (6.EE.3, 6.EE.4). With these skills and knowledge, they applied expressions to represent and solve geometric problems (i.e. perimeter of a polygon) (6.G.A). Students must be fluent working with integer and rational number operations prior to this unit.”
• The narrative for the teacher in the Unit Overview makes connections to current work. The materials state, “Unit 7 is the first time students will formally learn about the concept of probability. While students have likely discussed the concept in other classes informally or in their lives outside of school, unit 7 is meant to formalize their understanding of probability and teach students how to utilize probability models and organizational methods to make sense of chance events in the real world. While students have not learned about probability in previous units or grade levels, they draw on previous knowledge learned in fraction, decimal percent and ratio reasoning units from earlier in $$7^{th}$$ grade as well as from previous grade levels.”
• Each lesson includes a “Connection to Learning and Conceptual Understanding” section that relates to prior knowledge. In Unit 5, Lesson 1 states, “In 6th grade, students work with ratios, rates, and unit rates. This intro lesson to unit 5 draws on the work that students have done in 6th grade to be able to write a rate (comparison of two different units) and convert it into a unit rate (a rate in which a unit is compared to 1 of another unit). In this lesson, students determine the units that are being compared and write two different unit rates and describe them in the context that they are given.”
• 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 build off of what they learned about percents in 6th grade with setting up an equation (proportion) to solve problems. Now that they know how to solve equations algebraically, they can do so.”

### Indicator 1f

Materials foster coherence through connections at a single grade, where appropriate and required by the Standards i. Materials include learning objectives that are visibly shaped by CCSSM cluster headings. ii. Materials include problems and activities that serve to 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.
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Indicator Rating Details

The instructional materials for Achievement First Mathematics Grade 7 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 3, Lesson 6: “SWBAT write an expression representing an unknown real-world value; SWBAT recognize that there are multiple equivalent expressions that can represent the same scenario,” is shaped by 7.EE.A: Use properties of operations to generate equivalent expressions.
• The AIM for Unit 5, Lesson 1: “SWBAT compute unit rates associated with quantities in different units,” is shaped by 7.RP.A: Analyze proportional relationships and use them to solve real-world and mathematical problems.
• The AIMs for Unit 8, Lessons 3: “SWBAT make a prediction about an entire population based on data from a sample, Sub-AIM: SWBAT determine whether or not a prediction is possible based on the validity of the sample” and “SWBAT use multiple simulated samples of the same size to gauge the variation in estimates or predictions,” is shaped by 7.SP.A: Use random sampling to draw inferences about a population.

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. For example:

• In Unit 3 Curriculum Review, Problem of the Day 3.1, students solve real-world and mathematical problems involving the four operations with rational numbers (7.NS.A) when they analyze proportional relationships (7.RP.A). In the Problem of the Day 3.1, Day 1 states, “Mr. Milliken is baking a cake for his upcoming wedding! He is going to start with a layer of cake, then a layer of icing, and so on, until the cake is finished with a layer of icing.  When finished, each layer of cake is $$1\frac{1}{5}$$ inches tall and each layer of icing is $$\frac{2}{5}$$ of an inch tall. Mr. Milliken uses $$1\frac{1}{4}$$ cups of sugar for one layer of cake and one layer of icing together.  Ms. Nichols wants the cake to be 16 inches tall. How much sugar will Mr. Milliken use in creating this cake?”
• Unit 3, Lesson 4 connects 7.NS.A and 7.EE.B as students use the full range of rational numbers when they solve algebraic expressions and equations. In the Independent Practice, Question 10 (PhD level) states, “Are the expressions $$-4.5n+3\frac{1}{2}r-2.25r-(-2\frac{3}{4}n)$$ and $$1\frac{1}{4}(1.4n+r)$$ equivalent? Prove it using two different methods.”
• Unit 6, Lessons 11-13 connect 7.RP.A and 7.EE.B as students solve simple interest problems by using equations such as the formula I = prt. In Lesson 12, Independent Practice (Masters level) #2 states, “What is the amount of interest that Mike earns on the following: deposit is $780, interest rate is 3.2% each year, for 18 months?” • In Unit 9, Lesson 11, students construct geometric figures (7.G.A) which they use to solve mathematical problems involving surface area and volume (7.G.B). In the Independent Practice, Question 6 (PhD level) states, “A rectangular prism has dimensions 3 inches, 4 inches, and 5 inches. Find the dimensions of another rectangular prism with the same volume but less surface area. Prove your answer is correct showing all calculations.” ### Gateway Two ## Rigor & Mathematical Practices #### Partially Meets Expectations + - Gateway Two Details The instructional materials reviewed for Achievement First Mathematics Grade 7 partially meet the expectations for rigor and the Mathematical Practices. The materials meet the expectations for rigor as they develop conceptual understanding and procedural skill and fluency and balance the three aspects of rigor. The instructional materials partially meet the expectations for practice-content connections. The Standards for Mathematical Practice (MPs) are identified. The materials also prompt students to construct viable arguments and analyze the arguments of others and attend to the specialized language of mathematics. ### Criterion 2a - 2d Rigor and Balance: Each grade's instructional materials reflect the balances in the Standards and help students meet the Standards' rigorous expectations, by helping students develop conceptual understanding, procedural skill and fluency, and application. 7/8 + - Criterion Rating Details The instructional materials reviewed for Achievement First Mathematics Grade 7 meet the expectations for rigor and balance. The materials meet the expectations for rigor as they develop conceptual understanding and procedural skill and fluency and balance the three aspects of rigor. The materials partially meet the expectations for application due to a lack of independent practice with non-routine problems. ### Indicator 2a Attention to conceptual understanding: Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. 2/2 + - Indicator Rating Details The instructional materials for Achievement First Mathematics Grade 7 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 1, Lesson 1, THINK ABOUT IT!, students develop conceptual understanding of addition with integers by modeling problems using number lines. The materials state, “Model the expressions $$-2 + 9$$ and $$-2 + (-4)$$ by accurately labeling using the number lines below.” (7.NS.1b) • In Unit 3, Lesson 1, Partner Practice, Question 2 (Bachelor level), students develop conceptual understanding of equivalence of equations by expanding expressions to combine like terms. The materials state, “Expand the following expressions and then combine like-terms: a) $$4x + 6 + 2x + 3$$; b) $$2r + 3y + 4 + 5y$$; c) $$4n + 3f + 5 + 5f + 2n + 1$$.” (7.EE.A) • In Unit 10, Lesson 2, Independent Practice, Question 4 (Master), students develop conceptual understanding of angle congruence to find unknown angles. The materials state, "Why must vertical angles always be congruent? Draw a diagram to help explain your answer." (7.G.5) Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Examples include: • In Unit 2, Lesson 6, Independent Practice, Question 3 (Master Level), students demonstrate conceptual understanding of multiplying and dividing rational numbers by using a number line. The materials state, “Use multiplication to prove that the quotient of $$-12 ÷ 4$$ is negative.” Question 4 states, “Use your answer to question 3 for the following two steps: Step A: Model the expression on the number line provided below. (number line from -15 to 15 provided). Step B: Explain how your number line in Step A could also represent multiplication.” (7.NS.2) • In Unit 4, Lesson 1, Independent Practice, Question 5 (Master Level), students demonstrate conceptual understanding of reasoning about quantities in a simple equation by using a balance model. The materials state, “Model the equation $$9n+31=66$$ using a balance model and apply your model to solve for the variable arithmetically.” (7.EE.3, 7.EE.4) • In Unit 6, Lesson 2, Independent Practice, Question 8 (Master Level), students demonstrate conceptual understanding of using proportional relationships to solve percent problems by using a number line. The materials state, “Set up a double number line to write and solve an equation for the given problem. a) 40 is 80% of what number? b) 18 is what percent of 72?” (7.RP.3) ### Indicator 2b Attention to Procedural Skill and Fluency: Materials give attention throughout the year to individual standards that set an expectation of procedural skill and fluency. 2/2 + - Indicator Rating Details The instructional materials for Achievement First Mathematics Grade 7 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 3, Exit Ticket Question 2, students develop procedural skill and fluency by using operations with rational numbers. The materials state, “Which of the following expressions with have a positive sum? Select all that apply: a) $$-14 + (-42)$$; b) $$34 + (-24)$$; c) $$-7 + 10$$; d) $$-50 + 45$$; e) $$8 + 88$$; f) $$-6 + 7$$.” (7.NS.A) • In Unit 3, Lesson 3, Independent Practice, Question 6 (Masters level), students develop procedural skill and fluency by rewriting equivalent expressions. The materials state, “Write at least four different expressions that are equivalent to $$-18 + 6n$$.” (7.EE.2) • In Unit 4, Lesson 8, Independent Practice, Question 2 (Bachelor level), students develop procedural skill and fluency by solving word problems that lead to 2-step equations. The materials state, “A dog is starting a diet to get in better shape. The dog starts at 89.5 points and loses 0.5 points each week for a certain number of weeks. Halfway through the diet, the dog weighs 80 pounds. How many weeks has the dog been dieting for?” (7.EE.4a) The instructional materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade-level. Examples include: • In Unit 1, Lesson 7, Independent Practice, Question 5 (Master level), students demonstrate procedural skill and fluency by using operations with rational numbers. The materials state, “Evaluate the following expression: $$-42-(-23)+(-37-5)$$.” (7.NS.A) • In Unit 4, Lesson 2, Independent Practice Question 5 (Master level), students demonstrate procedural skill and fluency by solving word problems that lead to 2-step equations. The materials state, “Mari is twice as old as Harry. Jacob is three times older than Harry plus two years. Their combined age is 50. How old is each person?” (7.EE.4a) • In Unit 9, Skill Fluency, 9.2, Day 1, Question 3, students demonstrate procedural skill and fluency related by rewriting equivalent expressions. The materials state, “Which expression is equivalent to $$(4x-5)-(3x-2)$$? a) $$7x-7$$; b) $$7x-3$$; c). $$x-7$$; d) $$x-3$$.” (7.EE.2) ### Indicator 2c Attention to Applications: Materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade 1/2 + - Indicator Rating Details The instructional materials for Achievement First Mathematics Grade 7 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, Lesson 10, Exit Ticket, Question 2, students apply skills related to operations with rational numbers. The materials state, “An hourglass loses $$8\frac{1}{4}$$ oz of sand every five minutes. How much sand will be in the hourglass after a half hour if it starts with 50 oz and 20.4 oz are added at the end of the 30 minutes?” (7.NS.3) • In Unit 3, Lesson 7, Exit Ticket, Question 1, students solve real life problems such as football yardage using positive and negative numbers. The materials state, “Nquan is playing in a football game and rushes for 7.4 yards on the first play. For the next three plays, he loses y yards on each play. The last play he gains 1 yard. Write an algebraic expression to represent this situation and solve it if $$y=3\frac{1}{2}$$ yards.” (7.EE.3) • In Unit 4, Lesson 8, Independent Practice, Question 9 (PhD Level), students apply skills related to reasoning about quantities by constructing simple equations. The materials state, “Ben and Jerry saved up their pennies to buy a present for their dad’s birthday. By the end of the first week, Ben had saved$15 dollars and Jerry had saved d dollars. By the end of the second week, they had tripled their savings, and had $66 in total. How much did Jerry save in the first week? Show two methods for solving this problem.” (7.EE.4) • In Unit 4, Lesson 19, Independent Practice, Question 6 (Master Level), students solve real life problems such as finding the area of a circle. The materials state, “Brian’s dad wants to put a circular pool in their pool (yard). He can choose between pools with diameters of 15 ft, 17 ft, or 22 ft. Step A: Determine how much more space the pool with a diameter of 22 feet would take up compared to the 15 foot diameter pool. Step B: Determine how much more space the 15ft and 17ft pools combined would take up compared to the 22ft pool.” (7.G.4) • In Unit 6, Lesson 13, Independent Practice, Question 1 (Bachelor Level), students solve real life problems such as finding final costs using percent problems. The materials state, “A snowboard originally costs$260. The sports store is having a sale of 10% off of items less than $100 and 15% off of items above$100. The sales tax is 12%. What is the final price for the snowboard, including tax?” (7.RP.A)
• In Unit 9, Problem of the Day 9.1, Day 1, Question 1, students apply skills related to using proportional relationships to solve percent problems. The materials state, “Last year, a property manager bought five identical snow shovels and six identical bags of salt. The total cost of the snow shovels was $172.50, before tax, and each bag of salt cost$6.20, before tax. This year, the property manager bought two identical snow shovels and four identical bags of salt. The total cost of the snow shovels was $70.38, before tax, and the total cost of the bags of salt was$26.04, before tax. Determine the item with the greatest percent increase in the price from last year to this year. Be sure to include the percent increase of this item to the nearest percent.” (7.RP.3)

### Indicator 2d

Balance: The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the 3 aspects of rigor within the grade.
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Indicator Rating Details

The instructional materials for Achievement First Mathematics Grade 7 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 7, THINK ABOUT IT!, students use number lines to demonstrate conceptual understanding of subtracting integers. The materials state, “Model and evaluate the addition and subtraction expressions on an open number line. a) $$25 + (-37)$$; b) $$25-37$$. Explain a generalized rule that you could use to subtract integers without the aid of a number line.” (7.NS.1)

Fluency and Procedural Skill:

• In Unit 4, Lesson 18, Exit Ticket, Question 2, students demonstrate fluency regarding the area of a circle by both estimating and finding the exact measure. The materials state, “What is the exact and approximate area of a circle with a diameter of 6 feet? For the approximate area, round your answer to the nearest tenths place.” (7.G.4)

Application:

• In Unit 4, Lesson 10, Independent Practice, Question 2 (Bachelor Level), students apply their knowledge about multi-step real world problems to find the winner of the reading contest. The materials state, “Aaliyah and Yohance are having a competition to see who can read more pages over the coming weekend. Aaliyah has bet Ms. Solomon that she’ll read 50 more pages than Yohance. Both scholars read at an average rate of 40 pages per hour. Yohance says that he’s going to read for 7.5 hours this weekend. How many hours will Aaliyah need to read for in order to fulfill her promise of reading 50 more pages than Yohance?” (7.EE.4)

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 2, Lesson 5, Partner Practice, Question 2 (Master Level), students demonstrate both conceptual understanding and procedural skill as they use a number line and an expression to represent division of rational numbers. The materials state, “A submarine starts at the surface and then descends to a depth of 250 feet below sea level. It took the submarine 5 minutes to complete this dive. How many feet can the submarine dive in 1 minute? Draw a model and write an expression to solve.” (7.NS.2)
• In Unit 5, Problem of the Day 5.1, Day 2, students demonstrate fluency and application with operations on rational numbers. The materials state, “A water well drilling rig has dug to a height of –60 feet after one full day of continuous use. a) Assuming the rig drilled at a constant rate, what was the height of the drill after 15 hours? b) If the rig has been running constantly and is currently at a height of –143.6 feet, for how long has the rig been running? c) A snake is $$\frac{3}{4}$$ the current distance underground of the rig and a spider is $$\frac{4}{5}$$ of the same distance. How far away are the snake and the spider?” (7.NS.3)
• In Unit 3, Lesson 2, Independent Practice, Question 8 (PhD level), students apply their conceptual understanding of variables to write and solve equations in real-world situations. The materials state, “You and your friend made up a basketball shooting game. Every shot made from the free throw line is worth 3 points, and every shot made from the half-court mark is worth 6 points. Write an equation that represents the total amount of points, P, if f represents the number of shots made from the free throw line, and h represents the number of shots made from half-court. Explain the equation in words.”  (7.EE.4)

### Criterion 2e - 2g.iii

Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice
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Criterion Rating Details

The instructional materials reviewed for Achievement First Mathematics Grade 7 partially meet the expectations for practice-content connections. The Standards for Mathematical Practice (MPs) are identified. The materials also prompt students to construct viable arguments and analyze the arguments of others and attend to the specialized language of mathematics.

### Indicator 2e

The Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout each applicable grade.
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Indicator Rating Details

The instructional materials reviewed for Achievement First Mathematics Grade 7 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, MP8 is emphasized in two units, while MP2 is emphasized in 10.
• The Mathematical Practices are not always identified accurately. For example:
• At the unit level for Unit 3, MP6 is not identified as an emphasized practice. However, at the lesson level, 5 lessons identify it as connected. MP8 is bold (indicating emphasis) at the unit level, but isn’t connected in any of the lessons.
• In Unit 6, MPs 1, 2, 5, and 7 are bold; however, MP5 does not appear in any of the lessons in this unit.
• The Unit 8 Overview does not bold MP1 as an emphasis, yet 5 out of 9 of the lessons include this practice. Whereas MP4 is an emphasis, but is not bolded in any lessons.
• 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 2 Overview states, “Students can reason from here that the sign of a must be negative because the product is negative meaning the signs of the factors must be different and 5 is positive.”
• Unit 5 Overview states, “Students also work backwards and reason that if they are given a dependent value, then the value divided by the CoP will produce the corresponding independent value.”
• Unit 6 Overview states, “In both lessons, students will still be expected to use double number lines to make sense of the problem.”
• Unit 7 Overview states, “Simulation uses tools (coin, number cube, etc.) to generate outcomes that represent real outcomes that otherwise might have been difficult to do in real time.”
• In Unit 3, Lesson 7 prompts the teacher to ask if student answers are reasonable and shows a key strategy as “Check for reasonableness.” But there is no specific connection to MPs.

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:

• MP2 - Unit 1 Overview states, “To develop a conceptual understanding of addition and subtraction with integers, students return to the number line to model the operations (SMP2, SMP4). In the first four lessons, students move from using number lines to understand and perform addition of integers to applying abstract rules to add integers fluently.” Although SMP4 isn’t one of the emphasized practices in this unit.
• MP6 - Unit 1 Overview states, “For subtracting integers using rules (SMP6), students may utilize one of two paths. They may visualize subtraction on a number line. Or, they can understand subtraction as adding the additive inverse and apply the rules for adding integers.”
• MP7 - Unit 1 Overview states, “...Therefore, regardless of the order of the minuend and subtrahend, if one takes the absolute value of the difference, they will find the distance between the two values (SMP7).”

### Indicator 2f

Materials carefully attend to the full meaning of each practice standard
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Indicator Rating Details

The instructional materials reviewed for Achievement First Mathematics Grade 7 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 1, Lesson 5, Independent Practice, Question 3 (Master Level) states, “Evaluate and model both expressions by drawing your own number line. a) $$1-(-7)$$; b) $$-9-(-6)$$.” In this example, MP4 is a visual representation but students do not choose a strategy, nor is there any real-world connection.
• In Unit 4, Lesson 1, Exit Ticket, Question 2 states, “Use a balance model to represent and solve the equation $$3x+3=24$$. Explain your answer and how you kept the equation balanced.” Students do not choose a strategy, nor is there any real-world connection.
• In Unit 4, Lesson 10, Think About It states, “Ted is planning a vacation this summer. The resort he wants to stay at charges $125.25 per day plus tax. Ted books the trip for three days and it costs him$422.25. How much did Ted pay in taxes per day of his vacation? See below for a potential model.” A model is provided to the students, supplying a pathway for students so they do not choose strategies.
• 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.
• In Unit 1, Lesson 1, Independent Practice Question 3 (Bachelor Level) states, “Jamie’s bank account currently has a negative balance of -2 dollars. How much money will be in her account if she deposits 8 dollars?” Instead of being able to choose a tool that could be best used to solve this problem, a number line has been given, suggesting that this is the correct tool to use.
• In Unit 4, Lesson 16, THINK ABOUT IT! states, “Record the diameter and circumference of the given circle using the tools provided.” Students are given rulers and string to calculate circumference without an opportunity to choose tools.
• In Unit 10, Lesson 4, AIM states, “Construct line segments and angles given measurements using a ruler and protractor.” In the problems, students are given rulers and protractors specifically by the teacher and do not self-select tools.

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

• MP1: Make sense of problems and persevere in solving them.
• In Unit 1, Lesson 11, THINK ABOUT IT!, students use any estimation or an integer operation strategy to solve this problem and determine if their solution makes sense. The materials state, “Maggie said that she could determine the answer to the problem below by just estimating the answer and comparing it to the given answer choices. Do you agree with Maggie? If so, explain and use estimation to prove she is correct. If not, explain and solve the problem to determine the actual answer. Dominic jumped from a height of 14.3 feet above the surface of a pool.  He traveled 18.7 feet straight down into the water. From there he traveled up 25.55 feet to the top of the biggest water slide at the pool. What is the height of the tallest waterslide? a) -55.55; b) -29.55; c) 21.15; d) 55.55.”
• In Unit 4, Lesson 19, Independent Practice, Question 7 (PhD Level), students solve an unrehearsed and unfamiliar problem by decoding information to work backwards. The materials state, “Explain how you would be able to determine the area of a circle if you were given the circumference. Draw a diagram and provide an example in your explanation. Explain why it would be more difference to determine the circumference of a circle given the area.”
• In Unit 6, Lesson 4, Independent Practice, Question 6 (Master level), students already know the answer to the question, but must figure out how they got there. The materials state, “Tom Brady threw 52 completions in the Super Bowl XLL loss to the New York Giants. His completion rate of completed passes to total passes was 65%. How many total passes did Brady attempt?”
• MP2: Reason abstractly and quantitatively.
• In Unit 5, Lesson 3, Partner Practice, Question 2 (Bachelor Level), students determine what numbers and quantities mean in a relationship. The materials state, “The table below shows the relationship between the cost of renting a movie (in dollars) to the number of days the movie is rented. Read each statement below the table and determine if it is true or false. a) Dollars represents the independent variable; b) The relationship between the cost and the number of days is proportional because $$6\div2=3$$ and $$9\div3=3$$; c) The relationship between the cost and the number of days is not proportional because the values do not increase in order; d) The relationship between the cost and the number of days is proportional because there is a CoP of $$\frac{1}{3}$$.”
• In Unit 6, Lesson 16, Independent Practice, Question 3 (Master Level), students understand the proportional relationship between two figures and represent the math problem symbolically. The materials state, “If the area of the shaded region in the larger figure is approximately 21.5 square inches, write an equation that relates the areas using scale factor and explain what each quantity represents. Determine the area of the shaded region in the smaller scale drawing. Equation: ___ Area shaded region (smaller square): ___.” The shaded area is square and the unshaded area is an oval.
• In Unit 8, Lesson 4, Independent Practice, Question 1 (Bachelor Level), students analyze a dot plot, then find a solution using the data to determine if the quantity is reasonable. The materials state, “Consider the distribution below. Part A: Describe the distribution of the data Part B: Describe the variability of the data. Part C: What would you expect the distribution of a random sample of size 10 from this population to look like?”
• MP6: Attend to precision.
• In Unit 1, Lesson 2, students “represent and add integers (-100 to 100) using a horizontal or vertical number line.” The Anticipated Misconceptions and Errors section identifies several points to help students attend to the precision of the mathematics and set up equations correctly. The materials state, “Anticipated Misconceptions and Errors: Students might not draw arrows proportionally to each other resulting in incorrect signs; Students might move in the wrong direction when drawing their arrows; Students might start their q-value arrow at zero instead of at the p-value; Students might add or subtract incorrectly (carrying or borrowing) when given larger numbers.”
• In Unit 8, Lesson 8, Interaction with New Material, students must understand vocabulary in order to communicate an accurate explanation. The materials state, “Ex. 1) The distances that two different track teams run every day for practice are compiled. The team that runs fewer miles has a mean distance of 6.5 miles. The distance between the means between the two teams is 4 miles. Use the number lines below to visualize the problem based on Step A and B. Step A: Would you expect there to be any visual overlap of the data if both teams had a MAD of 2 miles? Step B: Could you still explain Part B if you didn’t know the mean of the slower team? Explain.”
• In Unit 9, Lesson 11, Interaction with New Material, students must understand the language of the problem in order to find surface area and volume, and calculate accurately in order to compare the two shapes. The materials state, “An airfreight company uses a box in the shape of a triangular prism to pack blueprints, posters, and other items that can be rolled up to fit inside the box. Each base is an equilateral triangle. The dimensions of the box are shown below. a) The freight company needs to first wrap the entire box in packing tape. How many square inches will be covered with tape? b) How many cubic inches of material can be packed within the prism? c) Another airfreight company uses a box shaped like a rectangular prism for the same purposes. The rectangular prism is also 38 inches long, and each of its square bases has a length of 3 inches.  Which box takes up more space?”
• MP7: Look for and make use of structure.
• In Unit 2, Lesson 6, the narrative describes the structure of fraction division which teachers guide students through before they have the opportunity to practice independently. The materials state, “Students understand that any division equation can be rewritten as a multiplication equation where the dividend of the division equation is equal to the product of the multiplication equation and the divisor and quotient are the factors of the multiplication equation. Students understand that because all division equations can be rewritten with multiplication that the rules for multiplying integers extends to dividing integers.”
• In Unit 3, Lesson 1, Test the Conjecture, Question 1, students use repeated addition and the commutative property within expressions to understand combining like terms. The materials state, “Write two expressions that are equivalent to the expression $$3x + 5 + 4x + 2$$ and indicate the expression that is in simplest form.” Teacher prompts include, “In order to help us combine like terms, we are going to expand each term in the expression that contains a variable. How could we expand this expression? How can we group the like terms so they are with each other?”
• In Unit 5, Lesson 4, Partner Practice, Question 3 (Master Level), students use structure by creating a table to discern the pattern of repeated addition. The materials state, “Mary is filling out a table to keep track of how much money is in her account. On the first day of the month, she has $50. On the third day she has$150. On the 4th day, she has $200. If her account continues the same way, write an expression to determine how much money she will have on the 9th day and how long it will take her to have$n in her account.”
• MP8: Look for and express regularity in repeated reasoning.
• Unit 1, Lesson 3, Exit Ticket, Question 1, students use repeated reasoning to add integers. The materials state, “Evaluate the expression: $$(-22) + 15 + (-9)$$ and explain how you used the generalized rules for adding integers.”
• Unit 4, Lesson 17, THINK ABOUT IT!, “For the circle below, Chandler says that there isn’t enough information to determine the circumference without measuring. Joey disagrees and says that he can write an equation to solve for the circumference. Who do you agree with and why?” Then the teacher states, “The circumference of a circle is equal to Pi multiplied by the diameter. What will we be able to do if our conjecture is true? We will be able to write an equation for the circumference of a circle and substitute to determine either the circumference or diameter.” Students use repeated reasoning about the relationship between circumference, diameter, and Pi.
• Unit 6, Lesson 3, AIM, students “develop the formula part = $$\frac{p}{100}×$$ total using a double number line diagram.” In THINK ABOUT IT!, the teacher prompts, “The number line below shows a general percent problem with the percent, part and whole. Write an equation and solve for the part. Using your equation, describe how you can find the percent of a number.” Then the teacher states,  “The percent of a number is equal to the percent (as a decimal) multiplied by the total.” Students use repeated reasoning to determine the percent of a number with a double number line.

### Indicator 2g

Emphasis on Mathematical Reasoning: Materials support the Standards' emphasis on mathematical reasoning by:
Narrative Evidence Only

### Indicator 2g.i

Materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards.
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Indicator Rating Details

The instructional materials reviewed for Achievement First Mathematics Grade 7 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 9, Independent Practice #7 (PhD Level), students add and subtract rational numbers. The materials state, “Using the multiple choice question below, determine which two answer choices that you can immediately eliminate without doing any calculations.  Explain how you were able to eliminate those answer choices. Evaluate:  $$-4.5-(-1\frac{4}{5})-(-2.1)+4\frac{1}{2}$$. a) $$-12.9$$; b) $$12.9$$; c) $$-5.7$$; d) $$5.7$$.” (7.NS.1d)
• Unit 2, Lesson 6, Independent Practice #6 (Master Level), students multiply and divide rational numbers. The materials state, “Jesse and Jahniece are trying to solve the expression -125 ÷ (-25). Jesse thinks that you can just divide normally and keep the sign negative since the other signs are negative. Jahniece doesn’t think it can be solved at all because you can’t split -125 into a negative number of groups. YOU PLAY THE TEACHER. How would you help both scholars to determine the correct answer without giving them the answer?” (7.NS.2)
• Unit 4, Lesson 13, Error Analysis Lesson, THINK ABOUT IT!, students use variables to create 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.” (7.EE.4)
• Unit 5, Lesson 14, Independent Practice #2 (Bachelor Level), students solve problems involving scale. The materials state, “Mark claims that he can multiply that area of Rectangle A by 4 to get the correct area of Rectangle B. Do you agree with him? Explain and prove your answer.” (7.G.1)
• In Unit 6, Lesson 13, Independent Practice #4 (Master Level), students use proportional relationships to solve percent problems. The materials state, “Justin wants to buy a new IPod that costs $250. When he gets to the Apple store, he sees that they are having a sale for 15% off all IPods. He then has a coupon that takes an additional 15% off the discounted price. Justin thinks that he can figure out the cost of the iPod by finding 30% of$250 and then subtracting that from \$250. Do you agree or disagree with his claim? Explain.” (7.RP.3)
• In Unit 7, Lesson 8, Independent Practice, Question 4 (Master Level), students approximate the probability of an event. The materials state, “Tishanna is experimenting with the same bag of pens. She randomly pulls a pen out of the bag 30 times, records the color, and replaces the pen. Her results are shown below. Step A: Now make a prediction for how many times Tishanna would pick a red pen, if she conducted 60 trials of the experiment. Step B: Which prediction are you more confident in – the prediction in question # 3, or the prediction you made in question #4? Why? Explain.” (7.SP.6)

### Indicator 2g.ii

Materials assist teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards.
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Indicator Rating Details

The instructional materials reviewed for Achievement First Mathematics Grade 7 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 1, Lesson 3, Interaction with New Material, students explore addition of integers, “Let’s look at another problem and determine if our previous key point will apply or if we need a new one. Example 1) Margo thinks that the sum of $$-25+40$$ will be equal to the sum of $$25+(-40)$$.  Prove Margo right or wrong and develop a generalized rule that can be used to solve without number lines.” Teacher prompts include, “Do you predict the sums will be the same? Is Margo correct? What additional key point could we form for adding integers? Was your prediction correct?” (7.NS.1)
• In Unit 5, Lesson 14, THINK ABOUT IT!, students explore scaling. Teacher prompts include, “Describe the first scholar’s strategy. Describe the second scholar’s strategy. What do you notice? Which scholar was correct?” (7.G.1)
• In Unit 6, Lesson 9, Error Analysis Lesson, Debrief, students compare exit ticket responses about a 20% mark down. Teacher prompts include, “Which scholar’s work did you agree with? What error did this scholar make? Does this answer seem reasonable? Why/why not? What did this scholar do to get this correct, and why was that helpful? If this had been a 20% mark-up, what would have been different? Why?” (7.RP.3)

### Indicator 2g.iii

Materials explicitly attend to the specialized language of mathematics.
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Indicator Rating Details

The instructional materials reviewed for Achievement First Mathematics Grade 7 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 3, Lesson 2, Opening, Debrief, FENCEPOST #1, students evaluate expressions. The materials state, “A number multiplying a quantity in parentheses must multiply all of the numbers in the quantity. This is called the Distributive Property. Show Call: Two different show calls, one with $$4(8) + 3 = 27$$ (incorrect) and one with $$4(8) + 4(3) = 44 or 4(11)$$ Which scholars’ work do you agree with?  Vote. CC. SMS: I agree with the scholar that got the answer 44 because if you do the order of operations then you add 8 and 3 to get 11 and then multiply by 4 to get 44. What mistake was made in getting 27?  CC. SMS:The scholar only multiplied 8 by 4 when they should have multiplied everything in the parentheses by 4. How can you explain this using the area model?  TT. CC. Discuss. SMS: In the area model, the width is 4 and the length is $$8+3$$ so you can think of it as adding two areas together $$4×8$$ and $$4×3$$. You have to multiply all terms or you aren’t finding the area of the entire model. Name the fencepost (teacher will have to name without using a question given that this is vocabulary): When a number is outside of parentheses like this, we can use something called the distributive property which you have just explained. A number multiplying a quantity in parentheses must multiply all numbers in the quantity.” (7.EE.1)
• In Unit 4, Lesson 14, Opening, Debrief, FENCEPOST #1, students write and interpret inequality symbols. The materials state, “Specific phrases indicate inequality symbols. Show Call: S work writes $$m<20$$ for the first inequality. Do you agree with this inequality? Vote. CC.  SMS: I agree because the problem uses the phrase “less than” so we have to use an inequality symbol to show that m is less than 20. Show Call: Two pieces of work; one with the “less than or equal to” symbol and one with the “less than” symbol. Which scholar wrote a correct inequality? Vote. TT. CC. SMS: The scholar with the less than or equal to symbol wrote the correct inequality because the problem says that her age plus four can be no more than 25. No more than means that it can’t be more than but it can be 25 so we have to use the less than or equal to symbol. BPQ – Can her age plus 4 equal 25? How do you know? Name the fencepost: What can we say about determining the inequality symbol from context? SMS: Specific phrases indicate inequality symbols.” (7.EE.4)
• In Unit 7, Lesson 1, Opening, Debrief, FENCEPOST #1, students use a spinner to determine probability. The materials state, “Probability measures how likely an event is from impossible to certain. Show Call: S work has correctly placed an “x” on impossible for a.) and certain for b.). Do you agree with this scholar? Vote. TT. CC. SMS: I agree with this scholar because for the first problem, there is no possible way for someone to spin the spinner and it to land on 5 because there isn’t a 5 on the spinner so it is impossible. For the second problem, if you spin the spinner it must land on 1, 2, 3, or 4 so it is certain that it will happen. Say: What you are calculating is called a probability. Probability is the likelihood of an event or outcome happening.  An event is an outcome in an experiment (in this case, the “experiment” is spinning the spinner and each number is an event. If you land on 1, that is an event. If you land on 2, that is an event. Etc.). Name the fencepost: What does probability measure? CC. SMS: Probability measures how likely an event is from impossible to certain. BPQ: Between what two possibilities does the probability of an event fall?” (7.SP.5)

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 in Unit 4, Lesson 4 “Key Vocabulary:
• Equation: two expressions set equal to one another.
• Variable: a letter used to take the place of an unknown value.
• Solution: the value that makes an equation true.
• Arithmetic approach: the approach to solving a problem that involves arithmetic only; numbers and operations.”
• The teacher is routinely prompted to use precise vocabulary such as Unit 5, Lesson 5, Connection to Learning. The materials state, “Students should understand that a graph is proportional if it is linear (i.e. forms a straight line) and passes through the origin because every value of x is multiplied by the CoP to produce the corresponding output. Students should understand that the point (0,0) must be a part of a proportional graph because no CoP can be multiplied by 0 to produce a non-zero output.”
• There is very little vocabulary emphasis in student-facing materials. For example, there is not a glossary for student reference.

## Usability

#### Not Rated

+
-
Gateway Three Details
This material was not reviewed for Gateway Three because it did not meet expectations for Gateways One and Two

### Criterion 3a - 3e

Use and design facilitate student learning: Materials are well designed and take into account effective lesson structure and pacing.

### Indicator 3a

The underlying design of the materials distinguishes between problems and exercises. In essence, the difference is that in solving problems, students learn new mathematics, whereas in working exercises, students apply what they have already learned to build mastery. Each problem or exercise has a purpose.
N/A

### Indicator 3b

Design of assignments is not haphazard: exercises are given in intentional sequences.
N/A

### Indicator 3c

There is variety in what students are asked to produce. For example, students are asked to produce answers and solutions, but also, in a grade-appropriate way, arguments and explanations, diagrams, mathematical models, etc.
N/A

### Indicator 3d

Manipulatives are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.
N/A

### Indicator 3e

The visual design (whether in print or online) is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.
N/A

### Criterion 3f - 3l

Teacher Planning and Learning for Success with CCSS: Materials support teacher learning and understanding of the Standards.

### Indicator 3f

Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.
N/A

### Indicator 3g

Materials contain a teacher's edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning.
N/A

### Indicator 3h

Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that contains full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge of the subject, as necessary.
N/A

### Indicator 3i

Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that explains the role of the specific grade-level mathematics in the context of the overall mathematics curriculum for kindergarten through grade twelve.
N/A

### Indicator 3j

Materials provide a list of lessons in the teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials), cross-referencing the standards covered and providing an estimated instructional time for each lesson, chapter and unit (i.e., pacing guide).
N/A

### Indicator 3k

Materials contain strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.
N/A

### Indicator 3l

Materials contain explanations of the instructional approaches of the program and identification of the research-based strategies.
N/A

### Criterion 3m - 3q

Assessment: Materials offer teachers resources and tools to collect ongoing data about student progress on the Standards.

### Indicator 3m

Materials provide strategies for gathering information about students' prior knowledge within and across grade levels.
N/A

### Indicator 3n

Materials provide strategies for teachers to identify and address common student errors and misconceptions.
N/A

### Indicator 3o

Materials provide opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.
N/A

### Indicator 3p

Materials offer ongoing formative and summative assessments:
N/A

### Indicator 3p.i

Assessments clearly denote which standards are being emphasized.
N/A

### Indicator 3p.ii

Assessments include aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.
N/A

### Indicator 3q

Materials encourage students to monitor their own progress.
N/A

### Criterion 3r - 3y

Differentiated instruction: Materials support teachers in differentiating instruction for diverse learners within and across grades.

### Indicator 3r

Materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.
N/A

### Indicator 3s

Materials provide teachers with strategies for meeting the needs of a range of learners.
N/A

### Indicator 3t

Materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.
N/A

### Indicator 3u

Materials suggest support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics (e.g., modifying vocabulary words within word problems).
N/A

### Indicator 3v

Materials provide opportunities for advanced students to investigate mathematics content at greater depth.
N/A

### Indicator 3w

Materials provide a balanced portrayal of various demographic and personal characteristics.
N/A

### Indicator 3x

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

### Indicator 3y

Materials encourage teachers to draw upon home language and culture to facilitate learning.
N/A

### Criterion 3aa - 3z

Effective technology use: Materials support effective use of technology to enhance student learning. Digital materials are accessible and available in multiple platforms.

### Indicator 3aa

Digital materials (either included as supplementary to a textbook or as part of a digital curriculum) are web-based and compatible with multiple internet browsers (e.g., Internet Explorer, Firefox, Google Chrome, etc.). In addition, materials are "platform neutral" (i.e., are compatible with multiple operating systems such as Windows and Apple and are not proprietary to any single platform) and allow the use of tablets and mobile devices.
N/A

### Indicator 3ab

Materials include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology.
N/A

### Indicator 3ac

Materials can be easily customized for individual learners. i. Digital materials include opportunities for teachers to personalize learning for all students, using adaptive or other technological innovations. ii. Materials can be easily customized for local use. For example, materials may provide a range of lessons to draw from on a topic.
N/A

Materials include or reference technology that provides opportunities for teachers and/or students to collaborate with each other (e.g. websites, discussion groups, webinars, etc.).
N/A

### Indicator 3z

Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices.
N/A
abc123

Report Published Date: 2021/03/11

Report Edition: 2020

## Math K-8 Review Tool

The mathematics review criteria identifies the indicators for high-quality instructional materials. The review criteria supports a sequential review process that reflect the importance of alignment to the standards then consider other high-quality attributes of curriculum as recommended by educators.

For math, our review criteria evaluates materials based on:

• Focus and Coherence

• Rigor and Mathematical Practices

• Instructional Supports and Usability

The K-8 Evidence Guides complements the review criteria by elaborating details for each indicator including the purpose of the indicator, information on how to collect evidence, guiding questions and discussion prompts, and scoring criteria.

## Math K-8

K‑8 Evidence Guide K‑8 Review Criteria

The EdReports rubric supports a sequential review process through three gateways. These gateways reflect the importance of alignment to college and career ready standards and considers other attributes of high-quality curriculum, such as usability and design, as recommended by educators.

Materials must meet or partially meet expectations for the first set of indicators (gateway 1) to move to the other gateways.

Gateways 1 and 2 focus on questions of alignment to the standards. Are the instructional materials aligned to the standards? Are all standards present and treated with appropriate depth and quality required to support student learning?

Gateway 3 focuses on the question of usability. Are the instructional materials user-friendly for students and educators? Materials must be well designed to facilitate student learning and enhance a teacher’s ability to differentiate and build knowledge within the classroom.

In order to be reviewed and attain a rating for usability (Gateway 3), the instructional materials must first meet expectations for alignment (Gateways 1 and 2).

Alignment and usability ratings are assigned based on how materials score on a series of criteria and indicators with reviewers providing supporting evidence to determine and substantiate each point awarded.

Alignment and usability ratings are assigned based on how materials score on a series of criteria and indicators with reviewers providing supporting evidence to determine and substantiate each point awarded.

For ELA and math, alignment ratings represent the degree to which materials meet expectations, partially meet expectations, or do not meet expectations for alignment to college- and career-ready standards, including that all standards are present and treated with the appropriate depth to support students in learning the skills and knowledge that they need to be ready for college and career.

For science, alignment ratings represent the degree to which materials meet expectations, partially meet expectations, or do not meet expectations for alignment to the Next Generation Science Standards, including that all standards are present and treated with the appropriate depth to support students in learning the skills and knowledge that they need to be ready for college and career.

For all content areas, usability ratings represent the degree to which materials meet expectations, partially meet expectations, or do not meet expectations for effective practices (as outlined in the evaluation tool) for use and design, teacher planning and learning, assessment, differentiated instruction, and effective technology use.