## Alignment: Overall Summary

The instructional materials reviewed for Achievement First Mathematics Grade 8 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 8 meet expectations for Gateway 1, focus and coherence. The instructional materials meet the expectations for focus by assessing grade-level content and spending at least 65% 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 8 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 8 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 4 Assessment, Question 3 states, “Consider the relation represented in the table below: (Given x inputs {2, 5, n} and y outputs {9, 7, 5}). Which of the follow(ing) statements is true? a) If $$n$$ represents any positive integer, the relation will represent a function. b) If the value of $$n$$ is any number other than 2 or 5, the relation will represent a function. c) If $$n$$ represent(s) any number, the relation will represent a function. d) If the value of $$n$$ is any number other than 5, 7, or 9 the relation will represent a function. Explain how you determine(d) which statement was true.” (8.F.1)
• In Unit 6 Assessment, Question 5 states, “A sandwich shop makes home deliveries. The average amount of time from when an order is placed until when it is delivered can be modeled by the equation $$y = 2.5x + 5$$, where y is the number of miles between the shop and the delivery location and $$x$$ is the number of minutes. According to this model, if it takes 17.5 minutes for the sandwich shop to deliver the sandwich to your house, how far away do you live? Show your work.” (8.SP.3)
• In Unit 7 Assessment, Question 7 states, “A system of linear equations is shown below. Without performing any calculations, determine the number of solutions to the system. Explain your reasoning. $$5x + 2y = 4$$ / $$5x + 2y = -1$$” (8.EE.8b)
• In Unit 8 Assessment, Question 6 states, “Glaciers advance at a rate of about 0.000003 of a meter per second. What represents the approximate rate at which glaciers advance in scientific notation? Explain why the exponent has the sign it does.” (8.EE.3)
• In Unit 10 Assessment, Question 9 states, “Heather walked 24 feet to the south and 32 feet to the east, but then she walked in a straight line back to where she started, as shown by the dotted line. How far did Heather walk in all?” (8.G.6)

### 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 8 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 8 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 6, Lesson 4, supporting standard 8.SP.3 enhances the major work of 8.EE.B. Students informally fit a straight line to data in a scatter plot, write an equation for the line, and make and justify a prediction using the equation. For example, Independent Practice, Question 1 (Bachelor level) states, “The scatterplot below shows the number of texts that a middle school scholar sends per day and their GPA. Which of the following statements would correctly describe the equation written from the line of best fit in the form $$y = mx + b$$? Select all that apply. a) m will be positive. b) b will be between 3.8 and 4.0. c) y will represent GPA. d) the slope will describe the change in number of texts sent per GPA point. e) If 100 is substituted into the equation for x, the resulting y value should be around 3.4.”
• In Unit 6, Lesson 5, supporting standard 8.SP.3 enhances the major work of 8.F.4. Students interpret the slope and y-intercept of the line of best fit given the context of the data to answer questions or to solve a problem. For example, Independent Practice Question 3 (Masters level) states, “Julie recorded the number of female students and male students in her school for the past 8 years in a table and graphed the data using a scatter plot where the x-axis represents the females and the y-axis represents the males. She wrote the equation $$y = 1.2x + 12$$ to represent the line of best fit. Step A:  What does the slope of the equation represent? Step B:  What does the y-intercept represent? Step C:  Draw a sketch of what you would expect the scatter plot to look like and explain why you drew the scatter plot in that way.”
• In Unit 9, Lesson 7, supporting standard 8.G.9 enhances the major work of 8.F.B. Students explore volume as a function of radius by graphing the relationship and identify the function as linear or non-linear and justify the identification. For example, Independent Practice, Question 4 (Master Level) states, “Determine a rule that could be used to explain how the volume of a cylinder or cone is affected as the radius changes.” Also, Independent Practice Question 7 (PhD level), “Predict if the relationship between volume and height for cylinders and cones is linear or non-linear. Explain your reasoning.”
• In Unit 10, Lesson 14, supporting standard 8.G.9 enhances the major work of 8.G.7. Students solve problems involving volume of cones, cylinders, and spheres by applying the Pythagorean Theorem. For example, Independent Practice, Question 6 (Master level) states, “An ice cream cone is 6 inches tall with a slant height of 7.5 inches. The opening of the cone is a circle.  What is the diameter of the opening of the cone?”

### 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 8 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 are three optional lessons: two before Unit 2, Lesson 1 and another in Unit 8 between Lessons 6.2 and 7. Since they are optional, they are not included in the total count.

According to The Guide to Implementing Achievement First Mathematics Grade 8, 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 8 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. Unit 3 states, “In fourth and fifth grades, scholars learn how to draw and classify shapes based on their lines, angles, and properties (4.G.A. and 5.G.B.). In the sixth grade, scholars continue to develop their understanding of two-dimensional figures and extend their understanding to the coordinate plane by learning how to ‘draw polygons in the coordinate plane given coordinates for the vertices’ (6.G.3) and determine the length of a side with joining points. Additionally, during Unit 2 in the eighth grade, scholars developed an understanding of congruence by investigating rigid transformations on and off the coordinate plane (8.G.A.).” The end of the Overview states, “In high school scholars formalize their understanding of similarity developed in middle school to defining it as rigid motions followed by dilations. In middle school, scholars will work with dilations centered around the origin or a vertex point on the figure (for figures with vertical and horizontal side lengths that can be counted on the coordinate grid), whereas in high school, scholars will learn how to perform dilations in the coordinate plane around a point other than the origin or a vertex point on specific types of figures.”
• Throughout the narrative for the teacher of the Unit Overview, there are descriptions of how the lessons will be used as the grade level work progresses. In Unit 5, Lessons 11 and 12 allow students to graph a line using a table and equation before progressing into writing equations from graphs, tables and points in Lessons 13-15. In Lesson 11, Exit Ticket Question 1 states, “Graph the equation $$y=-\frac{1}{2}x+2$$ by making a table of values.” In Lesson 15, Exit Ticket, Question 1 states, “Write an equation in slope-intercept form of a line with a slope of -3 that travels through the point (-5,4). Show all of your work.”
• The last paragraph of each narrative for the teacher in the Unit Overview describes the importance of the unit in the progressions. Unit 9 states, “Then, looking further ahead to high school, scholars ‘begin to formalize their geometry experiences from elementary and middle school, using more precise definitions and developing careful proofs’ (CCSS 74). More specifically, in high school, scholars will have to understand and apply theorems about circles, find arc lengths and areas of sectors of circles, explain volume formulas in addition to using them to solve problems, and visualize relationships between two-dimensional and three-dimensional objects.”
• 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 Expressions and Equations” progression document is included with the footnote, “Common Core Expressions and Equations Progression 6-8” by Common Core Tools.  Achievement First does not own the copyright in ‘CC Expressions and Equations Progression’ and claims no copyright in this material.”
• Each lesson includes a “Connection to Learning and Conceptual Understanding” section that describes the progression of the standards within the unit. In Unit 8, Lesson 13 states, “In the previous lesson, students developed a method for multiplying and dividing numbers in scientific notation by applying the commutative and associative property to group the coefficients and powers of ten to efficiently apply the product/quotient rule of exponents. In this lesson, students apply everything that they have learned about scientific notation to solve real-world problems by picking the correct operations to use in a problem-solving way.”
• In the Scope and Sequence Detail from the Implementation Guide, there are additional progression connections made. The Cumulative Review column for each unit provides a list of lesson components and the standards addressed. Prior (Remedial) standards are referenced with an “R” and grade level standards are referenced with an “O.” In Unit 1, Geometry states, “Skill Fluency (4 days a week): 7.NS.1 (R), 7.NS.2 (R), 7.NS.2d (R), 7.NS.3 ® Mixed Practice (3 days a week): 7.NS.3 (R), 7.EE.4 a(R), 8.G.3 (O), 8.G.1 (O).”

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

• In Unit 2, Lesson 4, Independent Practice Question 8 (PhD level), students establish facts about about the angles created when parallel lines are cut by a transversal. The materials state, “How could you use a transparency to prove that the angles created when a transversal passes over one line are identical to the angles created when the transversal crosses the other line if it is parallel to the first? How does this relate to rigid transformations?” (8.G.5)
• In Unit 3, Lesson 3, Think About It!, students understand congruence and similarity using transformations. The materials state, “Rectangle ABCD dilated by a scale factor of 3 about the origin and resulted in the image A’B’C’D’. Record the coordinates for the image and pre-image.  What relationship exists between the points on the image and pre-image?” (8.G.4)
• In Unit 7, Lesson 7, State Test Alignment, students analyze and solve simultaneous equations,. The materials state, “What is the solution to the system of equations below?  $$3x+4y=-2$$ and $$2x - 4y = -8$$.” Students choose from four answers. (8.EE.8)
• In Unit 8, Lesson 3, Independent Practice Question 6 (Master level), students apply the properties of integer exponents to generate equivalent numerical expressions,. The materials state, “Jose simplified the expression $$5^3×2^3$$ and wrote $$7^3$$. Did he simplify the expression correctly? How do you know? If Jose made an error, identify it, and fix the mistake.” (8.EE.1)

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 5 states, “Most of Unit 5 draws directly from Ratios & Proportions and Expressions & Equations in seventh grade math as well as Expressions & Equations and Geometry already studied thus far in eighth grade. In the seventh grade, scholars spend a majority of the year developing a deep understanding of ratios and proportions in different variations and contexts. In eighth grade, scholars continue to draw on this knowledge to extend their understanding of ratios to include rate of change in Unit 4 which directly aligns to slope in Unit 5. Additionally, during the seventh grade, scholars develop the ability to write equivalent expressions by manipulating the terms in an expression to simplify and/or expand; this ability will lend itself directly to helping eighth grade scholars manipulate linear equations written in different forms to be rewritten in the desired form (SMP4).”
• The narrative for the teacher in the Unit Overview makes connections to current work. Unit 1 states, “The start of this unit connects to the previous geometry work that students have done in grades 4,5, and 6. Students recall basic terminology such as line, line segment, polygon, etc. They build upon being able to draw and identify angles, and classify shapes by their angles and properties while classifying two-dimensional figures based on their properties. In this lesson, students discover that translations, reflections, and rotations are distance-preserving transformations which means that they create congruent images.”
• Each lesson includes a “Connection to Learning and Conceptual Understanding” section that relates to prior knowledge. In Unit 4, Lesson 1 states, “This is the first lesson of the new unit on functions, drawing heavily from prior knowledge that students have learned in ratios and proportions, and expressions and equations units in $$7^{th}$$ and $$8^{th}$$ grade. In $$7^{th}$$ grade, for example, students studied proportional relationships in depth: They analyzed these relationships using a variety of representations including the equation $$y=kx$$.”
• 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 2 states, “Lesson 6 builds on the understanding of using angle pair relationships formed by a transversal of two parallel lines and focuses on the common error of identifying congruent vs. supplementary angle pairs and using them to solve.”

### 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 8 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 AIMs for Unit 4, Lesson 2: “SWBAT determine and understand the definition of a function by analyzing the similarities and differences in the relationships between the dependent and independent variables in equations, tables, and graphs” and “SWBAT determine if a relationship represented as a verbal description, table, mapping diagram, graph or ordered pairs is a function by applying the definition,” are shaped by 8.F.A: Define, evaluate, and compare functions.
• The AIM for Unit 7, Lesson 1: “SWBAT define a system of linear equations and solve real world problems using two equations and tables by interpreting the x,y values as the solution,” is shaped by 8.EE.C: Analyze and solve linear equations and pairs of simultaneous linear equations.
• The AIM for Unit 9, Lesson 3: “SWBAT develop and apply the formula for the volume of a cone to solve real world and mathematical problems,” is shaped by 8.G.C: Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

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 2, Lesson 6 connects 8.G.A and 8.EE.C as students determine similarity of a missing corresponding angle measure in a transversal diagram by writing and solving an algebraic equation. In the Independent Practice (Bachelor Level), Question 2 states, “If $$\angle5=5x$$ and $$\angle1=100\degree$$, what is the value of $$x$$? Justify your reasoning by identifying any relevant angle pair relationships.”
• In Unit 5, Lesson 8 connects the concept of similarity (8.G.A) to work in defining slope (8.EE.B) as students compare triangles on a coordinate plane. In the Independent Practice, Question 2 (Master level) states, given a diagram of a line on a coordinate plane with two sizes of “slope triangles,” “A smaller triangle is inscribed inside a larger triangle. Use the triangles to prove that the slope between any two points on a line is equivalent to the slope between any other two lines. Your explanation should prove that the triangles are similar first.”
• In Unit 5, Lesson 13, students use functions to model relationships between quantities (8.F.B) to develop understanding about connections between proportional relationships, lines, and linear equations (8.EE.B). In the Independent Practice Question 10 (PhD level), students are instructed to, given a table of a proportional relationship, “Write an equation that represents the function in the table below. Explain how you were able to determine the slope and y-intercept.”
• In Unit 7, Day 2 connects the work between two major clusters 8.F.A and 8.F.B when students construct functions to model linear relationships and then compare them. In the Problem of the Day, Day 1 states, “Nathaniel is trying to expand his investment portfolio. His broker presents him with three shares that he can buy. Nathaniel is only looking to buy shares from one corporation during this first quarter and wants to buy the share that will give him the most profit over time. The first company, Apple, Inc. sells 15 shares for $225 and 30 shares for$450. From those shares, Nathaniel is likely to receive $15 for each share per month. Lush is selling their shares for$7.50 each. A previous shareholder of Lush says that she regularly got 200 dollars in dividends for her twenty shares. Lastly, GoPro is selling shares for 20 dollars each but there is an additional broker’s fee of 10 dollars for your first purchase of shares. However, reports from last year indicate that the largest shareholders of the company received a gross income of $15,000 in dividends for the 750 shares that they owned. Using the information above, help Nathaniel make an informed decision about the shares that he should buy to get the biggest bang for his buck. Be sure to mathematically justify your answer. (Dividends is the amount of money that you earn from buying a share monthly. Gross income is the amount of money that you earn before tax).” ### 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. 4/4 + - Criterion Rating Details The instructional materials reviewed for Achievement First Mathematics Grade 8, when used as designed, spend approximately 78% 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. 4/4 + - Indicator Rating Details The instructional materials reviewed for Achievement First Mathematics Grade 8 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 8 out of 10, which is approximately 80%. • The number of days devoted to major work of the grade (including assessments and supporting work connected to the major work) is 128 out of 140, which is approximately 91%. • The number of minutes devoted to major work (including assessments and supporting work connected to the major work) is 9815 out of 12,600 (90 minutes per lesson for 140 days), which is approximately 78%. 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 78% of the instructional materials focus on major work of the grade. ### Gateway Two ## Rigor & Mathematical Practices #### Partially Meets Expectations + - Gateway Two Details The instructional materials reviewed for Achievement First Mathematics Grade 8 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 8 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 8 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 rigid transformations by using manipulatives such as tesselation tiles. The materials state, “Thomas was playing with three tessellation tiles on his desk. He went to the bathroom and when he returned, he found that someone had moved all the tiles and put them in a different place (a before and after diagram is provided). Part A: Look at each tile before and after and describe how someone moved the tile using as much detail as possible.Triangle; Rectangle; Trapezoid. Part B: How could you prove that the triangle tile could be the exact same tile and someone didn’t switch it out for a larger or smaller tile?” (8.G.A) • In Unit 2, Lesson 4, Independent Practice, Question 8 (PhD Level), students develop conceptual understanding of angle relationships within parallel lines by using manipulatives and properties of transformations. The materials state, “How could you use a transparency to prove that the angles created when a transversal passes over one line are identical to the angles created when the transversal crosses the other line if it is parallel to the first? How does this relate to rigid transformations?” (8.G.A) • In Unit 4, Lesson 2, THINK ABOUT IT!, students develop conceptual understanding of functions by analyzing examples. The materials state, “The following input/output tables have been split into two categories; relations and relations that are also functions. Look for similarities and differences and write a definition for what a function is.” (8.F.A) Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Examples include: • In Unit 4, Lesson 7, Independent Practice, Question 2 (Bachelor Level), students demonstrate conceptual understanding of function definitions by organizing information in a table. The materials state, “Does the equation represent a linear function? Prove your answer by showing the constant ROC (rate of change) in a table.” (8.F.A) • In Unit 5, Lesson 8, Independent Practice, Question 2 (Master Level), students demonstrate conceptual understanding of slope by using similar triangles. The materials state, “A smaller triangle is inscribed inside a larger triangle. Use the triangles to prove that the slope between any two points on a line is equivalent to the slope between any other two lines. Your explanation should prove that the triangles are similar first.” (8.EE.B) • In Unit 10, Lesson 2, Independent Practice, Question 2 (Bachelor Level), students demonstrate conceptual understanding of rational numbers by justifying their classification, “Is 0.6666… rational or irrational? Justify in two ways.” (8.NS.1) ### 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 8 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 5, Lesson 12, Independent Practice, Question 4 (Master Level), students develop procedural skill and fluency by graphing functions. The materials state, “Graph the function $$y=3x-2$$ and explain the steps you used to create the graph based on the structure of the equation.” (8.F.5) • In Unit 7, Lesson 7, Partner Practice, Question 3 (Master level), students develop procedural skill and fluency by solving simultaneous equations using elimination. The materials state, “Solve the system of equations using elimination two different ways (addition and subtraction) and verify both methods produce the same solution. {$$4y+3x=22;-4y+3x=14$$}.” (8.EE.8b) • In Unit 10, Lesson 8, Partner Practice, Question 3 (Bachelor Level) states, “Which set of measurements are the side lengths of a right triangle? a) 7, 8, 12; b) 9, 12, 15; c) 10, 24, 26; d) 2.4, 3.4, 5.5.” (8.G.6) The instructional materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade-level. Examples include: • In Unit 1, Lesson 11, Independent Practice, Question 7 (PhD Level), students demonstrate procedural skill and fluency by using coordinates to describe transformations. The materials state, “Triangle ABC has vertices at A (3, 4), B (3, 9), and C (6, 4). What are the vertices of the image A’B’C’ if the triangle was rotated 180 degrees around the origin and translated four units up. Explain how you know.” (8.G.3) • In Unit 2, Lesson 2, Independent Practice, Question 2 (Bachelor level), students demonstrate procedural skill and fluency by solving multi-step linear equations and using substitution to check their answer. The materials state, “Solve the equation and check your solution using substitution. $$\frac{1}{5}b+3b=2b+42$$.” (8.EE.7b) • In Unit 8, Lesson 7, Exit Ticket, Question 3, students demonstrate procedural skill and fluency by expressing scientific notation. The materials state, “The length of a very fine grain of sand is about 0.0005 inches. Which of the following also show the length of the grain? Select all that apply. a) $$5×10^3$$ ; b) $$5×10^4$$ ; c) $$5×10^{-3}$$ ; d) $$5×10^{-4}$$; e)$$\frac{5}{10^{-4}}$$ f) $$\frac{5}{10^{4}}$$” (8.EE.3) ### 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 8 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 4, Lesson 14, Exit Ticket, students solve real life problems by creating a graph to show qualitative data. The materials state, “When Kaylee woke up at 6 a.m. for school, it was 53 degrees outside. By the time she left home an hour later to catch the bus the temperature had risen 4 degrees. It remained the same temperature until school began at 7:30. The temperature increased at a steady rate until she got to go outside after her fourth 60-minute class where she measured the temperature to be 80 degrees. It stayed at this temperature throughout her hour lunch break. During Kaylee’s last three hour-long classes, the temperature decreased to 71 degrees. From the end of school until she was done with lacrosse practice at 5:00, the temperature dropped an additional 2 degrees. Part A: Sketch a graph that models this situation. a) Determine the time period that the temperature was changing the fastest. b) When did the temperature change the slowest?” (8.F.B) • In Unit 6, Lesson 5, Independent Practice, Question 3 (Master Level), students interpret a scatterplot and it’s line of best fit. The materials state, “Julie recorded the number of female students and male students in her school for the past 8 years in a table and graphed the data using a scatter plot where the x-axis represents the females and the y-axis represents the males. She wrote the equation $$y=1.2x+12$$ to represent the line of best fit. Step A: What does the slope of the equation represent? Step B: What does the y-intercept represent? Step C: Draw a sketch of what you would expect the scatter plot to look like and explain why you drew the scatter plot in that way.” (8.SP.3) • In Unit 7, Lesson 14, Partner Practice, Question 2 (Master Level), students solve real life problems by using simultaneous equations to find pricing data. The materials state, “Two chocolate chip cookies and three brownies cost a total of$9.50. One chocolate chip cookie and two brownies cost a total of 6.00. What is the price of a chocolate chip cookie and a brownie?” (8.EE.8c) • In Unit 8, Lesson 13, Independent Practice Question 3 (Master Level), students use scientific notation to solve a real world problem. The materials state, “If one water molecule contains 2 hydrogen atoms and 1 oxygen atom, and 10 water molecules contain 20 hydrogen atoms and 10 oxygen atoms, how many hydrogen atoms and oxygen atoms are in $$6.02×10^{23}$$ water molecules? Show your work.” (8.EE.4) • In Unit 9, Lesson 4, Interaction with New Material, Example 2, students use volume to solve real life problems. The materials state, “A beach ball has a diameter of 1.5 feet. Approximately how many cubic feet of air are needed to inflate three beach balls?” (8.G.9) • In Unit 10, Lesson 13, Independent Practice, Question 6 (PhD Level), students use the Pythagorean Theorem to solve real-world problems. The materials state, “The typical ratio of length to width that is used to produce televisions is 4:3. A TV with length 20 inches and width 15 inches, for example, has sides in a 4:3 ratio; as does any TV with length $$4x$$ inches and width $$3x$$ inches for any number $$x$$. a) What is the advertised size of a TV with length 20 inches and width 15 inches? b) A 42” TV was just given to your family. What are the length and width measurements of the TV?” (8.G.7) ### 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. 2/2 + - Indicator Rating Details The instructional materials for Achievement First Mathematics Grade 8 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 8, Lesson 3, Exit Ticket, Question 2, students demonstrate conceptual understanding of properties of integer exponents when they explain why a rule is true. The materials state, “Explain why the rule $$a^5×b^5=(ab)^5$$ is true using the commutative and associative properties.” (8.EE.1) Fluency and Procedural Skill: • In Unit 1, Lesson 7, Independent Practice, Question 6 (PhD Level), students demonstrate procedural knowledge in order to determine which ordered pair represents a reflection. The materials state, “Which of the following describes the location of a point (x,y) reflected over the y-axis and reflected over the x-axis? a) $$(x,y)$$; b) $$(-x,y)$$; c) $$(x,-y)$$; d) $$(-x,-y)$$.” (8.G.3) Application: • In Unit 9, Lesson 6, Independent Practice, Question 3 (Bachelor Level), students apply their knowledge about volume to determine how much cheesecake they get. The materials state, “A round cheesecake has a diameter of 8 inches and a height of 3 inches. It is cut into 8 equal-sized slices. How many cubic inches does each slice take up in the cheesecake? Use 3.14 for pi and round your answer to the nearest tenth of a cubic inch.” (8.G.9) 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 3, Lesson 6, Independent Practice, Question 6 (Master Level), students engage in application and conceptual understanding about properties of transformations to prove that triangles are similar. The materials state, “Are the two triangles similar? Prove by graphing and a series of rigid transformations. Triangle A: (1, 2), (4,8), and (10, 5); Triangle B: (-4, -2), (-4, -3), and (-3, -5).” (8.G.4) • In Unit 7, Lesson 12, Independent Practice Question 3 (Master level), students demonstrate fluency by solving simultaneous equations in more than one way as they apply the mathematics to understand given data. The materials state, “In the fall, the math club and science club each created an Internet site. You are the webmaster for both sites. It is now January and you are comparing the number of times each site is visited each day. Science club: There are currently 400 daily visits and the visits are increasing at a rate of 25 daily visits per month. Math club: There are currently 200 daily visits and the visits are increasing at a rate of 50 daily visits per month. a) Write a system of linear equations to represent the situation. Then graph to determine the solution. b) Explain what the solution to the system means in the context of the problem.” (8.EE.8) • In Unit 8, Lesson 10, Independent Practice, Question 5 (Master Level), students apply their procedural fluency of operations with numbers expressed in scientific notation to real world scenarios. The materials state, Bubba’s Boot Barn is a favorite stop of visitors to Nashville’s downtown shopping area. Last year, $$2.42×10^5$$ people visited Bubba’s. This year it has become and even more popular venue, with $$2.53×10^6$$ visitors. Step A: How many total visitors did Bubba’s get over the two years? Step B: How many more visitors did Bubba’s get this year compared to last year?” (8.EE.4) ### Criterion 2e - 2g.iii Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice 8/10 + - Criterion Rating Details The instructional materials reviewed for Achievement First Mathematics Grade 8 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. 1/2 + - Indicator Rating Details The instructional materials reviewed for Achievement First Mathematics Grade 8 partially meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade level. The Mathematical Practices are listed in the Unit Overviews as well at the beginning of each lesson, however there is little direction provided about how the MPs enrich the content or make connections to enhance student learning. All eight MPs are clearly identified throughout the materials, with few or no exceptions, though they are not always accurate. For example: • In the Guide to Implementing AF Math, “Math Lesson Types” explains how different types of lessons engage students with the Mathematical Practice Standards. For example, Conjecture Based Lessons states, “Purpose: Through the use of investigation and guided inquiry, students develop conceptual understanding of math topics and strategies. They persevere by developing and proving mathematical conjectures. Structurally based off of the Standards for Mathematical Practice 3, these lessons push students to make viable arguments and critique the thinking of others to generate a conjecture that will then be tested. They must make connections to previously learned content, apply sound mathematical practices, and think flexibly.” • The MPs are listed at the beginning of each lesson in the Standards section. • All MPs are represented throughout the materials, though lacking balance. For example, MP5 is emphasized in two units, while MP2 is emphasized in 8. • The Mathematical Practices are not always identified accurately. For example: • At the unit level for Unit 1, MP6 is not identified as an emphasized practice. However, at the Lesson level, 13 of the 16 lessons identify it as connected. • The Unit 2 Overview does not bold MP6 as an emphasis, yet nine out of 10 of the lessons include this practice. Whereas MP5 is noted as an emphasis, but is not identified in any lessons. • In Unit 6, MPs 1, 2, 4, and 7 are bold; however, MP7 does not appear in any of the 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: • In Unit 1, Lesson 3 uses MP language in a teacher prompt, but doesn’t make the connection to the actual MP. The materials state, “How can we verify this with our tools?” • In Unit 3 Overview, Skills and Procedural Knowledge states, “Dilate two-dimensional figures on and off the coordinate plane given precise directions and a scale factor.” • Unit 5 Overview states, “Looking directly ahead to the next two units specifically, scholars continue their work with understanding linear equations to help them make sense of bivariate data in scatter plots to make formal mathematical predictions and to solve simultaneous equations by graphing, substitution, and elimination.” • Unit 7 Overview states, “This understanding will be formalized in lesson 10 and 11 when students analyze the structure of the equations to determine the number of solutions and the focus of lesson 3 should remain on the result of graphing.” • Some of the unit overviews make direct connections to help teachers understand how to emphasize the content to incorporate the MPs; however, they are still not always clear in their intent. For example: • MP1 - Unit 5 Overview states, “Since scholars will know that two triangles are similar if they have congruent corresponding angles and proportional corresponding sides by this point in the eighth grade, they should be able to logically conclude that the slope of a line must be the same between any two points given that the ratio of the triangles side lengths always remains proportional (SMP1).” • MP2 - Unit 7 Overview states, “Additionally, scholars must have a strong foundation from eighth grade math aligned to fluently writing, graphing, and solving linear equations (SMP2).” • MP5 - Unit 1 Overview states, “Students learn the properties of a translation and perform translations by performing translations along a specific vector using transparencies (SMP5).” ### Indicator 2f Materials carefully attend to the full meaning of each practice standard 1/2 + - Indicator Rating Details The instructional materials reviewed for Achievement First Mathematics Grade 8 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. • 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 7, Lesson 2, Independent Practice, Question 4 (Master Level) state, “Lori graphed two equations and determined that their lines would never intersect. The equations have the same slope, but different y-intercepts. What should she conclude about the solution to this system of equations? Explain.In this example, MP4 is a visual representation but students do not choose a strategy, nor is there any real-world connection. • In Unit 7, Lesson 11, students create a system of two linear equations to satisfy a given condition related to the number of solutions. There are no instances in this lesson where students use real-world problems to model their understanding. • In Unit 10, Lesson 10, Interaction with New Material, Example 1 states, “Edward walks along a path that goes directly from his house to school. His house is located at (1,2) and school is located at (-3, -3). Each gridline represents 1 kilometer. What is the exact distance that Edward travels between his house and school?” In the independent practice, students are provided with grids and points, which limits the options of how a student might approach the solution. Additionally, students do not have the opportunity to choose a strategy. • 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 4, Partner Practice, Question 1 states, “Translate trapezoid ABCD five units down and three units right.” Students are provided a graph with a trapezoid in quadrant II. Materials provided include rulers or straightedges. • In Unit 2, Lesson 4, Think About It states, “Lines $$L_1$$ and $$L_2$$ were cut by a third line, T. Use a protractor to measure all the angles that are formed when the two lines were crossed by line T. What do you notice about the angles you recorded? Is there a rule that exists?” Students are directed to use a protractor without an opportunity to choose tools. • In Unit 6, Overview states, "Between lessons 4 and 5, there is an opportunity for a technology-based lesson. If possible, given the availability of technology for certain schools, scholars will be provided with the opportunity to use graphing calculators to construct scatter plots, determine the equation for the line of regression, and make predictions about the data (SMP5). Schools that do not have graphing calculators available can utilize the Chrome Books to have scholars work on a Google Spreadsheet where they can input quantitative data and create a scatter plot with a trend line. Since the trend line will either be determined by a graphing calculator or a Google Spreadsheet, it will represent the most accurate possible line that fits the data. On a graphing calculator and on a Google Spreadsheet the equation for the line will be provided for the scholars, but the scholars will still need to interpret the meaning of the slope and y-intercept in context of the situation. Additionally, scholars can use the technology to determine the accuracy of the line drawn by looking at the r-squared value; scholars do not need to learn what an r-squared value means in eighth grade but they can make statements about the accuracy of the line dependent upon how close the r-squared value is to one." Students are given technology tools to create scatter plots without an opportunity to choose an appropriate strategy or tool. The following practices are connected to grade-level content and are developed to their full intent over the course of the materials. • MP1: Make sense of problems and persevere in solving them. • In Unit 5, Lesson 9, students extend their thinking of proportional graphs to determine the equation of a line that is not proportional. Students persevere throughout Think About It by developing and following a plan (Understand, plan, estimate/predict and solve) and building on prior knowledge. The materials state, “Interaction with New Material: What if the relationship is linear, but not proportional? How does the equation change? Derive the equation for a line that passes through the y-axis at value b instead of the origin. Use the given triangles and label the sides accordingly. Then answer the question: Does this equation also work for a proportional relationship?” • In Unit 7, Lesson 15, Exit Ticket, Question 1, students persevere as they determine the most appropriate strategy to solve a systems of equations word problem. The materials state, “Use the most appropriate strategy to solve the problem described below. Show all of your work. A hotel offers two activity packages. One costs192 and includes 3 hours of horseback riding and 2 hours of parasailing. The second costs \$213 and includes 2 hours of horseback riding and 3 hours of parasailing. What is the cost for 1 hour of each activity?”
• In Unit 10, Lesson 13, Partner Practice (Master Level), Question 4, students determine the most appropriate strategy to solve a systems of equations word problem. The materials state, Christopher has a garden in the shape of an isosceles trapezoid (pictured below). He wants to plant roses on $$\frac{1}{4}$$ of the garden and tulips on the other $$\frac{3}{4}$$ of the garden. How many more square feet will be covered with Tulips than Roses? Round your answer to the nearest hundredth.”
• MP2: Reason abstractly and quantitatively.
• In Unit 6, Lesson 3, Exit Ticket, Question 2, students reason about quantities in order to make predictions based on data points. The materials state, “Draw an appropriate line of best fit given the scatter plot below. Explain why the line you drew is an appropriate model for the graph by discussing the patterns of association present in the data. Use your line of best fit to predict the temperature at an elevation of 750 meters.”
• In Unit 6, Lesson 6, Exit Ticket, Question 1, students contextualize the meaning of the slope and y-intercept from a visual scatter plot graph. The materials state, “Justin drew a line of best fit in the scatter plot below represented by the equation $$y=2x+50$$ to determine how many hours he would need to study during the unit to earn a 100% on his upcoming test. Part A: What do the slope and y-intercept mean given the context of the scatterplot?”
• In Unit 8, Lesson 2, Think About It, students de-contextualize powers of exponents by expanding the problem out to determine the procedural process. The materials state, “Simplify the following exponential expressions by first expanding and then rewriting as a base raised to a single power. a) $$(2^3)^2$$ ; b) $$(h^2)^3$$”
• MP6: Attend to precision.
• In Unit 4, Lesson 2, Exit Ticket, Question 2, students are given the opportunity to determine a definition for a function versus a relation and apply that knowledge to various questions. The materials state, “Joshua claims that all relations are functions. Nathan claims that all functions are relations. Determine who is correct and provide an example that fully supports the claim.”
• In Unit 9, Lesson 2, Partner Practice, Question 2 (Bachelor Level), students use precision in computation and labeling to calculate volume. The materials state, "Find the exact volume of the cylinder pictured below."
• In Unit 10, Lesson 2, Interaction with New Material, Question 2, students use precise vocabulary as they classify irrational and rational numbers and their subsets.The materials state,  “Identify each of the following numbers as rational or irrational and write a one sentence explanation for your classification. If you identify a number as rational, then determine its most specific classification: rational, integer, whole, natural.”
• MP7: Look for and make use of structure.
• In Unit 2, Lesson 3, Think About It, students use the structure of proportions to solve linear equations. The materials state, “Solve the two equations using any methods and check to verify that your solution satisfies the equation. How was your method for solving the same or different between the two equations? $$\frac{4}{5}=\frac{52}{n}$$;$$\frac{4}{8}=\frac{2}{n+1}$$
• In Unit 5, Lesson 4, Independent Practice, Question 8 (PhD Level), students analyze the structure of equations in order to determine the number of solutions. The materials state, “Write two equations, one with no solution and the other with one solution, that requires combining like-terms to determine the number of solutions. Explain how you created the equations using the structure of the equations.”
• In Unit 7, Lesson 10, Exit Ticket, Question 1, students inspect equations and use the structure and components to identify how many solutions a system of equations has. The materials state, “Determine the number of solutions to the following system of two linear equations without performing any calculations. Explain how you were able to determine the number of solutions without performing calculations. $$3x - 2y = 5; 3x - 2y = -1$$.”
• MP8: Look for and express regularity in repeated reasoning.
• In Unit 2, Lesson 3, Exit Ticket, Question 2, students use repeated reasoning to describe an error in work about solving proportions. The materials state, “Explain the mistake made in the work below. Your explanation should include a description about solving proportions.
• In Unit 5, Lesson 1, Test the Conjecture #2, students use repeated reasoning to understand if equations have one solution, no solution, or many solutions. The materials state, “Determine the number of solutions for the following equation $$-(-4x - 6) + (-2x) = -4x - 5 + (-5)$$.”
• In Unit 8, Lesson 4, THINK ABOUT IT!, students use repeated reasoning to understand the value of a number raised to the zeroth power. The materials state, “Simplify the expression $$3^0×3^2$$ using the product rule. Simplify the expression $$a^0×a^4$$ using the product rule. What can you conclude about the value of a number raised to the zeroth power?” The teacher provides a conjecture to discuss: “Any number raised to the zeroth power is 1.”

### Indicator 2g

Emphasis on Mathematical Reasoning: Materials support the Standards' emphasis on mathematical reasoning by:
0/0

### 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 8 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.”
• Unit 2, Lesson 6, Error Analysis Lesson, Independent Practice #6 (Master Level), students investigate angles created when parallel lines are cut by a transversal. The materials state, “In the diagram below, $$\angle3=105\degree$$ and $$\angle8=5x$$.  Scholar A says that the value of $$\angle7=75\degree$$.  Describe the mistake that the scholar made and provide at least two different ways to prove the scholar wrong.” (8.G.5)
• In Unit 4, Lesson 8, THINK ABOUT IT!, students compare functions represented in different ways. The materials state, “Below are two different linear functions. Determine which function is changing the fastest using any methods you have learned. Justify why the function you choose is changing faster than the other.” Students are given a table and a graph to compare. (8.F.2)
• In Unit 5, Lesson 1, Independent Practice, Question 5 (Master level), students determine the number of solutions to an equation. The materials state, “Mark and Molly are debating over the solution to the equation $$11(x+10)=110$$. Mark says that there is no solution because the 110’s cancel out of the equation. Molly says that the solution x = 0 is a valid solution to the equation. Who do you agree with and why?” (8.EE.7a)
• Unit 7, Lesson 9, Exit Ticket Question 2, students analyze pairs of simultaneous linear equations. The materials state, “Abigail decided to solve the following system of two linear equations by graphing. Do you agree or disagree with Abigail’s decision? Why? $$\frac{6x+54=4}{x-5y=16}$$.” (8.EE.8)
• Unit 8, Lesson 5, Independent Practice, Question 6 (Masters level), students explore properties of integer exponents. The materials state, “Prove that any number raised to a negative exponent is equal to the reciprocal of the base raised to the opposite exponent. Use examples and explain.” (8.EE.1)

### 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 8 meet expectations that the instructional materials assist teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

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

• In Unit 2, Lesson 6, Error Analysis Lesson, Debrief, students compare exit ticket responses about angles created with a transversal. Teacher prompts include, “Which scholar’s work did you agree with? Turn and tell your partner who you chose and why. What error did this scholar make? Was Scholar B’s answer reasonable? Why/why not? What did this scholar do to get this correct, and why was that helpful?” (8.G.5)
• In Unit 7, Lesson 10, THINK ABOUT IT! Debrief, students solve systems by graphing. Teacher prompts include, “Do you agree with the first graph? How could you determine that the system has one solution by only looking at the equations? Do the y-intercepts help to determine if (the) system has one solution? What generalized rule can we say about determining if a system has one solution? What do these systems have in common?” (8.EE.8b)
• In Unit 10, Lesson 13, THINK ABOUT IT!, students use Pythagorean’s Theorem. Teacher prompts include, “Which scholar do you agree with? What did both scholars do correctly in their approach to the problem, and why does it make sense?” (8.G.7)

### 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 8 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 2, Lesson 5, Opening, Debrief, FENCEPOST #1, students prove if angles are equal. The materials state, “A translation can map one line to another line if they are parallel. Show Call: S explanation describes using a translation to map one line to the other. Do you agree with this scholar? Vote. CC. SMS: I agree because the scholar said that a translation can be used to map one line on to the other line.  We map $$L_1$$ onto $$L_2$$ using a translation and transversal T as the vector which can be shown using the transparency. Without the transparency, how do you know that the lines will completely map onto each other? TT. CC. SMS: We learned that translating a line along a vector produces a parallel line so if two lines are already parallel, we must be able to map one onto the other using a translation. BPQ – If a line is translated along a vector, what is the relationship between the original line and its image? BPQ – How could you use this property to prove that the lines will map to each other? Name the fencepost: A translation can map one line to another line if they are parallel.” (8.G.5)
• In Unit 5, Lesson 6, Opening, Debrief, FENCEPOST #1, students determine the rate of change. The materials state, “The slope of a linear function is its rate of change: $$\frac{\triangle y}{\triangle x}$$. Show Call: S work starts with the expression $$\frac{\triangle y}{\triangle x}$$. Do you agree with the formula this scholar used? Vote. CC. SMS: I agree because in unit four we learned that the rate of change between any two points is equal to $$\frac{\triangle y}{\triangle x}$$ which is the change in y over the change in $$x$$ in which the delta implies subtraction. BPQ – What does ‘change in y’ mean/imply? How would you define the relation graphed? CC. SMS: Since there is exactly one output for every input and the graph produces a straight line, this is a linear function. When working with linear functions and equations, the rate of change has a special name called the slope of a line and is often denoted using the variable $$m$$. [Planner’s note: To teach and reinforce this vocabulary, reveal permanent visual anchor with ‘slope’ defined and annotated on a graph, as well as calculated using two points and the formula]. Name the Fencepost:  What do we know then about slope? SMS: The slope of a linear function is its rate of change:$$\frac{\triangle y}{\triangle x}$$.” (8.EE.6)
• In Unit 10, Lesson 2, Opening, Debrief, FENCEPOST #1, students use Pi Day to discuss rational and irrational numbers. The materials state, “Rational numbers are repeating or terminating decimals that can be expressed as a fraction. Before we determine the difference between rational and irrational numbers, how would you define rational numbers? TT. CC. SMS: Rational numbers are integers and decimals that either terminate or repeat. What do all the integers and decimal in rational numbers have in common? TT. CC. SMS: The numbers are all written as a fraction first. This is a major key point and is the basic definition of a rational number. The official definition is a number that can be expressed as a ratio of two integers. Do these numbers meet that definition? CC. SMS: Yes, because a ratio of two integers is the same as a fraction and all these numbers are written as a fraction of two integers. By our definition, would 231 be a rational number? TT.  CC. SMS: 231 would be a rational number for two reasons. The first reason is that 231 is technically a terminating decimal and can be written as 231.0. The other reason is that we can express 231 as a ratio of two integers as $$\frac{231}{1}$$ which means that all integers are rational numbers because they can be written as that number over 1. BPQ – What does this mean about all integers? Name the fencepost:  How do we define rational numbers? SMS: Rational numbers are repeating or terminating decimals that can be expressed as a fraction.” (8.NS.1)

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:
• Independent Variable – a variable (often represented by x) whose variation does not depend on another variable.
• Dependent Variable – a variable (often represented by y) whose variation depends on another variable.
• Substitution – replacing a variable with a value or expression.
• Relation – any set of ordered pairs.
• Input – the independent variable, defines the function.
• Output – the dependent variable, changes based on change in the input.
• Function – a mathematical relationship where each input has a unique output.
• Rate of Change – a change in the dependent variable per a change in the independent variable; when comparing rates of change, you compare the magnitude of the rate of change, not the actual value.
• The teacher is routinely prompted to use precise vocabulary such as Unit 1, Lesson 1, Debrief. The materials state, “How would you describe how the individual tiles were moved? Mathematicians have specific names for these movements. We call a slide a translation, a flip is a reflection, and a turn is a rotation. All three of these are called rigid transformations. How did these rigid transformations change the figure?” Guidance within a possible student response, “Rigid transformations changed where the figure is sitting (T: We call this location) and which way it is facing (T: We call this orientation).”
• There is very little vocabulary emphasis in student-facing materials. For example, there is not a glossary for student reference.

## Usability

#### Not Rated

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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.
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### 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.
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### 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.
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### 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.