8th Grade - Gateway 1

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)

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.

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?”

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.”

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.”

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).”