8th Grade - Gateway 3

The materials reviewed for Achievement First Mathematics Grade 8 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development. Teacher guidance is found throughout the materials in the Implementations Guides, Unit Overviews, and individual lessons.

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

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

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

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

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. Each lesson includes anticipated challenges, misconceptions, key points, sample dialogue, and exemplar student responses. Examples from Unit 5, Expressions and Equations - Linear Equations, Lesson 6 include:

• “What do we want every student to take away or do as a result of this lesson? How will a teacher know if students have met this goal? Understand: The terms ‘slope’ and ‘rate of change’ can both be used to describe the way two variables change together. They are mathematically the same- determined by dividing the change in output by the change input at two distinct points of a function. For a straight line, the slope will be the same between any two points because for a function to be linear, the rate of change must be constant. Do: Students calculate the slope of a graphed line given points or having to pick their own points. Students prove that the slope of a line is the same regardless of the points that are chosen on the line by calculating multiple slopes using multiple points.”

• “Key Learning Synthesis - Conjecture: The slope between any two points in a linear function is equivalent to the slope between any other two points on the line.”

• Teacher prompts state, “What will we be able to do if our conjecture is true? Take 30 seconds to read and annotate the problem. What is the question asking us to do? How can we apply our conjecture to solve the problem? If we were being strategic, which two points would we pick? Model creating a quick input/output table and adding these points to it. How do I calculate the slope? How can we prove that our conjecture worked? With your partner, identify two more points and calculate the slope. Teacher can walk scholars through this if they are struggling with calculating the slope. So far, does our conjecture hold up?”

• “Anticipated Misconceptions and Errors: Students might mix up the x and y coordinates. Students might correctly determine Δx and Δy but write the final slope as Δx/Δy. Students might make a subtraction error when subtracting negative values. Students might not start with the same point for calculating the rise and run (e.g. for (1, 4) and (2, 6), scholars might find Δy as 4 - 6 = -2 but then switch the order and find Δx as 2 - 1 = 1).”

Each lesson includes a “How” section that lists the key strategies of the lesson and delineates what “top quality” work should include. Examples from Unit 5, Expressions and Equations - Linear Equations, Lesson 6 include:

• “Key Strategy: Write the equation Slope = \frac{Δy}{Δx}, Identify two points on the line and make an input/output table to record them. Calculate Δy or the rise. Calculate Δx or the run. Give final answer in the form of a fraction.”

• “CFS (Criterion for Success) for top quality work: Equation for slope is written. Coordinate pairs are annotated with x and y. Individual Δy and Δx are calculated. Final slope is written as the ratio of Δy/Δx.”

The materials reviewed for Achievement First Mathematics Grade 8 meet expectations for containing adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current grade so that teachers can improve their own knowledge of the subject. 

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

• The Unit Overview Appendix also often includes an excerpt from an unknown source which provides a teacher with an understanding of grade-level standards progression. Unit 10, Appendix A: Teacher Background Knowledge 8.NS, Unpacking the standard, “Students understand that Real numbers are either rational or irrational. They distinguish between rational and irrational numbers, recognizing that any number that can be expressed as a fraction is a rational number. The diagram below illustrates the relationship between the subgroups of the real number system. Students recognize that the decimal equivalent of a fraction will either terminate or repeat. Fractions that terminate will have denominators containing only prime factors of 2 and/or 5. This understanding builds on work in 7th grade when students used long division to distinguish between repeating and terminating decimals.”

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

• Unit 1 Overview, Geometry – Understanding Rigid Transformations and Congruence, Identify the Narrative, “Looking ahead to the remainder of the year, Unit 1 lends itself most immediately to the work that students do in Units 2 and 3. While the connection to the content in Unit 2 might not be immediately apparent, the learning from Unit 1 is necessary in Unit 2 for students to explain angle relationships between angles formed by a transversal cutting parallel lines. Specifically, students explain that vertical angles are congruent because one angle can be mapped to the other using a 180-degree rotation about the point of intersection. Alternate interior angles are congruent because one can be mapped to the other using a 180-degree rotation about a center. And, corresponding angles are congruent because one can be mapped to the other using a translation. During Unit 3, students extend their knowledge about transformations to develop an understanding of dilations given scale factor in the coordinate plane. Discovering dilations leads to the first time in eighth grade that students have the chance to discuss similarity. During Unit 1, scholars developed an understanding that any two figures that can be obtained from one another using any series of rigid transformations are congruent. In Unit 3, scholars expand upon that understanding to include the fact that any two figures that can be obtained from one another through a series of transformations that include a dilation result in similar figures. For the remainder of the year, similarity of figures in the coordinate plane becomes a repeating theme as students learn rate of change and slope. Students need to understand the concept of similar figures to apply that knowledge to similar triangles to use proportional reasoning in regard to rate of change and slope. Similarity also is an essential concept when helping students discover special right triangles when using the Pythagorean Theorem; if students are able to develop the skills to identify the patterns when solving problems involving special right triangles using the Pythagorean Theorem then they more readily are able to access the content. In high school, students formalize their middle school geometry experiences using more precise definitions and proofs. Students continue their work with congruence and similarity in high school geometry through a means of rigid transformations. Triangle congruence is studied in depth to establish ASA, SAS, and SSS criteria ‘using rigid motions … to prove theorems about triangles, quadrilaterals, and other geometric figures’ (CCSS 74). The foundational concepts discovered during Unit 2 are the building blocks for the entirety of the high school standard G-CO which addresses all possible outcomes involving congruence from experimenting with transformations in the coordinate plane to understanding congruence in terms of rigid transformations to proving geometric theorems involving rigid transformations. Without mastery of these foundational building blocks during middle school, students will be at an extreme disadvantage in a high school common core aligned geometry course.”

• Unit 5 Overview, Expressions and Equations – Linear Equations, Identify the Narrative, “For the remainder of eighth grade, it will continue to be imperative that scholars have a solid understanding of slope, y-intercept, and graphing, solving, and writing linear equations.  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. Additionally, at the end of the year when scholars are studying exponents, volume, and irrational numbers, the connection should be made back to the concept of linear versus non-linear functions so scholars are able to clearly state that functions in which the independent variable has an exponent of any value other than zero or one represent non-linear functions. Without an extremely solid foundation built around linear equations stemming from Unit 4 through Unit 5, it will be difficult for scholars to thoroughly understand much of the algebraic material taught through the remainder of the eighth-grade year that is essential for scholars to be successful in high school math. Looking further ahead to high school, scholars will continue to develop their understanding of linear equations and when linear equations represent functions versus when linear equations do not represent functions. Additionally, in high school, scholars will use function notation, f(x), to denote a function. As scholars deepen their understanding of functions in high school, they continue to model with and interpret linear functions, but they also extend their understanding to include quadratic, exponential, square root, trigonometric, one-to-one, and inverse functions. When studying non-linear functions, high school scholars will broaden their understanding of rate of change to include the average rate of change of a function over a specified interval (which lends itself to calculus in the future). Scholars learn the foundational concepts about functions in middle school that are necessary for them to be able to work fluently with linear and non-linear functions in high school; Unit 4 was the first step toward scholars developing these foundational understandings and Unit 5 is the second step in which scholars work toward solidifying their understanding of these concepts to be used in more abstract ways in the future.”

• Unit 8 Overview, Expressions and Equations – Integer Exponents and Scientific Notation, Identify the Narrative, “Looking ahead to the remainder of grade 8 and to high school, it is essential that scholars fluently understand the topics covered in Unit 8 in order to be successful in the future. During the next unit of eighth grade the scholars will be studying volume of cylinders, cones, and spheres which requires scholars to use their knowledge of exponents to find the cubes and cube roots numbers. Then, during the final unit, scholars will study irrational numbers and the Pythagorean Theorem which will require scholars to extend their understanding of integer exponents to include rational (fractional) exponents that result in radicals that cannot be simplified causing them to be categorized as irrational numbers.  Therefore, Units 8, 9, and 10 are closely connected to one another which implies that the baseline information learned in Unit 8 is essential for scholars to be successful for the remainder of the school year. Additionally, when scholars move on to high school, they will continue to develop their understanding of radicals by ‘extending the properties of whole-number exponents’ (CCSS p.58) to learn about square roots, cube roots, etc. More specifically, standards N-RN relates directly back to the material that scholars will have studied during Unit 8 and it will be essential that they have a solid understanding of properties of integer exponents so they are able to understand properties of rational exponents. In addition to understanding exponents for the sake of understanding radicals, scholars must also develop an understanding of exponents in eighth grade for a future understanding of exponential expressions and equations, polynomials, and rational expressions in high school algebra.  Therefore, it is imperative that scholars develop a strong foundation for understanding properties of integer exponents in the eighth grade so they are set up for success for the remainder of the year as well as their futures in high school.”

The materials reviewed for Achievement First Mathematics Grade 8 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.

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

• Guide to Implementing AF Grade 8, Program Overview, “Scope and Sequence Detail is designed to help teachers identify the standards on which each lesson within a unit is focused, whether on grade level or not. You will find the daily lesson aims within each unit and the content standards addressed within that lesson. A list of the focus MPs for each lesson and unit and details about how they connect to the content standards can be found in the Unit Overviews and daily lesson plans.”

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

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

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

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

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

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

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

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

• Unit 7, Lesson 6, Connection To Learning and Conceptual Understanding identifies previous skills for grade level related standards. “How does this lesson connect to previous lessons? In the previous lesson, students solved a system of equations using substitution when at least one of the equations in the system was solved for a variable making it easy to substitute the equivalent expression into the second equation for the same variable. In this lesson, students are given a system of equations where both equations are not solved for a variable and students must decide which equation to rewrite and for what variable in order to apply the substitution method of solving systems. Students also must solve more situations in which distributing after the first substitution is required.”

• Unit 9, Lesson 4, Connection To Learning and Conceptual Understanding identifies previous skills for grade level related standards. “How does this lesson connect to previous lessons? In the previous lesson, students compared the volumes of cylinders and cones with the same height and radius to determine that the formula for the volume of a cone is \frac{1}{3} of a cylinder with equal height and radius. In this lesson, students are given the formula for the volume of a sphere and use it to determine exact and approximate volumes of sphere in and out of context.”

The materials reviewed for Achievement First Mathematics Grade 8 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies.

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

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

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

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

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

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

The materials reviewed for Achievement First Mathematics Grade 8 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

The Program Overview, Guide to Implementing AF Math: Grade 8, Scope and Sequence Detail, Supplies List includes a breakdown of materials needed for each Achievement First Mathematics Program. Examples include:

• Grades 5-8 Math Instructional Materials Purchase List, Math Supplies, “ETA Hand2Mind Classroom Number Line (-20 to 100), 1 for each math classroom.”

• Grades 5-8 Math Instructional Materials Purchase List, Math Supplies, “Transparency sheets (clear page protectors), 1 pack should be sufficient.”

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

The materials reviewed for Achievement First Mathematics Grade 8 meet expectations for having assessment information included in the materials to indicate which standards are assessed. 

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

• Unit 3 Overview, Unit 3 Assessment: Understanding Dilations and Similarity, denotes the aligned grade-level standards and mathematical practices. Question 6, “Given that line = and > are parallel, are the two triangles formed by the intersecting transversals congruent, similar, or neither. Explain how you know.” (8.G.5, MP2, MP3)

• Unit 6 Overview, Unit 6 Assessment: Bivariate Data, denotes the aligned grade-level standards and mathematical practices. Question 1, “Describe the association of one set of ordered pairs below as linear or non-linear and cite evidence using the x- and y-coordinates. a. (2,25), (7,26), (11,25), (16,25), (19,25) b. (7,13), (8,18), (12,32), (14,41), (15,45) c. (3,56), (5,50), (11,31), (13,27), (17,14).” (8.SP.1, MP3, MP4, MP8)

• Unit 10 Overview, Unit 10 Assessment: Irrational Numbers & Pythagorean Theorem, denotes the aligned grade-level standards and mathematical practices. Question 15, “The rectangular prism below has a square base with a side length of 8 cm and a height of 4 cm. What is the length of HB?” (8.G.7, MP2, MP4)

The materials reviewed for Achievement First Mathematics Grade 8 meet expectations for including an assessment system that provides multiple opportunities throughout the grade, course, and/or series to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

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

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

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

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

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

Program Overview, Guide to Implementing AF Math: Grade 8, Differentiation, Unit-Level Errors, Misconceptions, and Response, “Every unit plan includes an ‘Evaluating and Responding to Student Learning Outcomes’ section after the post-unit assessment. The purpose of this section is to provide teachers with the most common 1-2 errors as observed on the questions related to each standard, the anticipated misconceptions associated with those errors, and a variety of possible responses that could be taken to address those misconceptions as outlined with possible critical thinking, strategic practice problems, or additional resources.” Examples include: 

• Unit 2 Overview, Unit 2 Assessment: Understanding Angle Relationships, Evaluating and Responding to Student Learning Outcomes, Suggestions for How to Respond, 8.G.5, “Suggested re-teach activities by question group: Q9-12 - Focus on known angle relationships created when a parallel line is cut by a transversal. For the question, have a student stamp the known angle relationship used to solve the problem. Then pose the question ‘what if I forgot this rule, how could I still solve this problem using what I do know about angle measures?’ Stamp patterns students may use that could help them solve future problems such as the ones below: Parallel lines cut by a transversal forms at most two distinct angles; if they are not both right angles, one will be acute and the other will be obtuse. Corresponding angles are congruent. If you know one angle, you can find all other angles as they are either congruent or supplementary (180 - known angle). Alternate means different sides of the transversal. Interior means on the inside and exterior means on the outside. Lessons for possible re-teach focus: Lesson 7- Focus on IP 4, 5, 6. Have students both solve the problems multiple ways and articulate/write out how they found solutions for the problems. Stamp known relationships about angles formed when parallel lines are cut by transversals. Q13-17 - As with the other set of questions, focus on the known rules for angle measures for triangles introduced in this unit and focus on walking students through the solution pathways and thought processes should they forget the rule. This will work to reinforce the reasoning behind the known rules or help students (with practice) learn the angle relationships for interior and exterior angles for triangles. To probe student thinking on these problems, ask the following: ‘What is the relationship between the angles? How do I know this?’ ‘How do I know this is an exterior angle? How could I use that to more easily solve this problem? What is the relationship to the other interior angles? How do we know that?’ ‘Why might I need to find the other angles in the triangle to solve for this unknown angle?’ Lessons for possible re-teach focus: Lesson 16- Choose select practice problems (such as IP3, 4, 5, or 7) and focus on students setting up and solving the equations, as well as annotating and writing the angle relationships that allowed them to set up those equations.”

• Unit 5 Overview, Unit 5 Assessment: Linear Equations, Evaluating and Responding to Student Learning Outcomes, Suggestions for How to Respond, 8.EE.7a, “Suggested re-teach activities by question group: Q1-2 - Use a show-call of an exemplary response and the most common error. Have students focus on what the exemplary response includes or shows (correct number of solutions identified and rationale). Have them write out the exemplary response and identify the criteria for success for identifying number of solutions: 1) Apply distributive property to remove grouping symbols. 2) Use inverse operations to move variable terms to one side, constants to the other. 3) If the equation is in the form x = a, it has one solution; a = a, infinite solutions, a = b no solutions Lessons for possible re-teach focus: Lesson 4 - Number of Solutions Complex Solutions → Have students focus on IP #4-7 and focus on both finding the number of solutions and ‘explain your reasoning.’ Beyond finding the form of the final solved solution, students should state that for equations with no solutions, there is no real value for x that would balance the equation and for infinite solutions, any real number will balance the equation.”

• Unit 10 Overview, Unit 10 Assessment: Irrational Numbers and Pythagorean Theorem, Evaluating and Responding to Student Learning Outcomes, Suggestions for How to Respond, 8.NS.1 / 8.NS.2, “Suggested re-teach activities by question group: Q1-3:- Show call the most common error and complete a live edit to build and exemplar together;, the debrief should will stamp the CFS below: For error 1: Rational numbers are whole numbers, fractions, terminating decimals, or repeating decimals. Irrational numbers are square roots of non perfect squares, pi (anything that doesn’t give a clean decimal) Rational numbers can be expressed as a ratio of two integers Irrational numbers cannot. The decimals that represent these are approximations. Lessons for possible re-teach focus: Lesson 2: Focus on IP #3, 4, 5, 7, 9. Push for complete responses in their explanations and use the CFS above to check student responses. Suggested re-teach activities by question group: Q4-7:- Show call the most common error and complete a live edit to build and exemplar together;, the debrief should will stamp the CFS below: Square roots give the factor which multiplies by itself to get the number under the symbol. For irrational roots (radicand = not perfect square), find the two perfect squares the radicand is between. The value of the irrational square root is between those two numbers. Lessons for possible re-teach focus: Lesson 3-6: Depending on the problems students missed the most, pull aligned masters and PhD problems from these units and use the points above to debrief.”

The materials reviewed for Achievement First Mathematics Grade 8 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series. There are a variety of question types including multiple choice, short answer, and constructed response. Mathematical practices are embedded within the problems. 

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

• The Unit 2 Assessment contributes to the full intent of 8.G.5 (Use informal arguments to establish facts about the angles created when parallel lines are cut by a transversal). Item 10, “Use the diagram below to answer equations for 9 and 10. Which statement is false? a) Angles 3 and 4 have a sum of 180° because they are supplementary. b) Angles 1 and 8 are congruent because they are alternate exterior angles. c) Angles 6 and 7 are congruent because they are vertical angles. d) Angles 1 and 4 are congruent because they are alternate interior angles.” 

• The Unit 7 Assessment contributes to the full intent of 8.EE.8b (solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations). Item 3, “Solve the following system of equations by graphing. Explain how you know that the coordinate pair identified represents the solution. $$y = -\frac{1}{4x} - 1$$ ; y = \frac{3}{4x} + 3”

• The Unit 8 Assessment contributes to the full intent of 8.EE.3 (use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities). Item 8, “The estimated number of chickens in the world is 1.9 X 10¹⁰. The estimated number of cows is 1.5 × 10⁹. The estimated amount of chickens in the world is about how many times greater than the estimated amount of cows? Show all of your work and  explain what operation was necessary to use to solve this problem.” 

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

• Unit 3 Assessment, Item 7, supports the full development of MP3 as students evaluate others' reasoning and construct arguments. “Aaliyah that rectangle EFGH is similar to rectangle IJKL because she can find a series of transformations that maps EFGH to IJK. She says that if she dilates EFGH by a scale factor of \frac{1}{2} around point E  and then reflects the resulting image across the x-axis followed by the y-axis, the image of EFGH will be exactly the same as IJKL. Jeremiah agrees with Aaliyah that the rectangles are similar because he can find a series of transformations that maps one rectangle to another, but he disagrees with the steps that she followed. Jeremiah says that he must dilate IJKL by a scale factor of 2 around point K and then rotate the resulting image 180 degrees around the origin. a. Who has correctly identified why these two rectangles are similar? Justify your answer.”

• Unit 7 Assessment, Item 11, supports the full development of MP1 as students must make sense of the problem to solve. “Michael and Lindsey are saving money. Michael begins with $20 and saves $5 per week. Lindsey begins with no money, but saves $10 per week. Determine the number of weeks it will take Lindsey and Michael to save the same amount of money. How much money will they each have when they have the same amount?“

• Unit 10 Assessment, Item 6, supports the full development of MP6 as students attend to precision. “Find a rational approximation of the value \sqrt{73} to the nearest tenth.”

The materials reviewed for Achievement First Mathematics Grade 8 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level mathematics.

Materials regularly provide strategies, supports, and resources for students in special populations to help them access grade-level mathematics as suggestions are outlined within each unit overview. According to the Program Overview, Guide to Implementing AF Math: Grade 8, Differentiation, Supporting Students with Disabilities, “Without strong support, students with disabilities can easily struggle with learning and not feel successful. Therefore, it is critical that strong curricular materials provide support for all student learners, but are created in a way to support students who have diagnosed disabilities. Our program has been designed to align to the elements identified by the Council for Learning Disabilities that should be used in successful curriculum and instruction: Specific and clear teacher models, Examples that are sequenced in level of difficulty, Scaffolding, Consistent Feedback, and Frequent opportunity for cumulative review. Unit Overviews and lesson level materials include guidance around working with students with disabilities, including daily suggested interventions in the Workshop Section of the daily lesson plan.” 

Examples of supports for special populations include: 

• Unit 1 Overview, Geometry – Understanding Rigid Transformations and Congruence, Differentiating for Learning Needs, “Previous Grade Content: Students have worked with the coordinate plane in Grade 6’s Unit 3 - The Number System Understanding and Representing Rational Numbers, where students focus on the coordinate system in 4 quadrants, reflections, and calculating distances on the coordinate plane. The following lessons may be useful for differentiation of pre-skill content: Grade 6, Unit 3 - Lesson 13, 14, 15. Further, students worked with triangles and congruency statements in Grade 7’s Unit 10 - Constructing with Angles. The following lessons may be useful for differentiation of pre-skill content: Grade 7, Unit 10 - Lessons 6, 7, 8, and 9.” Error Analysis, “The unit includes Lesson 10.2 - Error Analysis that focuses on common student misconceptions with 90 and 270 degree rotations. This can be included at any point within the unit, but is highly recommended between Lessons 10 and 11 to help students with additional practice with clockwise and counterclockwise 90 degree rotation before moving onto 180 degree rotations around the origin and/or not around the origin. Beyond this error-analysis lesson, it may be worthwhile to plan for an additional error-analysis lesson using exit ticket data from Lesson 15 - Series of All Transformations to allow students time to analyze and review common errors when putting together different types of transformations to prove two figures congruent or not.” Responding to Student Learning Outcomes, “See the Unit Assessment ‘Evaluating and Responding to Student Learning Outcomes’ at the end of the Overview for suggestions on unit-level common errors, misconceptions, and suggestions on how to respond. These can be useful for supporting struggling learners proactively throughout the unit.” Student Grouping Suggestions, “Pre-Test: Use the 8th Grade, Unit 1 Pre-test and Key to identify student strengths and weaknesses when it comes to understanding the coordinate plane with shapes, side lengths, area, and perimeter. Identify specific problems to sequence through cumulative review and create groupings of students for small group instruction during that period. Consider changing student seating so that students who struggled with the pre-test are seated next to students who had higher mastery for support throughout the unit or strategically group students who struggle together for teacher support or small group instruction. Exit Tickets: Closely analyze student mastery of the Exit Tickets for Lessons 4, 8, and 10 as these are critical for mastery of specific transformations (translation, reflection, and 90-degree rotations on the coordinate plane). Students who have struggled with these should be prioritized for small group instruction and/or more support during independent practice prior to the end of the unit where these skills will all need to have reached full mastery to move towards a series of transformations and congruence statements.”

• Unit 4 Overview, Functions – Understanding Functions, Differentiating for Learning Needs, “Visual Anchors: Throughout this unit, students will need to gain mastery of graphing linear and nonlinear functions using tables and equations. Teachers and students may find it useful to use clear visual anchors throughout the unit that show an exemplary problem completed for each transformation along with student-friendly criteria for success.” Differentiated Problems, “To ensure that all students, regardless of previous mastery level, can engage in regular and active participation in grade-level mathematics, teachers should prepare each lesson with a differentiated set of problems for students to complete based on their mastery either from previous, related content or based upon informal assessment of mastery from the Think About It and Test the Conjecture portion of the lesson. For example in Lesson 1 to ensure all students are prepared to show mastery on the Exit Ticket, students that are showing lower mastery could be assigned the following problems: - PP1, 2, IP 1, 2, 4. These problems would ensure students have had practice with representing relations in multiple representations and interpreting coordinates, or input and output values, in the context of the situation.”

• Unit 7 Overview, Expressions and Equations – Systems of Linear Equations, Differentiating for Learning Needs, “Visual Anchors: Throughout this unit, students will need to gain mastery of finding solutions to systems of equations using various methods and checking their solutions using substitution. Teachers and students may find it useful to use clear visual anchors throughout the unit that show an exemplary problem completed for each transformation along with student-friendly criteria for success.” Differentiated Problems, “To ensure that all students, regardless of previous mastery level, can engage in regular and active participation in grade-level mathematics, teachers should prepare each lesson with a differentiated set of problems for students to complete based on their mastery either from previous, related content or based upon informal assessment of mastery from the Think About It and Test the Conjecture portion of the lesson. For example in Lesson 5 to ensure all students are prepared to show mastery on the Exit Ticket, students that are showing lower mastery could be assigned the following problems: - PP1, 2, IP 1, 2, 3. These problems would ensure students have had practice with solving systems of linear equations using substitution.”

The materials reviewed for Achievement First Mathematics Grade 8 meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.

While there are no instances where advanced students do more assignments than classmates, materials do provide multiple opportunities for students to investigate grade-level content at a higher level of complexity. According to the Program Overview, Guide to Implementing AF Math: Grade 8, Differentiation, Supporting Advanced Students, “Part of supporting all learners is ensuring that advanced students also have opportunities to learn and grow by engaging with the grade level content at higher levels of complexity. A problem-based approach is naturally differentiated as students choose the strategies they use to model and solve the problem. Teachers highlight particular strategies for the class, but they always affirm any strategy that works, regardless of its level of complexity. In a classroom implementing the Achievement First Mathematics program, students are expected to work with a variety of tools and strategies even as they work through the same set of problems; this allows advanced students to engage with the content at higher levels of complexity. Daily lessons resources (DLRs) also provide differentiated problems labeled by difficulty. Teachers should differentiate for student needs by assigning the most challenging problems to advanced students while allowing them to skip some of the simpler ones, so that they can engage with the same number of problems, but at the appropriate difficulty level.” Independent Practice in each lesson provides three levels of rigor in the lesson for student work: Bachelor, Master, and PhD work. Examples include:

• Unit 7, Lesson 7, Independent Practice Bachelor Level, “Solve the system using the more efficient method-elimination or substitution, and check your solution.  2x + 3y = 15; 5x − 3y = 6.”

• Unit 7, Lesson 7, Independent Practice Master Level, “Solve the system of equations using elimination. Explain why elimination is a more efficient method to solve this system compared to substitution and graphing. -2x - 9y = -25; -4x − 9y = -23.”

Unit 7, Lesson 7, Independent Practice PhD Level, “Twice one number added to another number is 18. Four times the first number minus the other number is 12. Let x represent the first number and y represent the other number. Write two equations to represent the problem. Then solve the system using elimination”

The materials reviewed for Achievement First Mathematics Grade 8 meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

Guidance is consistently provided for teachers to support students who read, write, and/or speak in a language other than English, providing scaffolds for them to meet or exceed grade-level standards. According to the Program Overview, Guide to Implementing AF Math: Grade 8, Differentiation, Supporting Multilingual and English Language Learners, “The Achievement First Mathematics Program appreciates the importance of creating a classroom environment in which Multilingual and English language learners (MLLs/ ELLs) can thrive socially, emotionally, and academically. MLLs/ ELLs have the double-task of learning mathematics while continuing to build their language mastery. Therefore, additional support and thoughtful curriculum is often needed to ensure their mastery and support in learning. Our materials are designed to help teachers recognize and serve the unique educational needs of MLLs/ ELLs while also celebrating the assets they bring to the learning environment, both culturally and linguistically. Our three structural lesson types: conjecture-based lesson, exercise-based lesson, and error analysis lesson all build on the four design principles for promoting mathematical language use and development in curriculum and instructions outlined by Stanford’s Graduate School of Education, Understanding Language/SCALE.” The series provides the following principles that promote mathematical language use and development: 

• “Design Principle 1: Support sense-making - Daily lesson resources (DLRs) are designed to promote student sense-making with an initial ‘Think About It’ task that engages students with a meaningful task upon which they can build connections. Students have time to read and understand the problem individually and the debrief of these tasks include clear definitions of new terminology and/or key ideas or conjectures…Additionally, teachers are provided with student-friendly vocabulary definitions for all new vocabulary terms in the unit plan that can support MLLs/ELLs further.”

• “Design Principle 2: Optimize output - Lessons are strategically built to focus on student thinking. Students engage in each new task individually, have opportunities to discuss with partners, and then analyze student work samples during a whole class debrief…All students benefit from the focus on the mathematical discourse and revising their own thinking, but this is especially true of MLLs/ELLs who will benefit from hearing other students thinking and reasoning on the concepts and/or different methods of solving.”

• “Design Principle 3: Cultivate conversation - A key element of all lesson types is student discussion. Daily lesson resources (DLRs) rely heavily on the use of individual think/write time, turn-and-talks with partners, and whole class discussion to answer key questions throughout the lesson script. The rationale for this is that all learners, but especially MLLs/ELLs benefit from multiple opportunities to engage with the content. Students that are building their mastery of the language may struggle more with following a whole-class discussion; however, having an opportunity to ask questions and discuss with a strategic partner beforehand can help deepen their understanding and empower them to engage further in the class discussion….”

• “Design Principle 4: Maximize linguistic and cognitive meta-awareness - The curriculum is strategically designed to build on previous lesson mastery. Students are given opportunities to discuss different methods to solve similar problems and/or how these concepts build on each other. The focus of the ‘Think About It’ portion of the Exercise-Based lesson is to help students build on their current understanding of mathematics in order to make a new key point for the day’s lesson. The entire focus of the Test the Conjecture lesson is for students to create their own conjecture about the new learning and then to test this by applying it to an additional problem(s). Students focus on building their own mathematical claims and conjectures and see mathematics as a subject that involves active participation of all learners. By ending each lesson type with this meta-awareness, all learners, but especially MLLs/ELLs benefit by building deeper connections.”

The series also provides Mathematical Language Routines in each unit. According to the Program Overview, Guide to Implementing AF Math: Grade 8, Differentiation, Supporting Multilingual and English Language Learners, “Beyond these design principles, our program outlines for teachers in every unit plan the most appropriate mathematical language routines (MLRs) to support language and content development of MLLs/ELLs with their learning within the specific unit.” Examples include:

• Unit 6 Overview, Statistics & Probability – Bivariate Data, Differentiating for Learning Needs, Supporting MLLs/ELLs, 

  • Vocabulary: “MLLs/ELLs should be provided with a student-friendly vocabulary handout throughout the unit that is either completed for them and/or that they add to each day. All terms included in the ‘Vocabulary’ section below should be included. This scaffold can be incredibly helpful for other learners to help them see a verbal and visual definition for each term. Each of the terms, definitions, and examples should be translated into the students preferred language using Google Translate or a translator (Spanish in the example provided).  Below is a sample of how this can be completed for several terms within the unit.”

  • Sentence Frames: “MLLs/ELLs and all students can greatly benefit from specific guidance around sentence frames for standard justifications or explanation within the unit. For this unit, Lesson 5 focuses heavily on justifications of the meaning of the slope and y-intercept of the line of best fit within the context of the given bivariate data. Teachers can provide students with the following sentence frames to use throughout these problems: Justifying Slope and Y-Intercepts ‘In this context the slope represents ________ and the y-intercept represents _________.’ ‘The independent variable is __________ and the dependent variable is _________ which means the slope represents _________.’ ‘If the ________ increases by 1 _________ , the model predicts that _________ (increases/ decreases) by _________ .’”

  • Language Development Routines: “Throughout the unit, teachers should focus on student discussion and use of critical thinking when analyzing student work samples. See the ‘Implementing Language Routines’ of the Implementation Guide for the course for further detail on how these routines live within all lessons. Within this unit, students should specifically focus on the following Mathematical Language Routines.

    • MLR1: Stronger and Clearer Each Time - Students will focus in ALL lessons on analyzing student work and revising their thinking either during the Think About It or Test the Conjecture portion of each lesson. 

    • MLR2: Collect and Display - Throughout this unit students will focus on learning, capturing, and applying new vocabulary terms for the unit on bivariate data, lines of best fit and correlation. Capture the vocabulary and phrases students use as they describe the patterns they notice when interpreting data sets. Here students will be exposed to key vocabulary of the unit and will need to be able to reference these throughout the unit for appropriate justifications. 

    • MLR7: Compare and Connect - In Lesson 6, students will be determining whether a line of best fit represents a strong enough correlation by informally analyzing the accuracy of the line. Use this routine to provide students an opportunity to compare different approaches to informally creating and analyzing a line of best fit. 

    • MLR8: Discussion Supports - Students will focus in ALL lessons on class discussions to revise their thinking, different representations, and strategies during the Think About It, Interaction with New Material, or Test the Conjecture portion of each lesson.”

The materials reviewed for Achievement First Mathematics Grade 8 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

Manipulatives are described as accurate representations of mathematical objects in the narrative of the Unit Overviews, and although there is little guidance for teachers or students about the use of manipulatives in the lessons, the use of manipulatives can be connected to written methods.

For example, in Unit 1 Overview, “Students are introduced to multiple tools that can be used to analyze transformations: tracing paper, rules, and then software (https://www.geogebra.org/geometry) to explore the effects of flipping, rotating, and  sliding figures. These tools are available throughout the unit for students to use as they explore reflections and  rotations. Students add to their list of tools different rules and observations regarding patterns of behavior on the coordinate plane when reflecting and/or rotating.”