The instructional materials for Achievement First Mathematics Grade 8 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

The instructional materials develop conceptual understanding throughout the grade level. Materials include problems and questions that promote conceptual learning. Examples include:

• Unit 1, Lesson 1, THINK ABOUT IT!, students develop conceptual understanding of rigid transformations by using manipulatives such as tesselation tiles. “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)

• 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. “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)

• Unit 4, Lesson 2, THINK ABOUT IT!, students develop conceptual understanding of functions by analyzing examples. “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:

• Unit 4, Lesson 7, Independent Practice, Question 2 (Bachelor Level), students demonstrate conceptual understanding of function definitions by organizing information in a table. “Does the equation represent a linear function? Prove your answer by showing the constant ROC (rate of change) in a table.” (8.F.A)

• Unit 5, Lesson 8, Independent Practice, Question 2 (Master Level), students demonstrate conceptual understanding of slope by using similar 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.” (8.EE.B)

• 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)

The materials for Achievement First Mathematics Grade 8 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. Although there are not many examples to practice within a lesson, students are provided opportunities to practice fluency both with a partner and individual practice, especially within exercise based lessons and the skill fluency of the cumulative review section. 

The materials develop procedural skill and fluency throughout the grade level. Examples include:

• Unit 5, Lesson 12, Independent Practice, Question 4 (Master Level), students develop procedural skill and fluency by graphing functions. “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)

• Unit 7, Lesson 7, Partner Practice, Question 3 (Master level), students develop procedural skill and fluency by solving simultaneous equations using elimination. “Solve the system of equations using elimination two different ways (addition and subtraction) and verify both methods produce the same solution. {}.” (8.EE.8b)

• Unit 10, Lesson 8, Partner Practice, Question 3 (Bachelor Level), “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 materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade-level. Examples include:

• Unit 1, Lesson 11, Independent Practice, Question 7 (PhD Level), students demonstrate procedural skill and fluency by using coordinates to describe transformations. “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)

• 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. “Solve the equation and check your solution using substitution. $\frac{1}{5}b + 3b = 2b + 42$.” (8.EE.7b)

• Unit 8, Lesson 7, Exit Ticket, Question 3, students demonstrate procedural skill and fluency by expressing scientific notation. “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) f) $\frac{5}{10^{4}}$” (8.EE.3)

The materials reviewed for Achievement First Mathematics Grade 8 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Students are given multiple opportunities to engage in real-world applications especially within exercise based lessons as well as the problem of the day in each cumulative review. 

Materials include multiple routine and non-routine applications of the mathematics throughout the grade level. Examples include:

• Unit 2, Lesson 3, Day 3, Mixed Practice, Question 4, students use informal arguments to establish routine facts about the angles created when parallel lines are cut by a transversal. “Lines 1 and 2 are parallel, and lines 3 and 4 are parallel. (note: diagram shows transversals) Use your knowledge of angle relationships to find the measure of angle E. Explain how you determined your answer.” (8.G.5)

• ​​Unit 3, Lesson 4, Partner Practice, Question 2 (Master Level), students engage in a non-routine problem by using a series of transformations and dilation to prove similarity. “Are the two figures similar? Prove your answer.” (8.G.4)

• Unit 5, Lesson 4, Day 1, Mixed Practice, Question 5, students compare two routine functions represented in different ways. "The table and the equation below represent the proportional relationship between the time (x), in  seconds, and the distance (y), in meters, of two runners. Runner 1: y=6.8x; Runner 2; a time/distance table (which yields y = 7.5x) Which runner is moving faster? Explain your reasoning.” (8.F.2)

• Unit 7, Lesson 14, Partner Practice, Question 2 (Master Level), students solve routine real life problems by using simultaneous equations to find pricing data. "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)

Materials provide opportunities for students to independently demonstrate routine and non-routine applications of the mathematics throughout the grade level. Examples include:

• Unit 6, Lesson 5, Independent Practice, Question 3 (Master Level), students interpret a scatterplot and its line of best fit in a non-routine format. "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)

• Unit 8, Lesson 13, Independent Practice Question 3 (Master Level), students use scientific notation to solve a routine real-world problem. "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)

• Unit 9, Assessment Question 7, students solve a non-routine problem using the volume formulas for cones, spheres, and cylinders. The materials provide a graphic of the three bottles with dimensions. “The manager at Scents for Cents needs to order the new bottles for the perfume the store sells in order from least to greatest volume. a) Help the manager determine the order in which he needs to buy the new bottles. Explain your reasoning so that the manager will feel confident using your work to place his order. b) Change the dimensions of the cone and cylinder shaped bottles so that all three have  the same volume.” (8.G.9)

• Unit 10, Lesson 13, Independent Practice, Question 6 (PhD Level), students use the Pythagorean Theorem to solve routine real-world problems. “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)

The materials reviewed for Achievement First Mathematics Grade 8 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade. Overall, there is an emphasis on the application aspect with the conceptual component of rigor being slightly less represented; however, each aspect of rigor is demonstrated throughout the curriculum. The materials often demonstrate a combination of aspects of rigor within single lessons and even single problems.

All three aspects of rigor are present independently throughout the program materials. Examples include:

Conceptual Understanding:

• Unit 8, Lesson 3, Exit Ticket, Question 2, students demonstrate conceptual understanding of properties of integer exponents when they explain why a rule is true. “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:

• Unit 1, Lesson 7, Independent Practice, Question 6 (PhD Level), students demonstrate procedural knowledge in order to determine which ordered pair represents a reflection. “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:

• Unit 9, Lesson 6, Independent Practice, Question 3 (Bachelor Level), students apply their knowledge about volume to determine how much cheesecake they get. “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:

• 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. “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)

• 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. “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) 

• 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. “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)

The materials reviewed for Achievement First Mathematics Grade 8 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. The Standards for Mathematical Practice are identified and incorporated within mathematics content throughout the grade level. The Mathematical Practices are listed in the Unit Overviews as well at the beginning of each lesson. There are instances where the Unit Overview gives a detailed explanation of the MPs being addressed within the unit, but the lessons do not cite the same MPs.

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

• Unit 5, Lesson 10, Independent Practice, Question 3 (Master Level), students make sense of the problem by interpreting the graph. “Juan, Ronnie, and Justin all walk to school every day. Juan can walk to school, a distance of 0.75 miles, in 8 minutes. The graph below represents Ronnie’s walk to school. The table below the graph shows different distances Justin walks and how long it takes him. Who walks the slowest to school and how much faster is the faster boy?” 

• 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. “Use the most appropriate strategy to solve the problem described below. Show all of your work. A hotel offers two activity packages. One costs $192 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?”

• Unit 10, Lesson 13, Partner Practice (Master Level), Question 4, students determine the most appropriate strategy to solve a systems of equations word problem. “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.”

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

• Unit 6, Lesson 3, Exit Ticket, Question 2, students reason about quantities in order to make predictions based on data points. “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.”

• Unit 6, Lesson 6, Exit Ticket, Question 1, students contextualize the meaning of the slope and y-intercept from a visual scatter plot graph. “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?”

• Unit 8, Lesson 2, Think About It, students de-contextualize powers of exponents by expanding a problem out to determine the procedural process. “Simplify the following exponential expressions by first expanding and then rewriting as a base raised to a single power. a) $(2^3)^2$ ; b) $(h^2)^3$.”

The materials reviewed for Achievement First Mathematics Grade 8 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

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

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

• Unit 2, Lesson 6, Error Analysis Lesson, Independent Practice #6 (Master Level), students investigate angles created when parallel lines are cut by a transversal. “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.” 

• Unit 4, Lesson 8, THINK ABOUT IT!, students compare functions represented in different ways. “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. 

• Unit 5, Lesson 1, Independent Practice, Question 5 (Master level), students determine the number of solutions to an equation. “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?” 

• Unit 8, Lesson 5, Independent Practice, Question 6 (Masters level), students explore properties of integer exponents. “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.” 

• Unit 10, Lesson 13, THINK ABOUT IT!, students use the Pythagorean 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?”

The materials reviewed for Achievement First Mathematics Grade 8 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

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

• Unit 3, Lesson 2, Test the Conjecture #2, students attend to precision as they investigate the effect of a dilation on a figure. “Determine the side lengths and angle measurements if the figure was dilated by a scale factor of $\frac{1}{2}$ about the given point.”

• Unit 7, Lesson 7, Independent Practice, Question 6 (Master Level), students attend to precision as they solve systems of linear equations algebraically. “Write two equations that describe the problem below and solve the system with elimination. Check your solution.” 

• Unit 8, Lesson 1, Exit Ticket, Question 2, students attend to precision as they apply and explain the product of powers properties. “A scholar simplified the expression 4² × 4³ and wrote 4⁶. Did this scholar simplify the expression correctly? If yes, prove it. If no, explain the error in this scholar’s reasoning and provide the correct simplified expressions.”

• Unit 9, Lesson 2, Exit Ticket, Question 1, students attend to precision when working with volume and rounding to a specified place value. “Find the exact and approximate volume of the cylinder shown below by rounding the volume to the nearest hundredth.” 

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

• The teacher is routinely prompted to use precise vocabulary such as in Unit 1, Lesson 1, Debrief. “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 is included within a possible student response, which “Rigid transformations changed where the figure is sitting (T: We call this location) and which way it is facing (T: We call this orientation).”

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

• Unit 5, Lesson 6, Opening, Debrief, FENCEPOST #1, students determine the rate of change. “The slope of a linear function is its rate of change: $\frac{\triangle y}{\triangle x}$.” Teachers are prompted to show student work that starts with the expression $\frac{\triangle y}{\triangle x}$ and ask, “Do you agree with the formula this scholar used? Vote.” Students might say, “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.” Teachers follow up with, “What does ‘change in y’ mean/imply? How would you define the relation graphed?” Students might say, “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.”

The materials reviewed for Achievement First Mathematics Grade 8 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards.

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

• Unit 2, Lesson 3, Think About It, students use the structure of proportions to solve linear equations. “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? $\frac{4}{5} = \frac{52}{n}$; $\frac{4}{8} = \frac{2}{n+1}$”

• Unit 5, Lesson 4, Independent Practice, Question 8 (PhD Level), students analyze the structure of equations in order to determine the number of solutions. “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.”

• 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. “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. $3x - 2y = 5; 3x - 2y = -1$.”

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

• Unit 2, Lesson 3, Exit Ticket, Question 2, students use repeated reasoning to describe an error in work about solving proportions. “Explain the mistake made in the work below. Your explanation should include a description about solving proportions.”

• Unit 5, Lesson 1, Test the Conjecture #2, students use repeated reasoning to understand if equations have one solution, no solution, or many solutions. “Determine the number of solutions for the following equation $-(-4x - 6) + (-2x) = -4x - 5 + (-5)$.”

• Unit 8, Lesson 4, THINK ABOUT IT!, students use repeated reasoning to understand the value of a number raised to the zeroth power. “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.”

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. ; $y = \frac{3}{4x} + 3$”

• 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.”