8th Grade - Gateway 2

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

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

• In Unit 1, Lesson 1, THINK ABOUT IT!, students develop conceptual understanding of rigid transformations by using manipulatives such as tesselation tiles. The materials state, “Thomas was playing with three tessellation tiles on his desk. He went to the bathroom and when he returned, he found that someone had moved all the tiles and put them in a different place (a before and after diagram is provided). Part A: Look at each tile before and after and describe how someone moved the tile using as much detail as possible.Triangle; Rectangle; Trapezoid. Part B: How could you prove that the triangle tile could be the exact same tile and someone didn’t switch it out for a larger or smaller tile?” (8.G.A)
• In Unit 2, Lesson 4, Independent Practice, Question 8 (PhD Level), students develop conceptual understanding of angle relationships within parallel lines by using manipulatives and properties of transformations. The materials state, “How could you use a transparency to prove that the angles created when a transversal passes over one line are identical to the angles created when the transversal crosses the other line if it is parallel to the first? How does this relate to rigid transformations?” (8.G.A)
• In Unit 4, Lesson 2, THINK ABOUT IT!, students develop conceptual understanding of functions by analyzing examples. The materials state, “The following input/output tables have been split into two categories; relations and relations that are also functions. Look for similarities and differences and write a definition for what a function is.” (8.F.A) 

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Examples include:

• In Unit 4, Lesson 7, Independent Practice, Question 2 (Bachelor Level), students demonstrate conceptual understanding of function definitions by organizing information in a table. The materials state, “Does the equation represent a linear function? Prove your answer by showing the constant ROC (rate of change) in a table.” (8.F.A)
• In Unit 5, Lesson 8, Independent Practice, Question 2 (Master Level), students demonstrate conceptual understanding of slope by using similar triangles. The materials state, “A smaller triangle is inscribed inside a larger triangle. Use the triangles to prove that the slope between any two points on a line is equivalent to the slope between any other two lines. Your explanation should prove that the triangles are similar first.” (8.EE.B)
• In Unit 10, Lesson 2, Independent Practice, Question 2 (Bachelor Level), students demonstrate conceptual understanding of rational numbers by justifying their classification, “Is 0.6666… rational or irrational? Justify in two ways.” (8.NS.1)

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

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

• In Unit 5, Lesson 12, Independent Practice, Question 4 (Master Level), students develop procedural skill and fluency by graphing functions. The materials state, “Graph the function $$y=3x-2$$ and explain the steps you used to create the graph based on the structure of the equation.” (8.F.5)
• In Unit 7, Lesson 7, Partner Practice, Question 3 (Master level), students develop procedural skill and fluency by solving simultaneous equations using elimination. The materials state, “Solve the system of equations using elimination two different ways (addition and subtraction) and verify both methods produce the same solution. {$$4y+3x=22;-4y+3x=14$$}.” (8.EE.8b)
• In Unit 10, Lesson 8, Partner Practice, Question 3 (Bachelor Level) states, “Which set of measurements are the side lengths of a right triangle? a) 7, 8, 12; b) 9, 12, 15; c) 10, 24, 26; d) 2.4, 3.4, 5.5.” (8.G.6)

The instructional materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade-level. Examples include:

• In Unit 1, Lesson 11, Independent Practice, Question 7 (PhD Level), students demonstrate procedural skill and fluency by using coordinates to describe transformations. The materials state, “Triangle ABC has vertices at A (3, 4), B (3, 9), and C (6, 4). What are the vertices of the image A’B’C’ if the triangle was rotated 180 degrees around the origin and translated four units up. Explain how you know.” (8.G.3)
• In Unit 2, Lesson 2, Independent Practice, Question 2 (Bachelor level), students demonstrate procedural skill and fluency by solving multi-step linear equations and using substitution to check their answer. The materials state, “Solve the equation and check your solution using substitution. $$\frac{1}{5}b+3b=2b+42$$.” (8.EE.7b)
• In Unit 8, Lesson 7, Exit Ticket, Question 3, students demonstrate procedural skill and fluency by expressing scientific notation. The materials state, “The length of a very fine grain of sand is about 0.0005 inches. Which of the following also show the length of the grain? Select all that apply. a) $$5×10^3$$ ; b) $$5×10^4$$ ; c) $$5×10^{-3}$$ ; d) $$5×10^{-4}$$; e)$$\frac{5}{10^{-4}}$$ f) $$\frac{5}{10^{4}}$$”  (8.EE.3)

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

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

Conceptual Understanding:

• In Unit 8, Lesson 3, Exit Ticket, Question 2, students demonstrate conceptual understanding of properties of integer exponents when they explain why a rule is true. The materials state, “Explain why the rule $$a^5×b^5=(ab)^5$$ is true using the commutative and associative properties.” (8.EE.1) 

Fluency and Procedural Skill:

• In Unit 1, Lesson 7, Independent Practice, Question 6 (PhD Level), students demonstrate procedural knowledge in order to determine which ordered pair represents a reflection. The materials state, “Which of the following describes the location of a point (x,y) reflected over the y-axis and reflected over the x-axis? a) $$(x,y)$$; b) $$(-x,y)$$; c) $$(x,-y)$$; d) $$(-x,-y)$$.” (8.G.3)

Application:

• In Unit 9, Lesson 6, Independent Practice, Question 3 (Bachelor Level), students apply their knowledge about volume to determine how much cheesecake they get. The materials state, “A round cheesecake has a diameter of 8 inches and a height of 3 inches. It is cut into 8 equal-sized slices. How many cubic inches does each slice take up in the cheesecake? Use 3.14 for pi and round your answer to the nearest tenth of a cubic inch.” (8.G.9)

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. Examples include:

• In Unit 3, Lesson 6, Independent Practice, Question 6 (Master Level), students engage in application and conceptual understanding about properties of transformations to prove that triangles are similar. The materials state, “Are the two triangles similar?  Prove by graphing and a series of rigid transformations. Triangle A: (1, 2), (4,8), and (10, 5); Triangle B: (-4, -2), (-4, -3), and (-3, -5).” (8.G.4)
• In Unit 7, Lesson 12, Independent Practice Question 3 (Master level), students demonstrate fluency by solving simultaneous equations in more than one way as they apply the mathematics to understand given data. The materials state, “In the fall, the math club and science club each created an Internet site. You are the webmaster for both sites. It is now January and you are comparing the number of times each site is visited each day. Science club: There are currently 400 daily visits and the visits are increasing at a rate of 25 daily visits per month. Math club: There are currently 200 daily visits and the visits are increasing at a rate of 50 daily visits per month. a) Write a system of linear equations to represent the situation. Then graph to determine the solution. b) Explain what the solution to the system means in the context of the problem.” (8.EE.8) 
• In Unit 8, Lesson 10, Independent Practice, Question 5 (Master Level), students apply their procedural fluency of operations with numbers expressed in scientific notation to real world scenarios. The materials state, “Bubba’s Boot Barn is a favorite stop of visitors to Nashville’s downtown shopping area. Last year, $$2.42×10^5$$ people visited Bubba’s.  This year it has become and even more popular venue, with $$2.53×10^6$$ visitors. Step A: How many total visitors did Bubba’s get over the two years? Step B: How many more visitors did Bubba’s get this year compared to last year?” (8.EE.4)

The instructional materials reviewed for Achievement First Mathematics Grade 8 meet expectations that the instructional materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

Student materials consistently prompt students to both construct viable arguments and analyze the arguments of others. Examples include:

• The Guide to Implementing AF Math describes Error Analysis lessons as one way to address MP3. The materials state, “Purpose: Through the use of error analysis, guided questioning and discussion students will identify and fix a common misconception related to a skill they learned the previous day. These are sequenced so that after a particularly complex conceptual lesson or a lesson involving a skill that surfaces a common misconception, students get another focused at bat to either fix their misunderstanding or deepen their reasoning around key mathematical concepts and viable strategies to guide them away from making the same error. These lessons start with analyzing fictional student work and are structurally based off of the Standards for Mathematical Practice 3.”
• Unit 2, Lesson 6, Error Analysis Lesson, Independent Practice #6 (Master Level), students investigate angles created when parallel lines are cut by a transversal. The materials state, “In the diagram below, $$\angle3=105\degree$$ and $$\angle8=5x$$.  Scholar A says that the value of $$\angle7=75\degree$$.  Describe the mistake that the scholar made and provide at least two different ways to prove the scholar wrong.” (8.G.5)
• In Unit 4, Lesson 8, THINK ABOUT IT!, students compare functions represented in different ways. The materials state, “Below are two different linear functions. Determine which function is changing the fastest using any methods you have learned. Justify why the function you choose is changing faster than the other.” Students are given a table and a graph to compare. (8.F.2)
• In Unit 5, Lesson 1, Independent Practice, Question 5 (Master level), students determine the number of solutions to an equation. The materials state, “Mark and Molly are debating over the solution to the equation $$11(x+10)=110$$. Mark says that there is no solution because the 110’s cancel out of the equation. Molly says that the solution x = 0 is a valid solution to the equation. Who do you agree with and why?” (8.EE.7a)
• Unit 7, Lesson 9, Exit Ticket Question 2, students analyze pairs of simultaneous linear equations. The materials state, “Abigail decided to solve the following system of two linear equations by graphing. Do you agree or disagree with Abigail’s decision? Why? $$\frac{6x+54=4}{x-5y=16}$$.” (8.EE.8)
• Unit 8, Lesson 5, Independent Practice, Question 6 (Masters level), students explore properties of integer exponents. The materials state, “Prove that any number raised to a negative exponent is equal to the reciprocal of the base raised to the opposite exponent. Use examples and explain.” (8.EE.1)

The instructional materials reviewed for Achievement First Mathematics Grade 8 meet expectations that the instructional materials assist teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

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

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

The instructional materials reviewed for Achievement First Mathematics Grade 8 meet expectations that materials explicitly attend to the specialized language of mathematics.

The materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams, and symbols. The materials also use precise and accurate terminology and definitions when describing mathematics, and support students in using them. Examples of explicit instruction on the use of mathematical language include:

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

Examples of the materials using precise and accurate terminology and definitions: 

• At the beginning of each lesson plan, there is a section labeled “Key Vocabulary” for the teacher. For example in Unit 4, Lesson 4 “Key Vocabulary:
  • Independent Variable – a variable (often represented by x) whose variation does not depend on another variable.
  • Dependent Variable – a variable (often represented by y) whose variation depends on another variable.
  • Substitution – replacing a variable with a value or expression.
  • Relation – any set of ordered pairs.
  • Input – the independent variable, defines the function.
  • Output – the dependent variable, changes based on change in the input.
  • Function – a mathematical relationship where each input has a unique output.
  • Rate of Change – a change in the dependent variable per a change in the independent variable; when comparing rates of change, you compare the magnitude of the rate of change, not the actual value.
• The teacher is routinely prompted to use precise vocabulary such as Unit 1, Lesson 1, Debrief. The materials state, “How would you describe how the individual tiles were moved? Mathematicians have specific names for these movements. We call a slide a translation, a flip is a reflection, and a turn is a rotation. All three of these are called rigid transformations. How did these rigid transformations change the figure?” Guidance within a possible student response, “Rigid transformations changed where the figure is sitting (T: We call this location) and which way it is facing (T: We call this orientation).”
• There is very little vocabulary emphasis in student-facing materials. For example, there is not a glossary for student reference.