8th Grade - Gateway 2

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. {$$4y + 3x = 22;-4y + 3x = 14$$}.” (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)$$\frac{5}{10^{-4}}$$ 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.”