8th Grade - Gateway 2

The materials reviewed for i-Ready Classroom 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 lessons include problems and questions that develop conceptual understanding throughout the grade-level. Examples include:

• Unit 3, Lesson 9, Session 2, “Apply It”, Problem 6, students develop conceptual understanding by using similar triangles to explain why slope m is the same between any two distinct points on a non-vertical line in the coordinate plane. “Liam’s class is planting bamboo seedlings in the school garden. The line represents the average height of a bamboo plant after it has been planted. Write an equation in slope-intercept form that Liam could use to predict the height y of his bamboo after x days. Explain what the slope and the y-intercept mean in this situation.” (8.EE.6)

• Unit 4, Lesson 15, Session 2, “Model It”: Equations, Problem 3, students develop conceptual understanding by comparing linear and nonlinear functions (8.F.3). “Many functions can be represented by equations. An equation shows how to calculate the output, y, for the input, x. Determine whether each equation represents a linear function. Show your work. y=2x-1; y=-x^(2); y=-x b. Explain how you know that equations in the form of y=mx+b always represent linear functions.” The text includes a blank graph for students to graph the functions.

• Unit 6, Lesson 26, Session 2, Model It: Algebraic Proof, Problem 5, students develop conceptual understanding by explaining a proof of the Pythagorean Theorem. “What is the total area of all four triangles in Figure 1? What is the area of the unshaded shape? Use your answers to write an expression for the area of the shape with the side length a+b.” (8.G.6) 

The materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade through the use of visual models, real world connections, mathematical discussion prompts, concept extensions, and hands-on activities. Examples include: 

• Unit 2, Lesson 4, Session 3, “Apply It”, Problem 5, students demonstrate conceptual understanding about the relationship between congruence and similarity. (8.G.4) “Keisha says: All congruent figures are similar, but not all similar figures are congruent. Use the definition of congruent figures and similar figures to explain whether this statement is correct.”

• Unit 3, Lesson 8, Session 1, Prepare, Problem 3, students demonstrate conceptual understanding as they graph proportional relationships using the unit rate as the slope of the graph. (8.EE.5) “A marine biologist is studying how fast a dolphin swims. The dolphin swims at a constant speed for 5 seconds. The distance it swims is 55 meters. The relationship between time and distance for the trip is proportional. a. Make a graph showing the change in the dolphin’s distance over time. How far does the dolphin swim in 1 second? Show your work. b. Check your answer to problem 3a. Show your work.”

• Unit 7, Lesson 31, Session 3, Apply It, Problem 5, Math Journal, students demonstrate conceptual understanding as they construct and interpret data in two-way tables (8.SP.4). “A piano teacher asks their students how many minutes they practice each day. The piano teacher then notes the number of mistakes each student makes during a recital. What association can you see from the data in the table? What might the piano teacher recommend to their students?” The data table is set up with minutes of practice in the range of 0-30, 31-60 and total minutes across the top of the table and number of mistakes in the range of 0-25, 26-60 and total mistakes down the left side of the table.

The materials reviewed for i-Ready Classroom Mathematics, Grade 8 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. Within each lesson, there is a Session that provides additional practice for students to have in class or as homework. Additionally, many lessons include a Fluency & Skills Practice section. Examples include: 

• Unit 3, Lesson 10, Session 2, Practice, Problem 2, students develop procedural skill and fluency by solving linear equations with one variable (8.EE.7). “Solve the equation for t. Show your work. 5(2t-3)-7.5t=0.5(12-t).” 

• Unit 4, Lesson 15, Session 2, Practice, Problem 2, students develop procedural skill and fluency with functions by using a rule to determine outputs when given the input in a table, graphing the resulting ordered pairs, and stating if the rule is linear (8.F.1, 8.F.3). “Complete the table and graph for the function. Tell whether the function is linear or nonlinear. Explain your reasoning. Input x, a number; Output y, 6 divided by x.” Students are given a table with x values from 1-4 and a graph with axes labeled from 0-8. 

• Unit 6, Lesson 24, Session 2, Fluency Skills & Practice, students develop procedural skill and fluency with multiple problems to convert a decimal expansion which eventually repeats into a rational number (8.NS.1), such as Problem 12, “Write each repeating decimal as a fraction, 0.\bar{162}.”

Materials provide opportunities for students to independently demonstrate procedural skills and fluency throughout the grade level. Within each lesson, students engage with practice problems independently at different sections of the lesson. Examples include: 

• Unit 3, Lesson 13, Session 2, Apply It, Problem 8, students demonstrate procedural skill and fluency by solving systems of equations algebraically (8.EE.8). “Solve the system of equations. Show your work. 3x=6y-21; 6x-9y=-30”. 

• Unit 4, Lesson 16, Session 2, Apply It, Problem 7, students demonstrate procedural skill and fluency by constructing functions to model linear relationships (8.F.4). “Mr. Seda plans a field trip for one of his classes. He rents one bus for the whole class and purchases a museum ticket for each student. The equation y=11x+400 gives the cost of the field trip, y, as a function of the number of students in the class, x. What is the initial value of the function? What is the rate of change? What do these values tell you about the field trip?” 

• Unit 5, Lesson 19, Session 4, Apply It, Problem 3, students demonstrate procedural skill and fluency by using integer exponents to generate equivalent expressions (8.EE.1). “Which expression is equivalent to \frac{(15^4\cdot7^6)^3}{15^2} ? a. 15^2\cdot7^9 b. 15^2\cdot7^{18} c. 15^5\cdot7^9 d. 15^{10}\cdot7^{18}. Caitlin chose C as the correct answer. How might she have gotten that answer?”

The materials reviewed for i-Ready Classroom Mathematics, Grade 8 meet expectations for being designed so teachers and students spend sufficient time working with engaging applications of the mathematics. 

Engaging routine and non-routine applications include single and multi-step problems. Examples include:

• Unit 2, Lesson 7, Session 3, Develop, Try It, students engage with a routine application problem by solving problems involving similar triangles (8.G.4). “Jorge wants to draw two triangles that have the same angle measures and are not similar. Carlos says that is not possible to do. Make or draw two triangles that have the same three angle measures but different side lengths. Are the triangles similar?”

• Unit 6, Lesson 23, Session 3, Develop, Try It, students engage with a non-routine application problem by applying cube roots to solve problems (8.EE.2). “Carolina works at a museum. She is looking online for a storage case for some large fossils. She wants the case to have a volume of 27$$ft^3$$. Carolina would like the case to be a cube. What edge length should the case have?”

• Unit 7, Lesson 32, Session 3, Develop, Try It, students engage with a routine application problem by constructing and interpreting a two-way table summarizing data on two categorical variables collected from the same subjects (8.SP.4). “Twenty middle school students are asked whether they own cell phones. The table shows their responses and grade levels. Which grade has the highest percentage of students who own cell phones?” The data for 20 middle school students, by grade level, is included.

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

• Unit 3, Lesson 14, Session 4, Apply It, Problem 9, students demonstrate a routine application problem by analyzing and solving pairs of simultaneous linear equations (8.EE.8). “Cameron buys 4 notebooks and 2 packages of pens for $16. Olivia buys 5 notebooks and 1 package of pens for $14.75. Write and solve a system of equations to find the price of each notebook and the price of each package of pens. Tell what each variable represents and what each equation represents.”

• Unit 4, Math in Action, Session 2, Persevere on Your Own, students demonstrate a non-routine application problem by constructing a function to model a linear relationship between two quantities, and determine and interpret the rate of change in terms of the situation it models, and in terms of its graph or a table of values (8.F.4). “Kazuko is the visual effects (VFX) supervisor for the new action movie. She considers bids from two visual effects companies: Moonshot and Lemon Cloud. Read this email from Kazuko to her assistant, Xavier. Then help Xavier respond to Kazuko’s request.” The email includes the following information: “The companies bidding on the VFX for the museum chase sequence have sent us their pricing information. Unfortunately, the companies have used different formats to show their pricing. This makes it a little difficult to compare them.” The data for Moonshot VFX price per VFX time is included in a table. The data for Lemon Cloud VFX is included in a graph. “We estimate that we need between 120 seconds and 130 seconds of medium difficulty VFX for the chase sequence. What I need from you: convert the pricing information from each company to the same format (table, graph, or equation), recommend a company for the VFX, and explain your choice, and in case we go over our original time estimate, calculate the cost per second of VFX for each company.”

• Unit 6, Lesson 28, Session 3, Practice, Problem 4, students demonstrate a routine application problem by using the volume of cylinders to solve problems (8.G.9). “A cylindrical storage bin is filled with sunflower seeds. The seeds from a cone-shaped pile above the bin. How many cubic feet of sunflower seeds are in the bin? Show your work. Use 3.14 for \ pi.”

The materials reviewed for i-Ready Classroom 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. The Understand lessons focus on developing conceptual understanding. The Strategy lessons focus on helping students practice and apply a variety of solution strategies to make richer connections and deepen understanding. The units conclude with a Math in Action lesson, providing students with routine and non-routine application opportunities. 

All three aspects of rigor are present independently throughout each grade level. Examples include:

• Unit 2, End of Unit, Unit Review, Performance Task, students apply congruence and similarity using models to create angles when parallel lines are cut by a transversal (8.G.5). “City Council plans to add walkways and flower corners to a small park. The upper and lower boundaries of the park are fences that form parallel sides. City Council asks you to plan two diagonal walkways that intersect with each other and form transversals with the fences. They also ask you to identify at least three pairs of congruent angles formed by the walkways that can be used as flower corners. Be sure to include the following in your plan and description: A drawing to show where the diagonal, intersecting walkways will be located; Labels at vertices, so you can name the angles formed; At least three pairs of angles that you prove are congruent.”

• Unit 4, Lesson 15, Session 2, Connect It, Problem 4, students develop conceptual understanding of functions (8.F.1). Students must explain how graphs or equations can help to determine if a function is linear. “You want to determine whether a function is linear. How can making a graph or writing an equation help?” 

• Unit 7, Lesson 29, Session 5, Apply It, Problem 6, students practice procedural skills and fluency as they investigate patterns of association in bivariate data recognizing that straight lines are used to model relationships between variables (8.SP.2). “Tell whether each statement is true or false. a) A good line of fit must go through at least one data point. b) A line of fit is good if all of the data points are fairly close to it. c) A good line of fit almost always goes through the origin (0, 0). d) There is always a good line of fit for any set of data points.” 

Multiple aspects of rigor are engaged simultaneously to develop students' mathematical understanding of a single unit of study throughout the grade level. Examples include: 

• Unit 3, Lesson 10, Session 2, Connect It, Problem 3, students attend to procedural skill and fluency while developing conceptual understanding as they solve linear equations in one variable (8.EE.7). “Look at both of the “Model Its”. Finish the solution for each method. Does it matter which solution method you use? Explain.” The Model Its show how to solve the equation \frac{1}{4}+20=\frac{1}{2}(x+20). The first Model It shows how to multiply both sides of the equation by 4 first and then use the distributive property. The second Model It shows using the distributive property first. 

• Unit 4, Lesson 18, Session 2, Practice, Problem 7, students utilize conceptual understanding and apply their knowledge of functions as they qualitatively describe the functional relationship between two quantities by analyzing a graph (8.F.5). “The graph shows gasoline prices over time. Tell a story about how gas prices change over time.” 

• Unit 6, Lesson 27, Session 1, Prepare, Problem 3, students practice fluency with irrational numbers as they apply the Pythagorean Theorem to solve right triangles. (8.G.7) “The flag of Scotland consists of a blue rectangular background with two white diagonals. a. The dimensions of a Scottish flag are shown. If the flag’s diagonals are made of white ribbon, what length of ribbon is needed for both diagonals? Show your work. Round your answer to the nearest foot.” The dimensions of the flag are 5 ft by 3 ft.

The materials reviewed for i-Ready Classroom 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 Mathematical Practices are embedded within the instructional design, found in the Teacher Toolbox under Program Implementation. The Teacher’s Guide includes “Standard of Mathematical Practice in Every Lesson” which states, “Through a dedicated focus on mathematical discourse, the program blends content and practice standards seamlessly into instruction, ensuring that students continually engage in developing the habits of the mathematical practices.” The Table of Contents and the Lesson Overview both include the Standards for Mathematical Practice for each lesson. In the Student Worktext, the Learning Target also highlights the MPs that are included in the lesson. 

There is intentional development of MP1: Make sense of problems and persevere in solving them, in the Try It problems, where students are able to generate their own strategies to solve the problem. Teachers are provided with guidance to support students in making sense of the problem using language routines such as Co-Craft Questions and Three Reads. Examples include: 

• Unit 4, Lesson 16, Session 1, Try It, students determine if their answers are reasonable by making sense of the problem as they construct a function to model a linear relationship between two quantities. “A customer can use the menu above to call in a pizza order. They choose a size and then add toppings. The graphs and equations model the prices of the two sizes of pizza. y=1.5x+8 and y=2x+12. Which equation and which line model the price of a small pizza? Which equation and which line model the price of a large pizza?” 

• Unit 5, Lesson 19, Session 2, Try It, students make sense of problems by applying the properties of integer exponents to generate equivalent numerical expressions. “The population of a colony of single-celled bacteria doubles each day. On Day x, the population of the colony is 2^x. How many times as large is the population of the colony on Day 7 than on Day 4?” 

• Unit 6, Lesson 27, Session 2, Try It, students make sense of the Pythagorean Theorem to determine unknown side lengths in right triangles. “A firefighter tries to rescue a kitten from a tree. The firefighter leans a 13-foot ladder so its top touches the tree. The base of the ladder is 5 feet from the base of the tree. The tree forms a right angle with the ground. How high up the tree does the ladder reach?”

There is intentional development of MP2: Reason abstractly and quantitatively, in the Try-Discuss- Connect routines and in the Understand lessons. Students reason abstractly and quantitatively, justify how they know their answer is reasonable, and consider what changes would occur if the context or the given values in expressions and equations are altered. Additionally, teachers are provided with discussion prompts to analyze a model strategy or representation. Examples include: 

• Unit 2, Lesson 4, Session 2, Discuss It (sidebar question in Model It), students analyze the relationships of quotients of corresponding sides of two figures in the coordinate plane. “Why does the relationship between the scale factor and the quotients of corresponding side lengths make sense?” 

• Unit 4, Lesson 18, Session 1, Try It, students demonstrate understanding of the relationships between problem scenarios and mathematical representations. “Efia warms up and goes for a run. The graph shows her distance from home as a function of time. Which sections of the graph represent Efia stretching, walking, or running?“ 

• Unit 6, Lesson 23, Session 1, Additional Practice, Problems 2 and 3, students make connections between mathematical representations. Problem 2, “Which of the expressions at the right are examples of a product? Explain.” Choices: 2(10 + y), x⁴, -7a, (3 + 4)(3 - 4). Problem 3, “Garrett and his sister Safara cut square patches from their old sports jerseys. Each patch has an area of 1 ft.². They want to put some of the patches together to make a square quilt. a. Which of the areas listed below are possible areas of a square quilt Garret and Safara can make? What side length will each of those square quilts have? Show your work.”

The materials reviewed for i-Ready Classroom 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. In the Discuss It routine, students are prompted with a question and a sentence frame to discuss their reasoning with a partner. Teachers are provided with guidance to support partners and facilitate whole-class discussion. Examples include:

• Unit 1, Math in Action, Session 2, Persevere on Your Own, Reflect, Critique Reasoning, students critique a partner’s solution to using transformations of parts of a diagram to create a closed figure. “Do the transformations your partner described result in the formation of a closed shape? Explain.”

• Unit 2, Lesson 6, Session 1, Teacher Edition, Connect It, Facilitate Whole Class Discussion, provides guidance for teachers to help students critique the reasoning of others. “Call on students to share selected strategies. Remind students that one way to agree and build on ideas is to give reasons that explain why the strategy makes sense. Invite students to reword informal language with mathematical vocabulary.”

• Unit 3, Lesson 13, Session 2, Connect It, Problem 4, students construct an argument about solving systems of equations. “Will substitution always work when solving a system of equations? Explain.”

• Unit 5, Lesson 19, Session 1, Try It, students apply the properties of integer exponents to generate equivalent numerical expressions. “How can you write (10^3)^2 as a single power of 10? A single power has one base and one exponent.” In the Discuss It sidebar question, students are asked, “How did you find the exponent in your answer?” The teacher edition suggests having students respond with a partner.

• Unit 7, End of Unit, Unit Review, Performance Task, Reflect, Argue and Critique, students use evidence to support their argument about which fundraiser will have the greatest participation based on data collected. “What mathematical evidence supports your recommendation?” In the Performance Task, data is provided about how middle school students prefer to raise money for a new school garden. Students organize the data into a two-way table and answer, “Which fundraiser do you recommend? Explain your reasoning.”

• Teacher Toolbox Program Implementation Support, Teacher’s Guide, Standards for Mathematical Practice in Every Lesson, SMPs are integrated in the Try-Discuss-Connect routine. “Discuss It begins as student pairs explain and justify their strategies and solutions to each other. Partners listen and respectfully critique each other’s reasoning (SMP 3). To promote and support partner conversations, the teacher may share sentence starters and questions for discussions. During this time, the teacher is listening in to peer conversations and reviewing student strategies, identifying three or four strategies to discuss with the whole class in the next part of Discuss It.”

The materials reviewed for i-Ready Classroom Mathematics, Grade 8 meet expectations for supporting the intentional development of MP4: “Model with mathematics”; and MP5: “Use appropriate tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.” The materials identify MP4 and MP5 in most lessons and can be found in the routines developed throughout the materials. 

There is intentional development of MP4: “Model with Mathematics,” to meet its full intent in connection to grade-level content. Many problems present students with the opportunity to use models to solve problems throughout the materials. Examples include:

• Unit 1, End of Unit, Unit Review, Performance Task, students describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates to model a situation. “A computer software company asks you to design a program that can be used to create a video game character. The character moves, but its size and shape do not change. The company gives you a list of program requirements.

  • Create a character by graphing a geometric figure composed of straight-line segments in the coordinate plane. List the coordinates of the character.

  • Use each of the rigid transformations to describe a movement that the character makes when players press a certain key on the keypad. Identify the key and the transformation the character undergoes when the key is pressed.

  • List a sequence of keys pressed and transformations that will take place. Graph the sequence of transformations in the coordinate plane. List the coordinates of the final location of the character.

  Describe the transformations using words and the original and final coordinates of the character.” Students are prompted to check to see whether an answer makes sense in the Reflect: Model “How did you make sure you accurately showed each transformation in the sequence?”

• Unit 3, Lesson 12, Session 3, Apply It, Problem 4, Part C, students model a system of equations with an appropriate representation. “Kenji and his stepbrother Ramon run on a  cross country team. Kenji runs at a rate of 150 meters per minute. Kenji has already run 750 meters before Ramon starts running. Ramon runs at a rate of 300 meters per minute.” In Parts A and B, students are given the equations that represent the system and are asked to graph them, and find and interpret the solution in the context of the problem. Part C, “Describe a situation in which Kenji and Ramon are running cross country but are never the same distance from the starting point at the same time. Write a system of equations or draw a graph to model the situation. How many solutions does the system have?”

• Unit 4, Lesson 16, Session 4, Try It, students construct a function to model a linear relationship between two quantities. “Kadeem spends the afternoon reading a book he started yesterday. He reads 120 pages in 3 hours. One hour after Kadeem begins reading, he is on page 80. Write an equation for the page he is on, y, as a function of minutes spent reading, x. What page number was Kadeem on when he started reading today?” Students may use a graph to model this equation and solve the problem. 

• Unit 6, Lesson 27, Session 5, Apply It, Problem 8, students use the Pythagorean Theorem to solve a real-world problem. “Jose is mailing some baseball equipment to his cousin. He wants to include a baseball bat that is 32 in. long. Will the baseball bat fit completely in the box? Explain your reasoning.” A rectangular prism with dimensions 24 in. x 18. in. x 16 in. is included in the text.

There is intentional development of MP5: “Use appropriate tools strategically to meet its full intent in connection to grade-level content.” Many problems include the Math Toolkit with suggested tools for students to use. Examples include:

• Unit 2, Lesson 7, Session 1, Try It, students use informal arguments to establish facts about the angle sum and exterior angle of triangles. “An architect needs to know the angle measures of the roof shown in the photo. The triangle to the right models the shape of the roof. What is the sum of the angle measures of the triangle?” Students could use various tools to solve this question such as grid paper, straightedges.

• Unit 4, Lesson 15, Session 2, Connect It, Problem 4 students choose from either a graph or equation to determine if a function is linear. “You want to determine whether a function is linear. How can making a graph or writing an equation help?”

• Unit 6, Lesson 25, Session 3, Teacher Edition, Connect It, provides guidance to Facilitate a Whole Class Discussion around strategically selecting approximations of irrational numbers or exact values. “Ask: Why does using a value of \piwith more decimal places give a more accurate area?...When would it make sense to use an approximate value of an irrational number rather than the exact number in symbolic form?”

• Unit 7, Lesson 29, Session 1, Try It, students construct and interpret plots for bivariate measurement data to investigate patterns of association. “Twenty middle schoolers use an app to play a memory game. The app tracks data for two variables. The first variable is the number of hours per week that the middle schoolers spent on screen time. The second variable is the students’ scores on a memory test. Use “Data Set: Memory Game” to plot the data as ordered pairs. Describe the shape of the data.” Students will need several tools in order to complete this problem such as graph paper, graphing calculator. 

The materials reviewed for i-Ready Classroom 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: “Attend to Precision 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 2, Lesson 4, Session 1, Model It, Problem 1, students understand that a two-dimensional figure is similar to another if the second can be obtained from transforming the first. “In art class, Kennedy draws scale copies of figure A to make a rug pattern. She uses a scale factor of \frac{3}{2} to draw figure B. Complete figure B by drawing the missing sides.” 

• Unit 4, End of Unit, Unit Review, Problem 4 students sketch and label a graph to represent a relationship between two quantities. “Karina buys a painting. The price of the painting slowly decreases at a constant rate for 3 years. The price increases faster during the next 2 years, and then it stays the same price for 2 years. Then the price increases slowly at a constant rate for 3 years. Sketch the price of the painting as a function of time in the coordinate plane.” A blank graph is provided with the numbers 0 through 10 on the x-axis. The axes are not labeled.

• Unit 6, Lesson 24, Session 1, Try It students calculate accurately and efficiently to determine if fractions are terminating or repeating decimals. “Chantel sketches a design for a hip-hop concert poster. She uses art software to make a poster from her sketch. The software requires her to use decimal measurements. Are both fractions in Chantel’s poster repeating decimals? How do you know?” A diagram of the poster shows its dimensions as \frac{3}{8}m and \frac{15}{33}m.

i-Ready Classroom Mathematics attends to the specialized language of mathematics. The materials use precise and accurate mathematical terminology and definitions, and the materials support students in using them. All Units include a guide to academic and content vocabulary. The Collect and Display routine is “A routine in which teachers collect students' informal language and match it up with more precise academic or mathematical language to increase sense-making and academic language development.” Teacher’s guides, student books, and supplemental materials explicitly attend to the specialized language of mathematics. Examples include: 

• Unit 1, Lesson 3, Teacher Edition_, Overview, Language Objectives, materials attend to the specialized language of mathematics by including Language Objectives in each lesson. Objectives in this lesson are: “Describe a sequence of transformations that map a figure onto a given image; Use the term congruent to describe the images that are the result of one or more rigid transformations. Read the ≅ in text as is congruent to; Use the term prime to label and name transformed figures. Use prime notation; Describe the location and orientation of an image resulting from a sequence of transformations; Use the term reverse order to describe a sequence of transformations; Justify a response by giving reasons to explain a strategy.”

• Unit 4, Math in Action, Session 2, Reflect, Be Precise, students use mathematical language as they discuss, “Should you round up or round down to the nearest whole number when determining the greatest number of days the film crew can afford to film crowd scenes? Explain.“

• Unit 5, Lesson 25, Session 2, Teacher’s Edition, Develop Academic Language, teachers develop academic language. “Why?: Reinforce the meaning of approximation through suffixes and examples. How?: Have students tell what they do when they approximate. Then write -tion on the board. Explain that it forms a noun that names a condition or the result of an action. To help students form the noun, ask: What do you get when you approximate? Encourage students to find the words solve, suppose, and discuss in the session and use similar reasoning to form and explain the corresponding nouns solution, supposition, and discussion.”

The materials reviewed for i-Ready Classroom 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.

The MPs are embedded within the instructional design. In the Teacher’s Guide, Front End of Book, Standard of Mathematical Practice in Every Lesson, teachers are guided “through a dedicated focus on mathematical discourse, the program blends content and practice standards seamlessly into instruction, ensuring that students continually engage in developing the habits of the mathematical practices.”

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

• Unit 2, Lesson 5, Session 3, Teacher Edition, Differentiation: Extend, provides guidance to support students in using structure to describe sequences of transformations involving dilations. “Prompt students to consider the symmetry of figure ABCD and how it affects the type and number of transformations that could be used. Ask: Why were two reflections, one across the x-axis and one across the y-axis, both needed?...How could the sequence of transformations differ if figure ABCD was a rectangle and vertex C had coordinates (2,4)? Why?”

• Unit 3, Lesson 11, Session 3, Connect It, Problems 1 and 2, students look for structure to make generalizations about determining the number of solutions of an equation . In the Try It, Model It, and Analyze It sections, students explore what number will complete the following equation to create both no solution and infinitely many solutions: 3x+5=3+____. Problem 1, “Look at Analyze It. What must be true about the constant terms on each side of the equation if the equation has no solution? What must be true about the constant terms on each side of the equation if the equation has infinitely many solutions? How do you know?” Problem 2a., “Is there more than one number you could write on the line so the equation has no solution? Explain.” Problem 2b., “Is there more than one number you could write on the line so the equation has infinitely many solutions? Explain.”

• Unit 4, Lesson 15, Session 2, Model It, Problem 3 students look for structures to make generalizations about the form of linear functions. “Many functions can be represented by equations that show how to calculate the output y for the input x. a. Determine whether each equation represents a linear function. Show your work. y=2x-1, y=-x^2, y=-x, b. Explain how you know that equations in the form y=mx+b always represent linear functions.”

• Unit 6, Lesson 21, Session 1, Try It, Students learn how to express and estimate quantities using integer powers of 10. “According to the U.S. Mint, the number of quarters minted in 2018 was about 2\times10^9 . The mass of a quarter is about 6\times10^{-3}kg. Write the number of quarters and the mass of a quarter in standard form.” The Discuss It box with the problem asks: “How did you use the exponents in the powers of 10 to help you solve the problem?” 

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

• Unit 3, Beginning of Unit, Math Background, Insights on Systems with No Solution or Infinitely Many Solutions, provides guidance for using repeated reasoning to correct a common misconception around solutions of systems of equations. “Students may think that a system has infinitely many solutions any time the equations in the system have the same slope. If so, be sure to provide many examples of systems that represent parallel lines. Help students make the connection that if the slopes are equal but the y-intercepts are not equal, then the lines will never intersect, which means the system has no solution.”

• Unit 3, Lesson 12, Session 2, Connect It, Problem 4 asks students to use repeated reasoning to make a generalization about graphs of a system of linear equations. “Look at Problems 1-3. In each system of equations both lines have the same slope. Can two lines with the same slope ever intersect at exactly one point? Explain.” Problems 1-3 present examples of graphs of systems with no solution or infinitely many solutions, along with their equation.

• Unit 5, Lesson 19, Session 3, Teacher Edition, Differentiation: Extend, provides guidance in supporting students to make a generalization about powers of a product. “Ask: How does the area of the new design compare to the original?...How would the area of the new design compare to the original area if each side length was doubled?...How would the areas compare if each original side length was quadrupled?...What conjecture can you make about how the area of a square changes when each side length changes by the same factor?”

• Unit 6, Lesson 25, Session 2, Connect It, Problem 4, students use repeated reasoning to find rational approximation for irrational numbers. “Suppose you continue to approximate the value of the square root of 2 to the nearest hundredth, then the nearest thousandth, and so on. Could you continue this process forever or would it eventually end? Why?”