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 use 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 planning 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 that show how to calculate the output y for the input x. a. Determine whether each equation represents a linear equation. Show your work. $y=2x-1$; $y=-x^{(^2)}$; $y=-x$ b. Explain how you know that equations in the form of y=mx+balways represent linear functions.” The text includes a blank graph for students to graph the functions.

• Unit 6, Lesson 26, Session 2, Model It, Problem 5, students explain 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) “Mrs. Aba says: All congruent figures are similar, but not all similar figures are congruent. Use the definition of congruent figures and what it means for figures to be similar 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 construct and interpret data in two-way tables (8.SP.4). “A piano teacher asks her students the amount of time they spend practicing each day. She then notes the number of mistakes they make in a recital. What association can you see from the data in the table? What might the piano teacher recommend to her students?”  There is a data table 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 solve linear equations with one variable (8.EE.7). Guided instruction includes the steps to solve the linear equations. Students then practice the procedure for solving equations, “Solve the equation for t. Show your work. 5(2t-3)-7.5t=0.5(12-t)." 

• Unit 5, Lesson 20, Lesson Quiz, Problem 1, students demonstrate understanding of the properties of integer exponents to generate equivalent numerical expressions (8.EE.1). “Rewrite the expression using only positive exponents. Show your work.  $$5^{-12}⋅32^{-3}⋅9^{-15}$$.”

• Unit 6, Lesson 24, Session 2, Fluency Skills & Practice, contains multiple problems for students to convert a decimal expansion which repeats eventually into a rational number (8.NS.1). In Problem 12, students, “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 solve systems of equations algebraically (8.EE.8). “Solve the system of equations. Show your work. 3x=6y-21 6x-94= -30.” 

• Unit 4, Lesson 16, Apply It, Problem 7, students construct 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 as a function of the number of students who attend. 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 understanding of integer exponents to generate equivalent expressions (8.EE.1). “Which expression is equivalent to $$\frac{(15^4⋅7^6)^3}{15^2}$$ ? a. $$15^2⋅7^9$$ b. $$15^2⋅7^18$$ c. $$15^5⋅7^9$$ d. $$15^{10}⋅7^{18}$$. 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, Try It, routine problem, students solve 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, Try It, non-routine problem, students apply 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, Try It, routine problem, students construct and interpret 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, Refine, Problem 9, routine problem, students analyze and solve 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, non-routine problem, students construct 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 is considering 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 will probably 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, routine problem, students use 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 3, Lesson 14, Session 4, Refine, Problem 9, students develop fluency as they analyze and solve pairs of simultaneous linear equations using a strategy of their choice (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, 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 5, Lesson 20, Lesson Quiz, Problem 1, students apply the properties of integer exponents to generate equivalent numerical expressions (8.EE.1). “Rewrite the expression using only positive exponents. Show your work.  $$5^{-12}⋅32^{-3}⋅9^{-15}$$.”

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 It ” tasks completed. 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. Later in the Apply It, Problem 6, students solve an equation, “What is the solution of the equation? Show your work. $$2x+4-\frac{3}{2}x=\frac{1}{3}(x+5)$$ ”.

• Unit 4, Lesson 18, Session 2, Additional Practice Problem 7, students utilize conceptual 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, Additional Practice 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 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.”

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. MP1 and MP2 are identified in every lesson from 1-33.  

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 select 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. He or she chooses a size and then adds 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 models the price of a large pizza?” 

• Unit 5, Lesson 19, Session 2, Develop, Try It, students make sense of problems by applying the properties of integer exponents to generate equivalent numerical expressions . “ One population 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 is trying to rescue a kitten from a tree. He 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 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, some Strategy lessons further develop MP2 in Deepen Understanding. Teachers are provided with discussion prompts to analyze a model strategy or representation. Examples include: 

• Unit 2, Lesson 4, Session 2, Discuss 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. Try It, “Efia warms up and goes for a run. The graph shows her distance as a function of time. Which sections of the graph represent Efia stretching, walking, or running?“ A graph showing distance from home (mi) vs. time (min) is shown.

• Unit 6, Lesson 23, Session 2, Teacher’s Edition, Differentiation Extend, Deepen Understanding, Reasoning Quantitatively with Problems Involving Square Roots, provides guidance for teachers to engage students in MP2 as they find a square root to solve problems . Try It, “A worker on the Rio Grande Gorge Bridge in New Mexico drops a bolt that lands in the water 576 feet below. The equation $$\frac{d}{16}=t^2$$ represents the distance in feet, d, that an object falls in t seconds. How long does it take the bolt to hit the water?” The teacher’s edition, “Differentiation, Extend “Prompt students to think about how the distance after 3 seconds compares to the distance after 6 seconds. ASK: It takes 6 seconds for the bolt to fall 576 feet. After 3 seconds, had the bolt fallen half this distance? Explain. LISTEN FOR: No; substitute 3 for t in the formula $$\frac{d}{16}=t^2$$ and solve for d. The result is 144 feet, which is less than half of 576 feet. ASK: How do the distances after 3 seconds and after 6 seconds compare? Explain. LISTEN FOR: The distance after 3 seconds is $$\frac{1}{4}$$ of the distance after 6 seconds. $$\frac{1}{4}⋅576-144$$ ASK: How can you use the formula to show why the distance the bolt falls after 3 seconds is $$\frac{1}{4}$$ the distance it falls after 6 seconds? LISTEN FOR: Rewrite the formula as $$d=16t^2$$ by multiplying each side by 16. Substitute $$\frac{1}{2}⋅6$$for t to get $$d=(\frac{1}{2}⋅6)^2=16⋅\frac{1}{4}⋅36$$”

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.” In the Discuss It routine, students are prompted with a question and a sentence frame to discuss their reasoning with a partner. Teachers are further provided with guidance to support partners and facilitate whole-class discussion. Additionally, fewer problems in the materials ask students to critique the reasoning of others, or explore and justify their thinking.

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

• Unit 1, Math in Action, Session 2, 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, 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. “Do you think using substitution will always work when solving a system of equations? Explain.”

• Unit 5, Lesson 19, Session 1, Explore, students discuss reasoning of answers with other students to know and apply the properties of integer exponents to generate equivalent numerical expressions by solving the following, “How can you write (10³)² as a single power of 10? A single power has one base and one exponent.” Students are then asked the following question: “How did you find the exponent in your answer?” 

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

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 generally 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 it’s 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, Part C, students model a system of equations with an appropriate representation. Problem 4, “Kenji and Ramon are running cross country. 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 neve 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, Develop, 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 he on when he started reading today?” Students may use a graph to model this equation and solve this 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 33 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 it’s 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, Explore, 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. Possible tools: grid paper, straightedge, ruler. 

• 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, 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 $$\pi$$with 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, Explore, 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’ score 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. Some possible tools: graph paper, pencils, 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 it’s 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, students understand that a two-dimensional figure is similar to another if the second can be obtained from the first “In art class, Kennedy draws scale copies of figure A to make a rug pattern. She uses a scale factor of $$\frac{3}{3}$$ to draw figure B. Complete figure B by drawing the missing sides.” 

• Unit 4, Unit Review, Unit Review, Problem 4 students use and label graphs to represent a relationship between two quantities . “Katrina buys a painting. The price of the painting slowly starts to decrease at a constant rate for 3 years. It then increases faster during the next 2 years and 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 convert fractions into decimals to determine if they are terminating or repeating . “Chantel is designing a poster for a hip-hop concert. She uses art software to create a poster from her sketch. The software program requires her to use decimal measurements. Are both fractions in the 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 attend to the specialized language of mathematics. The materials use precise and accurate mathematical terminology and definitions, and the materials support students in using them. The Collect and Display routine is described as, “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, 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 in 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, Model It, 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 -ion 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, Deepen Understanding, provides teachers with 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 1and 2 students look for structures to make generalizations about determining the number of solutions of an equation . The Try It, Model It, and Analyze It in this session have students determine what number they can fill in the following equation to make it have 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. Introductory problem is “According to the U.S. Mint, the number of quarters minted in 2018 was about $2×10^9$ . The mass of a quarter is about $6× 10^{-3}$kg. Write the number of quarters and the mass of a quarter in standard form. “ Teachers are instructed to listen for understanding as students Discuss the Introduction problem.How did you use the exponents in the powers of 10 to help you solve the problem?” Students use structure in recognizing powers of 10 in various forms. 

There is intentional development of MP8 to meet it’s 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 teachers with guidance to use 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, Deepen Understanding, provides teachers with guidance in supporting students to make a generalization about powers of a product. “Ask: How does the area of the new rangoli compare to the original?...How would the area of the new rangoli 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 24, Session 2, Connect It, Students use strategies to understand how to write a repeating decimal as a fraction. “What is the decimal form of $\frac{1}{9}$? How can you use this answer to help you write $\bar{0.4}$ as a fraction?” As students work through the examples they will use different representations to reason about strategies for writing decimals as fractions.

The materials reviewed for i-Ready Classroom 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.

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

• The Program Overview provides the teacher with information on program components and description about i-Ready classroom Mathematics implementation.

• Each unit has a Math Background document that provides the teacher with information to unpack the learning progressions and make connections between key concepts.

• Each Unit has an Unit Opener that provides the teacher with Unit Big Ideas and describes the themes of the unit.

• Each Unit has a Unit Flow and Progression video that describes how concepts are developed in the unit.

• Each Unit has a Professional Development document that provides guidance on instructional strategies, such as Supporting Math and Academic Vocabulary Development, Establishing Classroom Environments That Support Mathematical Discourse for ALL Learners, Knowing and Valuing Every Learner: Culturally Responsive Mathematics Teaching.

• Each Unit has a Unit Overview that provides the teachers with pacing, objectives, standards, vocabulary and lesson-level differentiation for each of the lessons in the unit.

• The Teacher’s Guide provides in-class instruction and practice included in the teacher’s edition.

• The Teacher’s Guide for Assessments and Reports supports whole group/partner discussion, ask/listen fors, common misconceptions, error alerts, etc.

• DIfferentiation strategies are included before and during the unit/lesson for the teacher. There are recommended resources to support students’ learning needs that are highlighted in the Prerequisites report.

• Unit and Lesson Support includes information about prerequisite lessons to focus on, and identifies the important concepts within those lessons.

• On the Spot Teaching Tips suggest additional scaffolding to support students with unfinished prerequisite learning as they engage with on-level work. 

• Digital Math Tools contain support videos that explain how to use their digital tools.

• Ready Classroom Central is an online teacher portal with resources for professional support such as training videos, planning tools, implementation tips, whitepapers, and discourse support.

• Language Expectations identify examples of what English learners at each level of language proficiency can do in connection with a one grade-level standard. 

• The Unit Prepare For provides teachers with guidance to support students when completing the graphic organizer in the beginning of the unit, Prepare for Unit. There is additional guidance to Build Academic Vocabulary through the use of identified cognates and specified academic terms. 

• The Unit Review includes problem notes for teachers identifying the Depth of Knowledge level of each problem and the standard, along with suggested strategies, and possible misconceptions based on the selected answer. 

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. Throughout each lesson planning information, there is narrative information to assist the teacher in presenting student materials throughout all phases of the unit and lessons. Examples include:

• Unit 2, Lesson 6, Session 1, Connect It, Problem 1, “What is m⦟BCF? What types of angle relationships did you use to find m⦟BCF?” The materials provide a diagram of two lines cut by a transversal with some of the angles labeled with degree measures or algebraic expressions. ⦟BCF does not have any measurements or labels. The Teacher’s Edition provides guidance for the teacher, “Look for understanding that the relationship between vertical angles can be used to find x, and then of the expressions x + 15 or 2x, and the relationship between vertical or supplementary angles can be used to find m⦟BCF.”

• Unit 3, Lesson 12, Session 3, Apply It, Problem 1, “A system of linear equations has exactly one solution. What can you say about the slopes of the lines when the equations are graphed? How do you know?” The Teacher’s Edition provides guidance, “Look for understanding that the lines in a system of linear equations with exactly one solution will have different slopes. If two lines have the same slope, they will either not intersect at all, so the system has no solution, or they intersect at every point, so the system has infinitely many solutions.

• Unit 4, Math Background, Functions, identifies common misconceptions for teachers around functions and how to support students in addressing this misconception. “Students may mistakenly believe that if there is more than one input with the same output, then the relationship is not a function. Be sure to include many examples and discussion about this idea.”

• Unit 5, Math in Action, Session 2, Discuss Models and Strategies, teachers are instructed to “Present the Insulating Blanket problem and use Three Reads to help students make sense of it. Have different volunteers take turns reading aloud each section of information. Ask: What is this problem? Listen For: It is about determining the thickness, in inches, of a blanket used to insulate spacecraft from damaging heat and radiation from the sun. Invite volunteers to point out what is known and what they need to figure out. Ask: What do you need to include in your plan and solution? Listen For: The plan needs to include the number of pairs of reflector and separator layers that will be used, as well as the choices for the thicknesses of these layers and of the inner and out covers. The solution will need to include the total thickness of the blanket.”

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

In the Teacher’s Guide, a Lesson Progression table is provided that links each lesson within the current unit to a prior and future grade level lesson. Within the Math Background section, detailed explanations of the mathematical concepts in each lesson are provided. For example, in Unit 4,  Math Background, Understanding Content Across Grades, insights are provided for prior knowledge, current lesson, and future learning starting in Lesson 15:

• Prior Knowledge, Insights on: Slope-Intercept Form of an Equation. For example, “In addition to slope, the other main feature of the graph of a linear relationship is the y-intercept, which is the y-coordinate of the point where the graph intersects the y-axis.” Visual examples are included to show comparisons between proportional relationships and nonproportional relationships, and to understand the slope-intercept form of an equation.

• Current Lesson, Insights on Understanding Functions. For example, “A function is a dependence relationship. The value of the output, y, depends upon the value of the input, x. For example, the cost of renting a kayak is a function of the number of hours. Common Misconception: Students may mistakenly believe that if there is more than one input with the same output, then the relationship is not a function. Be sure to include many examples and discussion about this idea.” Examples include input/output tables that are functions and not functions. 

• Future Learning, Insights on Understanding and Interpreting Function Notation. “In high school, students continue to build their understanding of functions by using formal function notation and by relating what they know of input and output with the function and its domain.” Examples include writing an equation using function notation and evaluating and interpreting the functions.

The materials reviewed for i-Ready Classroom 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. 

• The Correlations Document describes lesson correlation to the CCSSM through multiple lenses. The document identifies the major and supporting areas of focus within the CCSSM and the lessons address those standards. There is a table correlating each lesson with the standards covered, designating standards as “Focus”, “Developing”, or “Applied” within each lesson. The Correlations Document also identifies the Standards of Mathematical Practice that are included in each lesson. One table is organized by MP while the other is organized by lesson. The Unit Review Correlation identifies the associated standard and lesson to each problem within the Unit Review, along with their Depth of Knowledge level. 

• The Program Overview provides teachers and explanations for how the standards are addressed in each unit. One section identified is the coherence section titled “Lesson Progression.” 

• At the beginning of each Unit, Lesson Progression shows how each standard connects to and builds upon the previous grade levels. Each standard is identified in each lesson. It is arranged in a flow chart, and connects lessons to future grade levels. 

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

Explanations of the role of specific grade-level mathematics are present in the context of the series. 

• Grade Level Support, Learning Progression identifies prerequisite skills for each lesson and their related standards for the two prior grade levels, when applicable, in a flow chart. For example, Unit 3, identifies a prerequisite skill for the unit as, “Solve multi-step equations”. The standards from grades 6 and 7 connected to the unit are, Understand Solutions of Equations (6.EE.5), Write and Solve One-Variable Equations (6.EE.7), and a designated essential skill, Write and Solve Multi-Step Equations (7.EE.4, 7.EE.4a). These standards are connected to Solve Linear Equations in One Variable (8.EE.7, 8.EE.7b).

The materials reviewed for i-Ready Classroom 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 Teacher’s Guide and the Program Implementation area in the digital platform contains a section “Understanding the Try-Discuss-Connect Routine.” This routine is embedded throughout the program. This document explains how the routine is used. “Ready Classroom Mathematics empowers all students to own their learning through a discourse-based instructional routine. Lessons are divided into Explore, Develop, and Refine sessions and are taught over the course of a week. In Explore and Develop sessions teachers facilitate mathematical discourse through a Try-Discuss-Connect instructional routine.” 

• “Using a Session” in the Teacher’s Guide describes the planning and support features within the Teacher’s Guide. This includes each component of the lesson and teacher’s guide and describes why it is important in the lesson. For example, “SMPs are infused throughout the instructional model. Deepen Understanding is a consistent opportunity to build understanding of a key lesson concept by extending mathematical discourse. The content connects a particular aspect of lesson learning to an SMP, showing how it might look in the classroom.”

• Integrating Language and Mathematics identifies and explains the six language routines embedded within the curriculum. It identifies each routine, why a teacher may use it, the process and what part of the Try-Discuss-Connect Routine it can be used within. For example, for Say It Another Way, “What: A routine to help students paraphrase as a way to process a word problem or other written text and confirm understanding. Why: Paraphrasing helps students figure out whether they have understood something they have read or heard...How: Students read or listen to a word problem or other written text. One student paraphrases the text. Other students give a thumbs-up to show that the paraphrase is accurate and complete.”

Materials reference relevant research sources. Examples include: 

• Boaler, (2016), Mathematical Mindsets

• Council of the Great City Schools, (2016), A Framework for Re-Envisioning Mathematics Instruction for English Language Learners

• Kersaint, (2016), Orchestrating Mathematical Discourse to Enhance Student Learning

• National Council of Teachers of Mathematics, (2010), Teaching and Learning Mathematics

• National Council of Teachers of Mathematics, (2014), Principals to Action

• National Council of Teachers of Mathematics, (2014), Using Research to Improve Instruction

• Richhart, (2009), Creating Cultures of Thinking

Materials include research-based strategies. Examples include: 

• “Collaborative learning (partner or small group) encourages students to present and defend their ideas, make sense of and critique the ideas of others, and refine and amend their approaches.” Examples include, “Ready Classroom Mathematics lessons provide multiple opportunities for collaborative learning, such as Discuss It prompts where students explain and justify their strategies to each other and Consider This prompts where students compare problem-solving approaches, solutions, and reasoning.” The research included to support this is, “Research tells us that when students work collaboratively, which also gives them opportunities to see and understand mathematics connections, equitable outcomes result.” (Boaler, 2016)

• Professional Development, contains an adapted excerpt from Reimagining the Mathematics Classroom, co authored by Dr. Mark Ellis for teachers. The excerpt explains “funds of knowledge” to teachers and how they can apply this knowledge using the materials. “Connect to Culture in the Teacher’s Guide for each lesson offers suggestions for tapping into students’ funds of knowledge and connecting the knowledge to Try It and other problems.”

The materials reviewed for i-Ready Mathematics Grade 8 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. The Teacher’s Guide includes an Activity Sheet in the Table of Contents which provides a list of printable tools and resources. “Dot Paper, Frayer Model 2, Fraction Bars are available to print and copy for each student.” Materials include a Manipulatives List by Lesson for each grade level. For example: 

• Unit 2, Lesson 7: 1 straightedge per student, 15 rulers for the class, and 1 protractor per student.

The materials reviewed for i-Ready Classroom 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

• At the end of the Lesson Quiz in the Teacher’s edition, there is a section for differentiation that provides suggestions for Reteach (Tools for Instruction), Reinforce (Math Center Activity), and Extend (Enrichment Activity). Reteach, “Students who require additional support for prerequisite or on-level skills will benefit from activities that provide targeted skills instruction.” Reinforce, “Students who require practice to reinforce concepts and skills and deepen understanding will benefit from small group collaborative games and activities (available on-level, below-level, and above-level versions).” Extend, “Students who have achieved proficiency with concepts and skills and are ready for additional challenges will benefit from group collaborative games and activities that extend understanding.” The digital platform contains these activities for each lesson.

• In Refine lessons, the teacher’s edition provides suggestions to Group & Differentiate, “Identify groupings for differentiation based on the Start and problems 1-3. A recommended sequence of activities for each group is suggested below. Use the resources on the next page to differentiate and close the lesson.” Resources are suggested for groups Approaching Proficiency, Meeting Proficiency, and Extending Beyond Proficiency. The resources are found in the digital platform (Reteach, Reinforce, Extend). The following pages also contain descriptions of additional activities in the teacher’s edition for Reteach, Reinforce, and Extend.

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

Refine sessions provide recommendations for students that demonstrate understanding “Extending Beyond Proficiency” to engage in problems for reinforcement and a challenge. The number of problems is the same as students who are considered to be “Meeting Proficiency”. Additional Enrichment Activities can be found online in the Small Group Differentiation Extend section.

In Explore and Develop sessions, the materials contain a Deepen Understanding section to extend understanding of the lesson’s key concepts through the use of discourse with students. The section contains teacher prompts and suggestions for what ideas to look for from students. Each Deepen Understanding is labeled with an embedded mathematical practice. Examples include: 

• Unit 4, Lesson 20, Enrichment Activity Grow and Shrink, students are provided with a challenge scenario at the beginning and multiple opportunities to explain their answer, “You are programming a video game in which the character grows or shrinks depending on what color token is used. A blue token makes the character double in size. A red token makes the character shrink to half its size. You must write exponential expressions and equations to represent the character’s changes in size.”

• Refine sessions include at least 1 problem where students utilize strategic thinking. In Unit 4, Lesson 18, Session 4, Problem 6, students explain and justify their thinking about graphs of functions. “In a graph that shows the distance a boat travels as a function of time, why would a vertical section not make sense?”

The materials reviewed for i-Ready Classroom 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.

Materials consistently provide strategies and supports for students who read, write, and/or speak in a language other than English to meet or exceed grade-level standards through regular and active participation in grade-level mathematics. Examples include:

• Each Lesson Session includes differentiated support for various levels of English proficiency with level 1-3, levels 2-4, and levels 3-5 identified. Support for Academic language is used during the “Try-Discuss-Connect Language” routines in each lesson. 

• In the Program Overview, language expectations charts are provided that describe the language English Learners can understand and produce in connection with students’ levels of English proficiency. Teachers can use the examples to help meet the needs of English Learners. 

• Each Unit Overview connects with one of the CCSS addressed in the unit and shows an example of how language expectations can help to differentiate instruction to meet the needs of English learners. 

• In the Program Overview, there is an Integrate Language and Mathematics section. “Scaffolded language support for a specific problem is outlined. These suggestions for scaffolding and amplifying language can be applied to other problems as well.” 

• Language objectives are included and students are expected to understand and produce language as they work on lesson objectives. Graphic organizers are used to help students access prior knowledge and vocabulary they build on in the lesson.

• Discourse cards are available in the Teacher Digital Experience under the Ready Classroom Mathematics Toolbox. These cards provide sentence starters and questions to help students engage in conversations with their partners, small groups or the whole class. 

• All classroom materials are available in Spanish.

• Multilingual Glossary is available in Arabic, Chinese, Haitian Creole, Portuguese, Russian, Tagalog, Urdu, and Vietnamese. There is a Bilingual glossary in the student textbook that includes mathematics vocabulary in English and Spanish.

The materials reviewed for i-Ready Classroom 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.

Digital tools are available for students. These tools include Counters and Connecting Cubes, Base-Ten Blocks, Number Line, Multiplication Models, Perimeter and Area, and Fraction Models. Geometry, Scientific Calculator and Graphing Calculator are also included but cannot be reviewed as these tools are powered by Desmos. Support videos are available for each of the digital tools, explaining how they may be used and their functions. For example:

• Grade 8 Standard Manipulative Kit includes the most commonly used manipulatives. Manipulatives include Algebra Tiles, plastic rulers, centimeter cubes, tangrams, geoboards, two color counters, and protractors. A la carte items are available. The materials state that these items may only be used once, may be common to classrooms, or print options are available. A la carte items include rainbow tiles, compasses, number cubes, tape measure, pattern blocks, base ten blocks, additional tangrams, and algebra tiles. 

• Visual models such as number lines, graphs, or bars, are also available but cannot be manipulated.

The Try-Discuss-Connect routine embedded throughout every lesson provides students the opportunity to connect and transition from the use of manipulatives to written methods. Inside of the digital platform, Program Implementation, Try-Discuss-Connect Routine Resources, Understanding the Try-Discuss-Connect Instructional Routine, the guide describes how the routine helps students transition from manipulatives to written methods. In the Try It activity, “students have access to a variety of tools and manipulatives to use to represent the problem situation. During the Discuss It activity, “Students present and explain their solution methods and listen to and critique the reasoning of others, models and representations.” “The class then looks at the strategies highlighted in the Picture It and Model It, and students make connections between strategies, their own strategies, and the strategies discussed as a class.” During the Connect It activity, “Students apply their thinking during the lesson to new problems.” The routine integrates the CRA model in the:

• Try It, “Students use concrete, representational, or abstract strategies to solve the problem, based on their understanding of the problem or mathematical concept.

• Discuss It, “Students who use more concrete approaches begin to make connections to representational or abstract approaches as they engage in partner discussions.”

• Connect It, “Through the Connect It questions, students connect concrete and representational approaches to more abstract understanding as they formalize their connections.”