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 $\pi$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, 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?”

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. Examples include:

• i-Ready Classroom Central, Preparing for a Unit of Instruction, “Before delivering each unit of instruction, make sure to peruse the unit-level resources in your Teacher’s Guide. Learn about the unit goals by reading the Unit Opener, take note of the vocabulary and language supports, and study the mathematics in the unit by watching the Unit Flow and Progression Video or reading the Math Background pages.” 

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

  • Plan is broken down into Unit, Lesson, and Session. 

  • Teach gives information on practice, and differentiation. 

  • Assess includes support for the diagnostic, reports, and data. 

  • Leadership informs the teacher on getting started, building routines, fostering discussions, making connections, and top leader actions. 

• Program Implementation includes numerous supports such as digital math tools, videos, discourse cards, vocabulary, language routines, graphic organizers, games, correlations with standards and practices, etc.

• Each unit has a Beginning of Unit document that provides the teacher with extensive information on Unit Flow and Progression, Unit Resources, Unit Opener, Unit Prepare For, Unit Overview, Lesson Progression, Prerequisites Report Overview, Professional Development, Understanding Content Across Grades, Language Expectations, Math Background, Cumulative Practice, Yearly Pacing for Prerequisites, and Unit Lesson Support. Examples include:

  • Unit Opener, Self Check, “Take a few minutes to have each student independently read through the list of skills. Ask students to consider each skill and check the box if it is a skill they think they already have. Remind students that these skills are likely to all be new to them and that over time, they will be able to check off more and more skills.”

  • Prerequisites Report Overview, “Diagnostic data generates the Prerequisites report, which helps you identify students’ prerequisite learning needs and provides guidance on how to best integrate prerequisite instruction into your grade-level scope and sequence for the year.” These are specific to current students and classes providing valuable data about entry points for students. 

  • Under the Prepare column, there is a Unit and Lesson Support document that provides multiple On-the-Spot Teaching Tips for each Unit. These tips provide information on what to reinforce from prior learning promoting scaffolding to current content.

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:

• Program Implementation, Teaching & Learning Resources, Discourse Cards, provides instruction on how to use the Math Discourse Cards. “These questions and sentence starters provide a way to engage all students in meaningful mathematical conversations. These cards will help students initiate, deepen, and extend conversations with partners, small groups, or the whole class. Each card has two questions or sentence starters on it-one on the front and one on the back. With each question, be sure to have students explain their reasoning for their response.”

• Unit 2, Lesson 6, Session 1, Teacher Edition, Connect It, Problem 1, “What is $m\angle BCF$? What types of angle relationships did you use to find $m\angle BCF$?” 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 one of the expressions $x+15$ or $2x$, and the relationship between vertical or supplementary angles can be used to find $m\angle BCF$.”

• Unit 3, Lesson 12, Session 3, Teacher Edition, 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, Beginning of Unit, Math Background, Functions, identifies common misconceptions for teachers around functions, “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, Teacher Edition, Session 2, Discuss Models and Strategies, teachers are instructed to “Present the Insulating Blankets 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 outer 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. 

The Beginning of Unit section for every unit provides an abundance of information for teachers, including sections to support teachers with adult-level understanding of the content:

• Math Background includes Unit Themes, Prior Knowledge, and Future Learning. In the Math Background, as well as throughout the teacher materials, there are insights on the concepts taught, Common Misconceptions, and Error Alerts to watch for when students are incorrectly applying skills. 

• Lesson Progression links each lesson within the current unit to a prior and future lesson so teachers know what students need to know to be successful with the current work as well as what the current work is preparing students for. This is important for a teacher’s complete understanding of how to scaffold and bridge the current content. For example, Unit 4, Lesson 17, Lesson Overview, Teacher Edition, Compare Different Representations of Functions - Full Lesson, Learning Progression:

  • “Earlier in Grade 8, students used the slope-intercept form of a linear equation to graph the equation. They learned that a function is a rule that assigns to each input exactly one output, and they used tables, graphs, equations, and verbal descriptions to interpret a function and identify an output for a given input. 

  • In this lesson, students compare functions represented in different ways (graphs, tables, equations, or verbal descriptions). Students solve problems by comparing initial values of functions, rates of change, and outputs that correspond to particular inputs.

  • Later in Grade 8, students will analyze graphs to qualitatively describe a relationship between two quantities. They will also sketch graphs of functions from a qualitative description.”

• Understanding Content Across Grades provides explanations of instructional practices as well as information about necessary prior knowledge and concepts beyond the current course for teachers to improve their own knowledge of the subject. For example, Unit 4,  Beginning of Unit, Understanding Content Across Grades related to Lesson 15:

  • Prior Knowledge, “Insights on: Slope-Intercept Form of an Equation. 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. 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 outputs with the function and its domain.” Examples include writing an equation using function notation and evaluating and interpreting the functions. 

• Each lesson includes a Reteach section with several pages called “Tools for Instruction” that provide explicit teacher guidance related to the current work and to prerequisite skills. These pages include adult explanations, step-by-step guidance for teaching, and check for understanding. For example, Unit 7, Lesson 30, Us e a Line of Fit to Make Predictions:

  • “Students have previously analyzed the association between two variables using scatterplots. They should be familiar with the terms positive association, negative association, nonlinear association, and no association… They should know that a linear equation can be written in the form $y=mx+b$… In this activity, students will combine these two concepts to model a good line of fit and use the linear equation to make predictions about two variables.”

  • “Step by Step: 1) Analyze association. (followed by three prompts) 2) Calculate the slope. (followed by two prompts) 3) Identify the y-intercept. (followed by two prompts) 4) Write the equation and interpret the slope and y-intercept in context. (followed by three prompts) 5) Make a prediction (followed by six prompts, including one for Support English Learners).”

  • “Check for Understanding: Have the student usd their equation to determine the amount of money a business needs to spend to get 25 new customers per week.” Then an error analysis chart is provided: “If you observe… the student may… Then try…”

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.

In Program Implementation, correlation information is present for the mathematics standards addressed throughout the grade level using multiple perspectives. For example: 

• The Correlations document for Content Focus in the Common Core State Standards (CCSS) describes lesson correlation to the CCSSM through multiple lenses. The document identifies the major and supporting areas of focus within the CCSSM and corresponding lessons that address those standards. Additionally, a table is provided that correlates each lesson with the standards addressed, 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, and another 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. 

• Digital Resource Correlations, Comprehension Check Correlations, and Cumulative Practice Correlations identify the lesson and a statement of the part of the standard it aligns to. 

• The WIDA PRIME V2 correlates the WIDA Standards Framework to examples in the material with descriptions of how they connect. 

• The English Language Arts Correlations provides a table that offers evidence of how the Common Core State Standards for English Language Arts are supported in every lesson and unit of the i-Ready Classroom Mathematics material.

In each Beginning of Unit section, there are numerous documents provided that contain explanations of the role of the specific grade-level mathematics in the context of the series. For example: 

• The Lesson Progression provides a flow chart delineating how each standard in the current lesson builds upon the previous grade levels and connects to future grade levels. This is developed in detail with examples in the Understanding Content Across Grades document. 

• There is a Unit Flow and Progression video for teachers that provides background about the content covered in the unit. 

• The Unit and Lesson Support document provides descriptions of the standards addressed in each unit with connections to prerequisites and teaching tips about prior knowledge. For example, Unit 3, Beginning of Unit, Unit and Lesson Support, the opening narrative provides the content of the unit, “In this unit, students use what they know about unit rate and proportional reasoning to explore the concept of slope and understand the significance of the y-intercept of a line. Then they build on their understanding of one-step and two-step equations to solve more complex one-variable equations and determine the number of solutions to one-variable equations. Next, students learn about systems of linear equations in two variables, and they solve systems of equations both graphically and algebraically.” The document continues with Instructional Support identifying specific lessons from prior grades to develop understanding, such as Unit 3, Lessons 10-14, “Prepare by reviewing how to solve one-variable equations with the variable on one side to support students with solving linear equations with the variable on both sides: Grade 7, Lesson 18.”

• In every teacher's Lesson Overview, the Learning Progression identifies how the standard is addressed in earlier grades, in the current lesson, next lesson, and in the next grade level. For example, Unit 4, Lesson 18, Overview, Learning Progression, “In Grade 6, students learned how to use variables to represent unknown quantities in expressions and equations. In Grade 7, students extended their work with algebraic expressions as they explored proportional relationships. Earlier in Grade 8, students learned to model linear relationships using tables of values, graphs, equations, and verbal descriptions. In this lesson, students explore what it means to say that an input-output rule is a function… In the next lesson, students will write equations for linear functions and interpret their rates of change and initial values. Later in Grade 8, students use functions to model and analyze real-world relationships. In high school, students will continue their work with functions, including quadratic, exponential, and trigonometric functions.”

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.

There are thorough explanations about the instructional approaches of the program. These are easily found under Program Implementation and in Classroom Central. For example:

• Program Implementation contains “Try-Discuss-Connect Routine Resources.” This routine is embedded throughout the program, “i-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.” In i-Ready Classroom Central, there are videos modeling the six steps of the Try-Discuss-Connect routine as well as an Exit Ticket.

• Program Implementation, User Guide, Protocols for Engagement describes multiple protocols and identifies the traits each protocol validates to help all students “feel accepted and included.” Further, “Protocols provide structure for activities so that all students have a chance to think, talk, and participate equally in classroom activities. Each protocol incorporates modes of communication common to one or more culture and leverages those behaviors for a particular instructional purpose.” For example, “Stand and Share: Students stand when they have something to share with the class. Validates: spontaneity, movement, subjectively, connectedness.” Protocols can be found in the Lesson Overview section of the Teacher Guide.

• Program Implementation, i-Ready Classroom Central, Building Community, Promote Collaborative Learning, has resources such as using Lesson 0 to introduce the Try-Discuss- Connect Routine and language routines, questions to support discourse, videos about sharing math ideas, ideas for promoting mathematical practices, and creating a positive mindset. 

• Program Implementation, i-Ready Classroom Central, has a link in the upper right under the search box called Explore the Resources page that has all of the additional resources organized in a list of links by category that provide abundant information, including a section called Program Overview.

Materials include relevant research sources. In Program Implementation, Supporting Research, “i-Ready Classroom Mathematics is built on research from a variety of federal initiatives, national mathematics organizations, and experts in mathematics.” A table describes 16 concepts that are embedded in the program with examples of how and where each is used, an excerpt from the research that supports it, as well as an extensive reference list. Examples include: 

• “The Concrete-Representational-Abstract (CRA) Model is a three-part instructional model that enhances students’ mathematical learning.” This model is built into all i-Ready Classroom Mathematics lessons in the Try It, Discuss It, Connect It, and Hands-On Activities. “Using and connecting representations leads students to deeper understanding. Different representations, including concrete models, pictures, words, and numbers, should be introduced, discussed, and connected to support students in explaining their thinking and reasoning.” (Clements and Sarama, 2014)

• “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.” Lessons provide multiple opportunities for collaborative learning during Discuss It and Pair/Share. “Research tells us that when students work collaboratively, which also gives them opportunities to see and understand mathematics connections, equitable outcomes result.” (Boaler, 2016)

• “An instructional framework supports students in achieving mathematical proficiency and rigor within a collaborative structure to develop greater understanding of how to reason mathematically.” The Try-Discuss-Connect instructional framework is foundational in this program. “Instructional routines are situated in the learning opportunity itself, providing students with a predictable frame for engaging with the content…”  (Kelemanik, Lucenta, & Creighton, 2016)

• Program Implementation, User Guide, Routines that Empower Students identifies 9 research-based language routines. Each routine includes the purpose, the process, and which part of the Try-Discuss-Connect Routine it can be used with. For example, Say It Another Way is used with Try It, “Why: This routine helps students paraphrase a word problem or text so they know if they have understood it. It provides an opportunity to self-correct or to ask for clarification and ensures that the class hears the problem or story more than once and in more than one way.”

• Program Implementation, i-Ready Classroom Central, Explore the Resources page (near search box), Program Overview,Building Effective Mathematics Teaching Practices within Classrooms, explains how NCTM’s Effective Teaching Practices are integrated into i‑Ready Classroom Mathematics. “When teachers use the program with integrity… they naturally engage in the Top Teacher Actions. This ensures students have access to high-quality mathematics instruction every day… The intentional design of the Teacher’s Guide provides teachers with the opportunity to consistently implement NCTM’s Teaching Practices each day.”

The materials reviewed for i-Ready Mathematics, Grade 8 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

The Lesson Overview for the teacher provides a Materials Required column for each lesson on the Pacing Guide; additional materials are listed in the Differentiation column. Any materials that need to be printed are also provided in the Overview, such as grid paper or double number lines. For example:

• Unit 3, Lesson 13, Session 2, “Materials tab: Math Toolkit graph paper, straightedges, Presentation Slides. Differentiation tab: algebra tiles (at least 10 each of x- and y-tiles and 20 1-tiles).”

Under Program Implementation, a Manipulatives List provides a document identifying manipulatives needed for each lesson K-8. For example:

• “Manipulatives List, Unit 3, Lesson 13, identifies a set of algebra tiles - 1 per student.” 

Program Implementation also includes digital math tools, discourse cards and cubes, activity sheets, data sets, and graphic organizers.

The materials reviewed for i-Ready Classroom Mathematics, Grade 8 meet expectations for having assessment information included in the materials to indicate which standards are assessed.

In the Teacher Toolbox, each lesson includes Assess which provides Lesson Quizzes & Unit Assessments. Lesson Quizzes, Teacher Guide lists information for each problem: tested skills, content standards, mathematical practice standards, DOK levels, error alerts, problem notes, Short Response Scoring Rubric with points and corresponding expectations, and worked out problems. For example:

• Unit 5, Lesson 20, Lesson Quiz, Problem 3, “DOK 1, 8.EE.A.1, SMP 7.”

Assess, End of Unit, Unit Assessments, Teacher Guide, Forms A and B are provided and include the content item with a solution. Form A includes Problem Notes, completed problems, DOK levels, content standards, mathematical practices, Scoring Guide, Scoring Rubrics, and Responding to Student Needs. Form B appears to parallel all of the correlations provided for Form A, though it is not labeled. It is noted in the Scoring Guide, “For the problems in the Unit 4 Unit Assessments (Forms A and B), the table shows: depth of knowledge (DOK) level, points for scoring, lesson assessed by each problem, and both the CCSS standard and Mathematical Practice Standards addressed.” For example:

• Unit 2, End of Unit, Assess, Unit Assessment, Form A, Scoring Guide, Problem 1, “DOK 2, 8.G.A.5, SMP 2.”

• Unit 4, End of Unit, Assess, Unit Assessment, Form A, Scoring Guide, Problem 5, “DOK 2, 8.F.A3, SMP 1.”

Digital Comprehension Checks “can be given as an alternative to the print Unit Assessment. For this comprehension check, the table below provides the Depth of Knowledge (DOK), standard assessed, and the corresponding lesson assessed by each problem.” While the Comprehension Checks identify the content standards, they do not identify the mathematical practices. For example:

• Unit 4, End of Unit, Assess, Comprehension Check Correlation Guide, Problem 5, “DOK 1, 8.F.B.5.”

The materials reviewed for i-Ready Classroom Mathematics, Grade 8 meet expectations for including an assessment system that provides multiple opportunities throughout the grade, course, and/or series to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. 

The assessment system provides opportunities to determine students’ learning that include teacher support for interpreting student performance in the Problem Notes and Rubrics provided, though the rubrics are generic rather than specific to the lesson. Examples include:

• Problem Notes for each problem in the Lesson Quizzes and Form A of the Unit Assessment provide guidance on steps to solve the problem and what students may have done incorrectly. For example:

  • Unit 3, Lesson 12, Assess, Lesson Quiz, Problem 2, “B is correct. Students could solve the problem by noticing that the lines intersect at exactly one point, (4, 4). A is not correct. This answer represents two lines that do not intersect. C is not correct. This answer represents misinterpreting two lines as having two solutions; lines can either intersect at zero, one, or infinitely many points, having no solution, one solution, or infinitely many solutions. D is not correct. This answer represents two lines that intersect at all points.”

  • Unit 5, End of Unit, Assess, Unit Assessment, Form A, Problem 7, “Students could first identify the food items with the least powers of 10 and then compare the single digits for each expression. Students could also write each expression in standard form and compare values.”

• Lesson Quizzes contain a Fill-in-the-Blank/Multiple Select/Choice Matrix Scoring Rubric and a Short Response Scoring Rubric. The Fill-in-the-Blank/Multiple Select/Choice Matrix Scoring Rubric states: “2 points if all answers are correct, 1 point if there is 1 incorrect answer, and 0 points if there are 2 or more incorrect answers.” The Short Response Scoring Rubric states: 2 points if the “Response has the correct solution and includes well-organized, clear and concise work demonstrating thorough understanding of mathematical concepts and/or procedures.” 1 point for “Response contains mostly correct solution(s) and shows partial understanding of mathematical concepts and/or procedures.” 0 points if the “Response shows no attempt at finding a solution and no effort to demonstrate an understanding of the mathematical concepts and/or procedures.”

• Unit Assessments contain the Extended Response Scoring Rubric (if there is an extended response question included in the assessment), Short Response Scoring Rubric, and a rubric for Fill-in-the-Blank/Multiple Select/Choice Matrix. For example, the Extended Response Scoring Rubric, a response should earn 4 points if, “Response has the correct solution(s) and includes well-organized, clear and concise work demonstrating thorough understanding of mathematical concepts and/or procedures.” This same expectation scores a 2 on the Short Response Scoring Rubric. The Fill-in-the-Blank/Multiple Select/Choice Matrix Scoring Rubric is the same as the Lesson Quizzes.

The Lesson Quizzes and Unit Assessments provide sufficient guidance to teachers to follow-up with students, although there is no follow-up guidance for the Comprehension Checks. The follow up suggestions reference previous work rather than new material. For example:

• Unit 4, Lesson 16, Assess, Lesson Quiz provides three types of differentiation for possible follow up depending on student performance: Reteach, Reinforce, and Extend. “Reteach: Tools for Instruction, Students who require additional support for prerequisite or on-level skills will benefit from activities that provide targeted skills instruction. Grade. Reinforce: Math Center Activity, Students who require practice to reinforce concepts and skills and deepen understanding will benefit from small group collaborative games and activities (available in on-level, below-level, and above-level versions). Extend: Enrichment Activity, 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 Reteach section directs teachers back to Lesson 16, Write the Equation of a Function. The Reinforce section directs teachers back to Lesson 16, Find the Function. The Extend section directs teachers back to Lesson 16, Springy Springs.

• Unit 4, End of Unit, Assess, Unit Assessment, Form A, provides a section called Responding to Student Needs. This section directs teachers back to the relevant lessons for review and where teachers can access the Review, Reinforce, and Extend options. “For students who answer problems incorrectly on the Unit Assessment, choose from the following resources on the Teacher Toolbox for additional support.””Reteach: Tools for Instruction, Linear and Nonlinear Functions (Lesson 15), Write the Equation of a Function (Lesson 16), Compare Slope and Initial Value of Functions (Lesson 17), Analyze Qualitative Graphs (Lesson 18)””For students who exceed proficiency on the Unit Assessment, choose from the following activities on the Teacher Toolbox.””Extend: Enrichment Activities, Springy Springs (Lesson 16), Comparing Functions (Lesson 17), Tell Tales (Lesson 18).”

The materials reviewed for i-Ready Classroom Mathematics, Grade 8 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series. 

There are formative and summative assessments provided as PDFs as well as comparable assessments provided online. Lesson Quizzes and Unit Assessments provided include a variety of item types for students to demonstrate grade-level expectations. For example:

• Fill-in-the-blank

• Multiple select

• Matching

• Graphing

• Constructed response (short and extended responses)

• Technology-enhanced items, e.g., drag and drop, drop-down menus, matching 

Throughout the lessons, there are opportunities to demonstrate critical thinking, develop arguments, or apply learning in a performance task, though these are not typically on the assessments.

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.

• i-Ready Classroom Central, Teach, Differentiate, has information to support the teacher in planning for all special populations. Address Unfinished Learning provides multiple links to guidance resources, data analysis resources, and instructional resources. i-Ready Personalized Instruction has resources for students who have taken the Diagnostic and will have access to online learning and instructional paths tailored to their individual needs to reinforce prerequisite skills and build grade-level skills. Support Every Learner, “Every student can excel in mathematics with the right supports. Access these resources to find ideas and strategies for organizing groups and adapting your instruction to meet the unique needs and learning styles of all students.” There are seven links related to grouping students and adapting instruction. For example:

  • Reference Sheet: Supporting Differentiated Needs before the Unit or Lesson is a 4-page document with numerous strategies including addressing prerequisites, integrating assessments, supporting English learners, and differentiating that links to practice, games, enrichment, literacy, tutorials, and more.

  • Reference Sheet: Supplemental Tools for Accessible Mathematics Instruction is a 5-page table that provides support ideas for every aspect of the lesson. For example, during Try It, a suggested support is, “Offer multiple means of representation, engagement, and action and expression such as: highlight important numbers, words, and phrases; Invite volunteers to act out the problem for the class; Offer options for how students express their ideas.” During Discuss It, “Use hand signals to agree, disagree, or share an idea.”

• In Refine, the last session of each lesson, 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. 

• 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.” 

• Program Implementation, i-Ready Classroom Central, Differentiate, Support Every Learner, Reference Sheet: Supplemental Tools for Accessible Mathematics Instruction, Accessibility and Accommodations with i-Ready Classroom Mathematics, Accessibility and Accommodations Update document which states, “To make i-Ready Classroom Mathematics accessible to the widest population of students, we offer a range of accessibility supports that may also meet the requirements of a number of student accommodations.” The table provided lists the Universal Supports, Designated Supports, and Accommodations that are both embedded and not embedded in the program. For example, embedded supports include audio support, closed captioning, calculator, zoom in/out, highlighting, and more. Available non-embedded supports include native language translation of directions, noise buffer, alternate response options, scribe, and more.

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.

• Each lesson has an Extend: Enrichment Activities column that provides a challenge task. For example, Unit 4, Lesson 20, Extend, Grow and Shrink, students are provided with a challenge scenario, “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,” followed by related tasks such as, “To pass Level 1, characters must grow to 16 times their original size. Write the change needed as a single power with base 2. Then write 3 possible token combinations using both red and blue tokens, and write the related exponential expressions that will result in the correct character size. Explain.”

• Refine sessions at the end of each lesson 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 the work for students who are considered to be “Meeting Proficiency.” Additional Enrichment Activities can be found online in the Small Group Differentiation Extend section. In addition, Refine sessions include at least one problem identified as DOK 3 where students utilize strategic thinking. 

• In Explore and Develop sessions in each lesson, the materials contain Differentiation: Extend, Deepen Understanding or Challenge for the lesson’s key concepts through the use of discourse with students. For example, Unit 4, Lesson 18, Session 4, Teacher Guide, Differentiation: Extend, Challenge, “Write a description of the movement of a roller coaster along a track. Then sketch a graph to represent the height of the roller coaster as it travels along the track compared to ground level as a function of time.”

The materials reviewed for i-Ready Classroom Mathematics, Grade 8 meet expectations for providing strategies and support for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics. Examples include:

• Program Implementation, i-Ready Classroom Central, Differentiate, Support All Learners, Reference Sheet: Supports for English Learners explains where to find and how to use all of the supports built into the Teacher Guide for every lesson to “address the strengths and needs of ELs” such as, Build Your Vocabulary, Connect Language Development to Mathematics, Language Objectives, Connect to Community and Cultural Responsiveness, and Connect to Language Development.

• Program Implementation, Program Overview, Integrate Language and Mathematics shows where teachers can access tips for targeted support using Language Routines in the Teacher Guide for every lesson.

• Program Implementation, Program Overview, Language Development and Discourse Support provides “support at the word/phrase, sentence, and discourse levels so that all students can engage in rigorous mathematics and communicate effectively.”

• Program Implementation, User Guide, Resources for Language Development describes nine features that are embedded in the teacher materials to “build academic language of all students, especially English learners. These supports help students learn how to communicate effectively across the language domains.”

• Program Implementation, User Guide, Routines that Empower Students provides nine language routines. “While these routines support English learners, they are designed to be used by all students as they access mathematical concepts and their growing mathematical understanding.” Three routines, in particular, are differentiated for English Learners: Act it Out, Co-Constructed Word Banks, and Stronger and Clearer Each Time. 

• Program Implementation, User Guide, Support for Academic Discourse describes “a variety of ways to support students in communicating with academic and math-specific vocabulary and language.”

• Program Implementation, Discourse Cards provide sentence starters and questions to help students engage in conversations with their partners, small groups, or the whole class such as “Did anyone get a different answer?; What would you add to what was said?”

• All classroom materials are available in Spanish.

• Program Implementation, 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.

• Beginning of Unit, Language Expectations is a chart that “shows examples of what English Learners at different levels of English language proficiency can do in connection with one of the Common Core State Standards (CCSS) addressed in this unit. As you plan for this unit, use these examples of language expectations to help you differentiate instruction to meet the needs of English Learners.”

• Beginning of Unit, Unit Prepare For, Build Academic Vocabulary includes a chart of academic words for the units paired with their Spanish cognates. There are three routines provided in Professional Development to support vocabulary development: Academic Vocabulary, Cognate Support, and Collect and Display. 

• Each lesson in the Lesson Overview, Teacher Guide’s Full Lesson includes Language Objectives, Connect to Culture, and Connect to Language. 

• Session 1 of every lesson uses graphic organizers to help students access prior knowledge and vocabulary they will develop in the lesson. Support for Academic language is used during the “Try-Discuss-Connect Language” routines in each lesson. 

• All sessions throughout the lesson embed support including references back to previously listed items.

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.

Students have access to both virtual and physical manipulatives throughout the program. For example:

• Program Implementation, 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 Tool, Scientific Calculator, and Graphing Calculator. Also in Program Implementation, support videos are available for each of the digital tools, explaining how they may be used and their functions. 

• Program Implementation, Manipulative Kit includes 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.  

• Program Implementation, Manipulative List by Lesson has specific manipulatives listed for each lesson. For example, Unit 2, Lesson 6: 1 ruler and 1 protractor per student. There is also a Manipulative Suggestions for At-Home Use document that provides ideas for using items commonly found at home or easily created that could be used in place of the actual manipulative (e.g. Attribute Buttons and Connecting Cubes could both be replaced with Lego bricks). 

• Program Implementation, Activity Sheet Resources includes a 52-page document full of visual models such as number lines, graphs, grid paper, nets, graphic organizers, etc. These are also provided as a link in lessons where they would be expected to be a helpful resource. 

Program Implementation, Try-Discuss-Connect Routine Resources, Understanding the Try-Discuss-Connect Instructional Routine, the foundational “Try-Discuss-Connect” routine is designed to “encourage proficiency and rigor within a collaborative structure.” A primary purpose is to “expose students to a number of representations and approaches” to help them make connections, develop mathematical language and thinking, and improve written and oral communication skills. This routine helps students transition from manipulatives to written methods. For example: 

• 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.” During the Connect It activity, “Students write their answers to Connect It questions independently (or in pairs to support language production, as needed) to solidify understanding and make further connections.” 

• “Tip: If students are struggling with writing responses…. have multiple students share answers orally while writing key words or phrases on the board. Have students use these key words and phrases to write their own response to the question in their worktexts.”

• “Tip: Encourage students to represent and solve problems in more than one way to build flexibility in their thinking.”

The “Try-Discuss-Connect” routine also integrates the Concrete-Representational-Abstract (CRA) model, for example:

• Try It, “Students may 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.”