The materials reviewed for i-Ready Classroom Mathematics, Grade 7 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 2, Lesson 7, Session 1, “Model It”, Problem 1, students solve addition and subtraction problems with rational numbers, with teacher support (7.NS.A). “Neva plays a video game. On her first turn, she earns 3 points. On her second turn, she loses 3 points.The expression 3 + (-3) represents her score after the two turns. You can use integer chips to find the sum of 3 and -3. a. The sum of any number and its opposite is 0. Another term for opposites is additive inverses. Since the sum of 1 and -1 is 0, 1 and -1 form a zero pair. Circle the zero pairs in the model. b. How many points does Neva have after her second turn? c. What is 3 + (-3)?”

• In Unit 4, Lesson 17, Session 2, “Model It”: Equations, Problem 3, students develop conceptual understanding of writing equations, comparing models, and reasoning about equations (7.EE.4). “a. Complete the equation to model 3 times the sum of k and 8 is 36. b. You can think of $k+8$ as the unknown quantity. How could you find the value of $k+8$? What is the value of $k+8$? c. How could you use the value of $k+8$ to find the value of k? d. How can you check that the value of k is correct?”

• Unit 7, Lesson 30, Session 2, “Model It”, Problems 1 and 2, students develop conceptual understanding of probabilities being a number between 0 and 1 (7.SP.5). “A bag contains 6 red marbles, 6 green marbles, and 12 blue marbles. Paloma reaches into the bag and selects a marble without looking.” Students are asked to name impossible, unlikely, equally likely as not, likely and certain events based on this scenario. Problem 2 connects this same scenario to a number line and fractions. “Draw a line from each event to show the probability of that event.” Students then associate each event with its place and proximity to 0,$$\frac{1}{2}$$ and 1 on the number line and “Explain why an event that is equally likely as not has a probability of $\frac{1}{2}$.”

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 9, Session 1, “Model It”, Problem 5, students extend previous understandings of addition and subtraction to demonstrate conceptual understanding of adding and subtracting rational numbers (7.NS.1). After completing Problems 3 and 4 involving representations of expressions using number chips, students are asked to compare the two problems to explain how subtracting a negative number is the same as adding a positive number. “Compare the models in problem 3 and 4. How do they show that $-5-5-(-2)$ is the same as $-5+2$?”

• Unit 3, Lesson 11, Session 3, “Apply It”, Problem 5, students demonstrate conceptual understanding of multiplication of positive and negative integers (7.NS.2). “Think about multiplying two integers. When will the product be less than 0? When will the product be greater than 0?”

• Unit 6, Lesson 29, Session 3, Practice, Problem 1, students demonstrate conceptual understanding by reasoning about geometric shapes with given conditions as they create different triangles using angle measurements or side lengths (7.G.2). “Consider the triangles at the right. a. Are the triangles the same? Explain your reasoning. b. How could you form a different triangle with a $30\degree$ angle, a $40\degree$ angle, and a side with length 5 units?” The text includes an image with two congruent triangles in different orientations with a $30\degree$ angle, a $40\degree$ angle, and a 5-unit side length.

The materials reviewed for i-Ready Classroom Mathematics, Grade 7 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 1, Lesson 6, Session 2, Apply It, Problem 9, students develop procedural skill and fluency by examining the relationship between circumference and area of a circle (7.G.4). “The diameter of a gong is 20 inches. Find the approximate circumference of the gong, using 3.14 for $\pi$ . Then find the exact circumference of the gong. Show your work.”

• Unit 3, Lesson 12, Session 2, Apply It, Problem 7, students develop procedural skill and fluency by extending their understanding of multiplication and division of fractions to rational numbers. “A peregrine falcon dives for prey. Its elevation changes by an average of 211.5 meters per second. The dive lasts for 3.2 seconds. What rational number represents the change in the falcon’s elevation? What does this number mean in the context of the problem? Show your work.” (7.NS.3) 

• Unit 4, Lesson 18, Session 4, Apply It, Problem 9, students develop procedural skill and fluency by rewriting an expression in different forms to shed light on the problem and how the quantities in it are related. (7.EE.2) “Edita says the equations $0.8x=0.8=1.6$ and $\frac{4}{5}(x-1)=1\frac{3}{5}$ are the same. How can she show this, without solving the equations?”

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 2, Lesson 10, Fluency and Skills Practice, Problems 2 and 9, students demonstrate procedural skill and fluency by subtracting positive and negative fractions and decimals (7.NS.1). Problem 2, “$$-8.2-4.2$$” and Problem 9, “$$-8\frac{5}{6}-3\frac{2}{3}$$”.

• Unit 4, Lesson 15, Sessions 4, Apply It, Problem 6, students demonstrate procedural skill and fluency with equivalent expressions by expanding expressions (7.EE.1). “Which expressions are equivalent to $\frac{1}{5}x(5y+60)$? Select all that apply. a. $\frac{1}{5}(2xy+20x+3xy+40x)$  b. $xy+60x$ c. $y+12x$ d. $25y+300x$ e. $13xy$ f. $x(y+12)$.”

• Unit 5, Lesson 20, Session 1, Prepare, Problem 3, students demonstrate procedural skill and fluency through solving multi-step percent problems. (7.RP.3) “Last year, a rapper performed 40 times. This year, the rapper performs 125% of that number of times. a. How many times does the rapper perform this year? b. Check your answers to problem 3a. Show your work.”

The materials reviewed for i-Ready Classroom Mathematics, Grade 7 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 10, Session 3, Develop, Try It, students engage with a routine application problem by adding and subtracting negative and positive numbers (7.NS.1) to solve real-world problems. “Mei celebrates the Lantern Festival with her family. She releases a lantern from a point that is 0.5m below sea level. The lantern rises 913.9m. Then the candle in the lantern goes out. The lantern comes down 925.2m to land on the surface of a lake. What is the elevation of the lake relative to sea level?”

• Unit 5, Lesson 20, Session 2, Develop, Try It, students engage with a routine application problem by using proportional relationships to solve multistep ratio and percent problems (7.RP.3). “Dario borrows $12,000 to buy a car. He borrows the money at a yearly, or annual, simple interest rate of 4.2%. How much more interest will Dario owe if he borrows the money for 5 years instead of 1 year?”

• Unit 7, Lesson 31, Session 3, Develop, Try It, students engage with a non-routine application problem by approximating the probability of a chance event by using data from a previous trial, and predict the approximate relative frequency given the probability (7.SP.6). “One day, Luis sets his music app to play a certain playlist on shuffle. His app tracks how many songs are played in each genre, as shown. The next day, Luis plays the same playlist on shuffle again and this time plays 130 songs. Based on the previous day’s results, predict the number of country songs that will play.” There is a picture included that shows the results of the previous trial (Hip-Hop 5, Pop 9, Rock 12, and Country 14).

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

• Unit 1, Lesson 5, Session 3, Apply It, Problem 3, students demonstrate a routine application problem by identifying the constant of proportionality and reason about the quantities in the problem to write an equation and solve a real- world problem (7.RP.2 and 7.RP.3). “Deyvi goes to a carnival with $20.00. He spends $2.00 to get in and the rest on ride tickets. Each ticket is $1.50. How many tickets does Deyvi buy? a. 9 tickets, b. 12 tickets, c. 13 tickets, d. 14 tickets. Bruno chose C as the correct answer. How might he have gotten that answer?”

• Unit 3, Math in Action, Session 2, Persevere on Your Own, students demonstrate a non-routine application problem by solving a multi-step real-life problem posed with rational numbers in any form including whole numbers, fractions, and decimals (7.EE.3). “Captain Alita’s next flight will travel from Los Angeles to Chicago. Her plane will carry cargo in addition to passengers and their baggage. Look at the information about Captain Alita’s flight and the cargo that needs to be shipped from Los Angeles to Chicago. Decide which cargo should go on Flight 910. Take all volume and weight restrictions into account, and try to carry as much cargo as possible. Make a plan for the cargo that Flight 910 should carry. Include the types of cargo and the number of containers of each type. Show that your plan meets the volume and weight restrictions for the flight. Explain how you decided which cargo to include on the flight.” Given information: Maximum payload (weight of passengers + bags + cargo): 44,700 lb, Weight of passengers + carry-on bags: 28,196 lb, Weight of checked baggage: 3,7,57 lb, Total volume of cargo holds: 1,041 $ft^3$ and volume of checked baggage: 747 $ft^3$. Airline restrictions: flights should carry no more than 80% of their maximum payload and checked baggage travels in the cargo holds, but carry-on bags do not.” There is also a chart included with the type of cargo, number of containers, volume of each container ($$ft^3$$), and weight of each container (lb) for each type of cargo. The values include fractions and decimals.

• Unit 5, Lesson 21, Session 3, Apply It, Problem 6, students demonstrate a non-routine application problem by applying information in different contexts to find percent error (7.RP.3). “The proper air pressure for the front tire of Kimi’s handcycle is 150 pounds per square inch (psi). The percent error in the front tire’s current air pressure is 15%. What are the possible amounts that the front tire’s current air pressure could be? Show your work.” Problem 7, “Jaylen estimates that she will take 8.5 h to read a book. It actually takes Jaylen 10 h to read the book. What is the percent error in Jaylen’s estimate? Show your work.”

The materials reviewed for i-Ready Classroom Mathematics, Grade 7 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, Lesson 8, Session 2, Connect It, Problem 5, students develop conceptual understanding of addition and subtraction of rational numbers  (7.NS.1). “The sum of two integers can be positive, negative, or zero. Why?”

• Unit 4, Lesson 19, Session 2, Apply It, Problem 7, students practice procedural skills and fluency as they construct simple equations and inequalities to solve problems by reasoning about the quantities, (7.EE.4). “The sum of 43.5 and a number, n, is no greater than 50. What are all possible values of n? Show your work.” 

• Unit 5, Lesson 24, Session 3, Practice, Problem 1, students apply informal inferences as they use sample data to compare two populations. (7.SP.4) “Refer to the data in the Example. a) There are about 200,000 dentists in the United States. About how many dentists in the United States can you infer use Mint Madness? Show your work. b) Mint Madness wants to advertise their toothpaste as “chosen by at least 50,000 dentists.” Based on the data, is this claim definitely true? Explain.”

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, End of Unit, Unit Review, Performance Task, students attend to conceptual understanding, procedural skill and fluency, and application as they solve real world problems involving operations with decimal numbers (7.NS.3). “A soap maker tracks costs of olive oil, coconut oil, and sodium hydroxide. The table below shows the cost of each ingredient last week and the change in cost per ounce. Find the new cost per ounce and complete the table. Round all the values to the nearest cent. To make a batch of soap, she needs the following ingredients: 30 oz olive oil, 7 oz coconut oil, 7 oz sodium hydroxide. Each bath makes 14 bars of soap. The soap maker can buy each ingredient in any amount. She will make as many whole batches as possible using this week’s costs and a budget of $150. Choose a sales price for one bar of soap and determine how much profit the soap maker with make this week if she sells all the bars that she makes.” 

• Unit 6, Lesson 25, Session 2, Fluency and Skills Practice, Problem 7, students attend to procedural skill and fluency and conceptual understanding as they solve mathematical problems involving area using unknown side lengths of polygons (7.G.6). In Problem 6, students find the length of one side of an irregular figure when given the area. In Problem 7, students use their understanding of area to apply skills in a new context, “Suppose for problem 6, the unknown side length was the side labeled 34 feet. Could you still solve for x? Explain.”

• Unit 7, Lesson 33, Session 5, Apply It, Problem 8, students attend to conceptual understanding and application as they find probabilities of compound events by designing a simulation to generate frequencies (7.SP.8). “Ellie has a playlist on shuffle. Her list is made of an equal number of country, hip-hop, and pop songs. When three pop songs play in a row, Ellie thinks her music player is broken. Do you agree or disagree? Explain.”

The materials reviewed for i-Ready Classroom Mathematics, Grade 7 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 3, Lesson 13, Session 1, Try It, students rewrite one quantity so they can compare two quantities as fractions or as decimals. “Lupita and Kevin walk to school. Lupita walks $\frac{3}{5}$mi. Kevin walks 0.65 mi. Who walks a greater distance to school? How much greater?” There is a graphic provided with the picture of students walking to school. 

• Unit 4, Math in Action, Session 2, students use multiple strategies to make sense of problems to solve multi-step word problems leading to inequalities of the form $px+q>r$ or $px+q<r$. “Jorge and Liam want to rent a van for their band to use on a tour around Texas. Read through their notes, and help them finalize their plans.” Information included in the problem is “Rental Company Info table (Company, Daily Rate, Fee for Extra Miles, and Van Gas Mileage) for three companies. Other Info: The tour starts and ends in Houston. The drawing shows the distance we will drive each day between cities on the tour. Right now, gas in Texas ranges from $2.39 to $2.63 per gallon. Our budget for renting the van, including gas, is $1,100.”  “What we need to do: Choose a rental company. Determine how many miles we can drive without going over budget for a 5-day tour. Figure out if we can afford to keep Dallas as the last show on our tour or if we should end the tour a day early. If our last show is in Waco, we will drive about 215 miles back to Houston on Day 4.” A map is included showing the route of the 5-day tour with the distances between each city on the tour. In the Reflect section, students discuss how to make sense of the problem. “Make Sense of the Problem - What costs or fees contribute to the total amount the band will pay for the van during the tour?”

• Unit 5, Lesson 20, Session 5, Apply It, Problem 3, students use proportional relationships to solve percent problems. “Which items have the same percent discount? a) sweater and shorts only; b) sweater, shorts, and shirt only; c) jeans and shirt only; d) sweater, shorts, and jeans only. Tokala chose C as the correct answer. How might he have gotten that answer?” Students are given a table with the four items, their original price, and the sale price. 

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 1, End of Unit, Unit Review, Performance Task, students represent proportional relationships symbolically and make sense of relationships between problem scenarios and mathematical representations. “Janice wants to have the interior of her house and office painted. The total area she needs painted is 3,480$$ft^2$$. She wants to choose one company to paint 2,880 ft² at her house and a second company to paint 960$$ft^2$$ at her office. Janice finds pricing information from four different painting companies, shown below.” Information for each company is provided. One provides a table with area painted in square feet (0, 50, 100, 150, and 200) and cost for each, a second company charges $47.00 for every 20 $$ft^3$$, a third company charges $2.80 per square foot, and a fourth provides three examples of an area in square feet and the corresponding cost in dollars. “Write an equation to represent each company’s cost per square foot. Then decide which two companies Janice should choose for the lowest total cost. Finally, calculate Janices’ total cost for having her house and office painted.” Students are provided guidance to help them make sense of the relationships between the equations and the numbers in the problem In the Reflect, Use Mathematical Practices section, “Use Reasoning, How is the information from each company related to the equations you wrote?”

• Unit 2, Lesson 8, Session 2, Try It, students reason about previous understandings of addition and subtraction to add and subtract rational numbers to solve, “Normally, the freezing point for water is $32\degree$F. A city treats its streets before a snowstorm. On the treated streets, the freezing point for water is changed by $-38\degree$F. What is the new freezing point for water on the treated streets?” 

• Unit 4, Lesson 17, Session 1, Model It, Problem 4, students explain what the numbers or symbols in a multi-step equation represent. “Think about the equation $4w-8=32$. a. The value of $4w$ is 40. How do you know that is true? b. The value of w is 10. How do you know this is true?”

The materials reviewed for i-Ready Classroom Mathematics, Grade 7 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 2, Lesson 9, Session 3, Apply It, Problem 2,  students critique the reasoning of a claim involving subtracting integers. “Patrick thinks that when a is a negative integer and b is a positive integer, each of the following statements is always true. Read the statements below and decide whether they are true or false. For statements that are true, give an example to support Patrick’s claim. For statements that are false, give a counterexample.a. $a-b$ is positive. b. $b-a$ is positive. c. $a-(-b)$ is negative.”

• Unit 3, Lesson 12, Session 2, Try It, Teacher’s edition, Differentiation Extend in Model It, provides guidance for teachers to engage students in MP3 as they critique an argument about multiplying integers. “Display this claim: If you can find the product of two positive numbers, then you can find a product that involves one or both of their opposites by factoring -1 from the expression.” A series of questions for the teacher include, “How can you rewrite the expression $-0.32(2.5)$ so that it has a factor of -1? Why can you make your first step in simplifying $-1(0.32)(2.5)$ multiplying the two positive factors? How does this show that the claim is reasonable? How can you show this claim is reasonable when multiplying two negative numbers?”

• Unit 5, Lesson 22, Lesson Quiz, Problem 3, students construct an argument about the reasonableness of a conclusion based on a random sample. “Alissa surveys a random sample of 50 students at her school about the country they would most like to visit. The table shows her results. Based on the sample, can Alissa conclude that there are probably fewer students at her school who want to visit Japan than Australia? Explain your reasoning.” A table with the data collected from the random sample is included.

• Unit 6, Lesson 25, Session 1, Teacher Edition, Connect It, Facilitate Whole Class Discussion, provides guidance for teachers to help students reason about problem solving strategies. “Call on students to share selected strategies. As they listen to their classmates, have students evaluate the strategies and agree and build on them. Remind students that one way to agree and build on ideas is to give another example.” 

• 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 7 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 4, Lesson 19, Session 4, Teacher’s Edition, Differentiation: Extend, provides guidance for teachers to engage students in MP4 as they discuss using inequalities to model situations. The Model It shows students how to model a situation using an inequality. “Prompt students to recognize how problem statements can be expressed as mathematical statements and provide information for interpreting the situation. Ask: Why is the situation represented by an inequality. Listen For: There is more money than the cost for the possible number of stoles that Cameron can buy. He can have money left over. Ask: If Cameron did not buy the frame, what inequality could model the problem? Listen For: The problem could be modeled by the inequality $24x<200$. Ask: Why is the solution to this problem only integers and not other rational numbers? Listen For: Cameron can only buy whole numbers of stoles. Generalize: Encourage students to describe how they might choose an appropriate model when solving a problem. If the solution is a single value, they might choose to model the problem with an equation. If the solution allows multiple values for the solution they might choose to model the problem with an inequality. If they represent the solution on a number line, they have to consider which values are acceptable for the situation.”

• Unit 5, Lesson 23, Session 2, Teacher Edition, Differentiation: Extend, gives teacher guidance for supporting students to consider how they can use double number lines to model data. “Prompt students to think about changing the model to answer different questions. Ask: How could you use the model to make an inference about a similar population with a different number of total students?...How could you change the model to find the number of students who take the bus?...How could you change the model to find the number of students who do not take the subway?”

• Unit 6, Lesson 26, Session 2, Try It, students model using the volume of a right square prism to find the volume of a right triangular prism. “Troy uses colored sand to make sand art. The storage container for his sand is shaped like a right square prism. He pours some of the sand into a display container shaped like a right triangular prism. When he is done, the height of the sand left in the storage container is 4 in. What is the height of the sand in the display container?” A storage container with a remaining cube of sand is pictured, along with an empty display container with given length and width dimensions. 

• Unit 7, End of Unit, Unit Review, Performance Task, students design a probability model. The text presents an online game where students click a button to determine their next move in the game. The moves are: go forward 1 space, go backward 2 spaces, lose a turn, and take another turn. “Delara plays the game and notes the actions she takes for 350 turns. Find the experimental probability for each action in the online game using the table below.” A table with the data is included. “Delara wants to create her own version of the game. For her version, she wants to determine the action for each turn by using either a spinner with 12 equal parts or a deck of 40 cards. She wants the theoretical probability for each action in her version to be similar to the experimental probability for each action in the online version. Determine the number of spaces on the spinner and the number of cards for each action that Delara can use to make her version. Use your data to describe whether Delara’s version should use a spinner or a deck of cards to match the experimental probabilities of the online version as closely as possible. Explain your reasoning.”

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 1, Lesson 2, Session 2, Connect It, Problem 6 engages students in MP5 as they reflect on the models and strategies in the Try It to find and compare unit rates associated with ratios of fractions. “Think about all the models and strategies you have discussed today. Describe how one of them helped you understand how to solve the Try It problem.” The teacher’s edition includes guidance to teachers, “Have all students focus on the strategies used to solve the Try It. If time allows, have students discuss their ideas with a partner.”

• Unit 2, Lesson 8, Session 1, Try It, students have a selection of tools to choose from to solve problems involving adding integers. “The temperature at a mountain weather station is -3℉ at sunrise. Then the temperature rises $5\degree F$. What is the new temperature?” The math toolkit includes: grid paper, integer chips, number lines.

• Unit 4, Lesson 18, Session 2, Try It, students select a tool to solve a problem resulting in a multi-step equation. “Noah is designing a set for a school theater production. He has 150 cardboard bricks. He uses some of the bricks to make a chimney. To make an arch, he needs to use 4 times as many bricks as he uses for the chimney. He also saves 15 bricks in case some get crushed. How many cardboard bricks can he use to make the arch?” The Math Toolkit suggests algebra tiles, grid paper, number lines, and sticky notes. 

• Unit 5, Lesson 23, Session 3, Teacher’s Edition, Differentiation: Extend, provides guidance for teachers to engage students in MP5 as they discuss using data displays to make inferences from a sample. In Analyze It, there is an example of how to use dot plots and box plots to determine mean and median to make inferences about data. “Prompt students to think about what information can be understood about the data set by using different displays; such as a dot plot and a box plot. Ask: How does each plot show outliers in the data set? Listen For: In the dot plot, there will be numbers that have few or no dots between the main set of data and a number at either end that has dots. The box plot shows a longer whisker between the box and the endpoint when there is an outlier. Ask: What aspects of the data set are easier to see in the dot plot? In the box plot? Listen For: The number of data points, the symmetry of the data, and the data points that occur most often are easier to see in a dot plot. The median number in the data and the way the data are distributed are easier to see in a box plot. Generalize: Encourage students to describe when they might choose each plot to model and solve a problem. Knowing how the data will be analyzed to solve the problem will help them determine which model would be easier to use to find that information.”

The materials reviewed for i-Ready Classroom Mathematics, Grade 7 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 3, Math in Action, Session 2, students reflect as they use precise units to describe a plane’s descent. “Why is it important to label each part of your final solution with units?” 

• Unit 5, Lesson 23, Session 2, Practice, Problem 1, the materials attend to the specialized language of mathematics as students complete problems about making inferences from samples about populations. “Kosumi conducts another survey of students in the school in the Example. This time, he surveys a random sample of 30 students. a. In Kosumi’s sample, 24 students plan to vote for Garrett. Based on this sample, about how many students in the school should Garrett expect to vote for him? Show your work.” A vocabulary box includes “random sample” with the definition.

• Unit 6, Lesson 25, Session 1, Prepare, Problem 2 asks students to attend to precision when evaluating the reasonableness of an expression to find surface area. “Muna claims that the expression $(8)(16)+(8)(12)+(16)(12)$ represents the surface area, in square inches, of the right rectangular prism shown. Is Muna correct? Explain.” 

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 4, Session 2, Teacher Edition, Discuss It, provides guidance for teachers to support students in attending to precision by correcting a common misconception. “Listen for students who misinterpret the meaning of a specific point in the graph, such as interpreting the point (3,9) as 9 pounds of peppers that cost $3. As students share their strategies, ask them to name the point on the graph using both the value and the unit: 3 pounds of peppers at a cost of 9 dollars. Have students discuss the meaning of the value, the unit, and the point.”

• Unit 4, Lesson 18, Session 2, Teacher Edition, Develop Academic Language, teachers are provided with guidance to attend to the specialized language of mathematics by developing understanding of the phrase isolate the variable. “In the second Model It, students explore solving an equation by isolating the x-term. Ask students to use prior knowledge to give a rough definition for isolate. Provide the synonym for separate. Read the second Model It and have students turn and talk with a partner about the steps used to isolate the x-term.”

• Unit 5, End of Unit, Vocabulary Review, Problem 1, provides practice with specialized mathematical vocabulary. Students are provided with a word bank of 14 math and academic vocabulary from the unit including commission, percent decrease, markup, and simple interest. “Use at least three math or academic vocabulary terms to describe a year-end sale at a store. Underline each term you use.”

The materials reviewed for i-Ready Classroom Mathematics, Grade 7 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 9, Session 2, Teacher Edition, Close: Exit Ticket (with Connect It, Problem 5), provides teachers with guidance using structure to correct a common misconception around subtraction of negative numbers. “If students think that a subtraction problem with negative numbers always results in a negative answer, then have them simplify $-4-4-(-3)$ and $-3-(-4)$ and compare the answers. 

• Unit 3, Lesson 11, Session 1, Model It, Problems 4, 5, and 6 students use patterns to generalize rules for multiplying integers. Problem 4 asks students to complete three sets of equations. Sets a, b, and c multiply the numbers 3, 2, 1, 0, -1, -2, and -3 by 2, 3, and 4, respectively. Problem 5, a. $-4(2)=$____, b. $2(-4)=$____, c. Does the order of the factors change the product when multiplying negative integers? Justify your answer.” Problem 6, “You have explored how to multiply two integers when one is positive and the other is negative. Is the product of a positive integer and a negative integer always positive or negative? Explain?”

• Unit 4, Lesson 18, Session 3, Connect It, Problem 5 students explain the structure within algebraic equations to a strategy for solving. “Consider the equation $12=b(2.5x+15)$. What values of b might make you want to start solving the equation by distributing b? What values of b might make you want to start solving the equation by dividing by b?”

• Unit 6, Lesson 29, Session 2, Fluency and Skills Practice, Problem 13, students make use of structure in order to determine if, and how many, triangles could be constructed. “A triangle has side lengths of 7 cm and 18 cm. If the length of the third side is a whole number, how many possible triangles are there? Explain your answer.” 

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

• Unit 1, Lesson 6, Session 2, Try It students notice repeated calculations to make generalizations about the relationship between the circumference of a circle and its diameter . “Look at the circumference, C, of each of the circles below. What do you think would be the circumference of a circle with diameter 1 cm?” There are four circles with the circumference and diameter labeled: D: 2 cm, C$$\approx$$6.28 cm, D: 3 cm, C$$\approx$$9.42 cm, D: 4 cm, C$$\approx$$12.56 cm, D: 2.5 cm, C$$\approx$$15.70 cm

• Unit 3, Lesson 13, Session 3, Differentiation: Extend, provides guidance to support students in using repeated reasoning to understand repeating decimals. “Prompt students to analyze remainders to understand why some rational numbers can be expressed as repeating decimals. Ask: When dividing by 7, what non-zero remainders can you get in any step? Why?...Can you stop dividing after you bring down a zero for any of these remainders? How does this tell you it will be a repeating decimal?...How do you know that when you express a mixed number or fraction with a denominator of 7 as a decimal it will be a repeating decimal?”

• Unit 6, Lesson 26 , Session 3, Teacher Edition, Differentiation: Extend, Teachers are prompted to ask students to apply repeated reasoning to finding the volume of different figures. Ask: “What type of prism is the storage bin? How do you know? …Alita wants to design a storage bin in the shape of a rectangular prism. How can she make the volume the same in both designs?…Generalize: Guide students to discuss how the shape of a prism affects its volume.”

• Unit 7, Lesson 33, Session 3, Teacher’s Edition, Differentiation: Extend, provides guidance to engage students in repeated reasoning to make generalizations about finding the sample space for a compound event: guessing a 4-digit passcode that only uses 0 and 1 on the first try. “Prompt students to identify the structure and patterns in the list of possible passcodes. Ask: What pattern do you see in the outcomes with three 0s? The outcomes with two 0s?...How does using patterns help you find the sample space for a compound event?...There are 16 possible passcodes. How could you use multiplication to show this number?”

The materials reviewed for i-Ready Classroom Mathematics, Grade 7 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 of 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 1, Lesson 4, Session 1, Teacher Edition, Discuss It, “Support Partner Discussion: After students work on Try It, have them respond to Discuss It. Listen for understanding of the relationship between distance and time and the use of ordered pairs to model a relationship between quantities. Facilitate While Class Discussion: Guide students to Compare and Connect the representations. If discussion lags, ask students to turn and talk about the ways they found the distances that Fiona travels.”

• Unit 2, Lesson 9, Session 1, Teacher Edition, Model It, provides Differentiation guidance in a Re-teach or Reinforce sidebar after students have solved problems including adding and subtracting with negative integers. “If students are unsure that subtracting a negative integer has the same result as adding the opposite of that integer, then use this activity to reinforce understanding of the concept.” The activity uses integer chips to model. 

• Unit 4, End of Unit, Teacher Edition, Unit Review, Problem 1 includes teacher notes to support assessing student knowledge. “A, B, D are correct. Students could solve the problem by writing an equation for the perimeter of the square and comparing the different forms of the equations to find which are equivalent.” The materials go on to explain why the other answer choices are incorrect. 

• Unit 5, Lesson 22, Session 2, Teacher Edition, Develop, Discuss It, students work with random sampling. Teachers are asked to support partner discussion. “Support Partner Discussion: After students complete problems 1 and 2, have them respond to Discuss It with a partner. Support as needed with questions such as: Why is it important that every member of the population has an equal chance of being selected? If some individual or group did not have a chance of being selected, how would that affect the resulting sample? Facilitate Whole Class Discussion: Ask why are random samples likely to be representative of the population?”

The materials reviewed for i-Ready Classroom Mathematics, Grade 7 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 7, Lesson 33, Lesson Overview, Teacher Edition, Solve Problems Involving Compound Events - Full Lesson, Learning Progression:

  • “Earlier in Grade 7, students determined experimental probability by identifying the favorable outcomes among a set of trials. They also found the theoretical probability of simple events by identifying the sample space for probability models such as spinners, coin tosses, and number cubes. They compared experimental probability with theoretical probability and explained why they might differ. 

  • In this lesson, students extend their understanding of probability by finding the theoretical probability of compound events. They also describe, analyze, and conduct simulations to find the experimental probabilities of compound events and explore both the usefulness and limitations of simulations for identifying probabilities. 

  • In later grades, students will describe events as subsets of a sample space, using characteristics of the outcomes, or as unions, intersections, or complements of other events. They will recognize that when two events are independent, the probability of the events occurring together is the product of their probabilities, and they will use permutations and combinations to find probabilities of compound events.”

• 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 1, Beginning of Unit, Understanding Content Across Grades related to Lesson 1:

  • Prior Knowledge, “Insights on: Understanding Multiplication as Scaling. Resizing a quantity (growing or shrinking) by multiplication is called scaling… Common Error - Students may assume that when they multiply the answer will always be a greater number. Experience with visual models of scaling situations will help them see that multiplying by a fraction means taking a fractional part of the starting amount. If you multiply by a fraction less than 1 whole, you will find a part less than the whole of the starting amount.” This information is included with a visual model of scaling. 

  • Current Lesson, “Insights on: Scale and Scale Drawings. A scale describes the relationship between lengths in the original figure and lengths in the scale drawing. A scale factor is the number you multiply an original length by to the corresponding length in the scale drawing.” This information is included with a visual example of scale drawings and scale factors. 

  • Future Learning, “Insights on… Representing Proportional Relationships. Building on Grade 6 work with rations, students learn that when the ratios of corresponding values of two quantities are equivalent, the quantities have a proportional relationship… Making connections between tables and graphs of a proportional relationship helps students understand why the graph of a proportional relationship must pass through (0,0) and (1, r), where r is the unit rate.”

• 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 4, Lesson 18, Solve Equations prerequisite:

  • “When solving equations with variables, students need to be able to explain to themselves the meaning of the variable in the context of the problem. The understanding will then help them make sense of the problem, solve it, and then assess the reasonableness of their solution… This “what number goes here” approach to solving equations with variables is a critical first step to being able to solve more complex equations using a process of performing operations on both sides of the equal sign.”

  • “Step by Step: 1) Write an addition equation. Instruct the student to write a numerical addition equation on a piece of paper but not show it to you. Have the student replace one of the addends with a variable and write the new equation (which should now be in the form $x+p=q$ or $x+p=q$) on another piece of paper to give to you. 2) Solve the Equation. (followed by two prompts) 3) Write a multiplication equation. (followed by two prompts) 4) Solve the Equation. (followed by two prompts) 5) Take turns writing and solving. (followed by two prompts) 6) Challenge. Consider including fractions or decimals in the equations. (followed by two examples)”

  • “Check for Understanding: Have the student solve the following problems and explain the steps. Make sure the student explains their thinking. $x+13=20$; $7x=63$” Then an error analysis chart is provided: “If you observe… the student may… Then try…”

The materials reviewed for i-Ready Classroom Mathematics, Grade 7 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 1, Beginning of Unit, Unit and Lesson Support, the opening narrative provides the content of the unit, “In this unit, students build on the idea of equivalent ratios to understand and calculate scale factors and make scale drawings. They use prior knowledge of unit rates and division with fractions to understand, interpret, and represent proportional relationships in tables and graphs. They go on to solve ratio problems. Finally, students derive formulas for circumference and area of a circle using proportional reasoning to understand the relationship between them, including the constant of proportionality, pi, and use them to solve problems.” The document continues with Instructional Support identifying specific lessons from prior grades to develop understanding, such as Unit 1, Lessons 1-2, “These lessons build on students’ understanding of equivalent ratios and unit rates from Grade 6, Units 3 and 4: Grade 6, Lesson 13 Find Equivalent Ratios.”

• 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 17, Overview, Learning Progression, “In Grade 6 students learned to solve one-step equations involving positive rational numbers using hanger diagrams, reasoning, and algebraic steps… Earlier in Grade 7 students learned how to add, subtract, multiply, and divide negative rational numbers… In this lesson students use hanger diagrams and reasoning to understand the relationship between a variable and other quantities in an equation… In the next lesson students build on their understanding of the structure of equations that have two addends or two factors to use algebraic steps to solve equations in the form px + q = r and p(x+q) = r… In Grade 8 students will build on this reasoning to solve equations with variables on both sides of the equal sign and to solve systems of equations.”

The materials reviewed for i-Ready Classroom Mathematics, Grade 7 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 Classroom Mathematics, Grade 7 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 1, “Materials tab: Math Toolkit base-ten blocks, fraction bars, hundredths grids, number lines, place-value charts, Presentation Slides. Differentiation tab: 2 copies of Activity Sheet Hundredth Grids.”

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 base 10 blocks (17 ten rods, 10 ones units) - 1 per pair.”

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 7 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 2, Lesson 7, Lesson Quiz, Problem 2, “DOK 1, 7.NS.A.1a, SMP 4.”

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, complete 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 4, End of Unit, Assess, Unit Assessment, Form A, Scoring Guide, Problem 6, “DOK 2, 7.EE.B.4, SMP 3.”

• Unit 6, End of Unit, Assess, Unit Assessment, Form A, Scoring Guide, Problem 12, “DOK 2, 7.G.A.2, SMP 2.”

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 6, End of Unit, Assess, Comprehension Check Correlation Guide, Problem 12, “DOK 2, 6.G.B.6.”

The materials reviewed for i-Ready Classroom Mathematics, Grade 7 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 6, Lesson 26, Assess, Lesson Quiz, Problem 2, “C is correct. Students could solve the problem by dividing the right prism into two rectangular prisms and one triangular prism, and then finding the sum of the volumes. (work shown) A is not correct. This answer represents the volume of a prism with the same base but a height of 4 in. B is not correct. This answer represents the confusion between the volume and the surface area of a figure. The surface area of the figure is 444 in$$^2$$. D is not correct. This answer represents the sum of the volumes of a rectangular prism with a 4 in-by-7in base and a height of 5 in, and an enclosing rectangular prism with an 8 in-by-10 in base and a height of 5 in.”

  • Unit 7, End of Unit, Assess, Unit Assessment, Form A, Problem 13, “Students could also calculate each probability and compare the results. The probability of selecting blue or yellow paint is $\frac{1}{2}$, and the probability of selecting green or red paint is $\frac{1}{2}$.”

• 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 7, Lesson 33, 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 33, Compound Events. The Reinforce section directs teachers back to Lesson 33, Compound Event Bingo. The Extend section directs teachers back to Lesson 33, Design a Simulation.

• Unit 7, 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, Understand Probability (Lesson 30), Probability Experiment (Lesson 31), Compare Theoretical and Experimental Probabilities (Lesson 32), Compound Events (Lesson 32)””For students who exceed proficiency on the Unit Assessment, choose from the following activities on the Teacher Toolbox.””Extend: Enrichment Activities, Fair Game (Lesson 30), Buffon’s Needle (Lesson 31), Try, Try, Try Again (Lesson 32).”

The materials reviewed for i-Ready Classroom Mathematics, Grade 7 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 7 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 7 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 1, Lesson 4, Extend, Constant Graphing, students are provided with a challenge question, “Can you find the constant of proportionality and the equation of a proportional relationship that passes through a given point on a graph?” followed by analysis tasks such as, “What do you notice about the line that passes through points D and F? Explain what this means.”

• 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 2, Lesson 8, Session 4, Teacher Guide, Differentiation: Extend, Challenge extends student thinking at the end of a lesson on adding with negative numbers, with a multi-step addition problem. “Students extending beyond proficiency will benefit from solving multi-step problems involving unknown addends. Describe this problem: An airplane flies 82.5 ft above sea level. It descends about 3 ft and then rises about 8 ft. Its new height is 87.1 ft above sea level. What could have been the two exact changes in height? ...Have students propose and then solve similar problems.”

The materials reviewed for i-Ready Classroom Mathematics, Grade 7 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 7 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, base ten blocks, number cubes, geometric solids, 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 fraction tile sets, compasses, geoboards, metersticks, transparent circle spinners, and transparent counters. 

• Program Implementation, Manipulative List by Lesson has specific manipulatives listed for each lesson. For example, Unit 5, Lesson 24 lists 1 yardstick, 40 counters per pair. 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. 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.”