7th Grade - Gateway 2

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. “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\degreeF. A city treats its streets before a snowstorm. On the treated streets, the freezing point for water is changed by -38\degreeF. 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?”