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 gets 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”, Problem 1, students develop the conceptual understanding of describing probabilities involving numbers (7.SP.5). Students first describe events in words. “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.

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 develop 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-(-2) is the same as -5+2?

• Unit 3, Lesson 11, Session 3, “Apply It”, Problem 5 is an opportunity for students to independently engage in writing while developing 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 reason about geometric shapes with given conditions  as they create different triangles using the 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 5-unit side length?” 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 between the two angles.”

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 examine the relationship between circumference and area of a circle (7.G.4). Students solve, “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 extend their understanding of multiplication and division of fractions to rational numbers as they solve, “A peregrine falcon dives for prey. Its elevation changes by an average of -11.5 meters every second. The dive lasts for 3.2 seconds. What is the change in the falcon’s elevation? What does this mean in the context of the problem? Show your work.” (7.NS.3) 

• Unit 4, Lesson 18, Session 4, Apply It, Problem 9, students rewrite an expression in different forms to shed light on the problem and how the quantities in it are related. (7.EE.2) “Damita 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, Fluency and Skills Practice, Problem 2 and 6, students subtract positive and negative fractions and decimals (7.NS.1).  Problem 2 “-8.2-4.2” and Problem 6, “$$7\frac{3}{4}-4\frac{1}{4}$$”

• Unit 4, Lesson 15, Sessions 4, Apply It, Problem 6, students explore 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. 25xy+300x  e. 13xy f. x(y+12).”

• Unit 5, Lesson 20, Session 1, Practice, 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, Try It, routine problem, students add and subtract negative and positive numbers (7.NS.1) to solve real-world problems. “Mel releases a lantern for the Lantern Festival. She stands in a field 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, Try It, routine problem, students use 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, Try It, non-routine problem, students approximate 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). “Luis sets his music app to play a certain playlist on shuffle. His app tracks the genre of each song played. Luis plays the same playlist on shuffle again and this time plays 130 songs. Based on the previous 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, Refine, Problem 3, routine problem, students identify 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.”

• Unit 3, Math in Action, Session 2, non-routine problem, students solve 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.” The book includes the following information for students to use: 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 $$ft^3$$. “The airline restrictions are that 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, non-routine problem, students apply information in different contexts to find percent error (7.RP.3). “The proper air pressure for Caitlin’s bicycle tire is 30 pounds per square inch (psi). The percent error in Caitlin’s current tire pressure is 15%. What are the possible current tire pressures for Caitlin’s tires? Show your work.” Problem 7, students find the percent error, “Jaime estimates it will take 8.5 hours to read a book. It actually takes Jaime 10 hours to read the book. What is the percent error in Jaime’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, Problem 5, Fluency Skills & Practice contains multiple problems for students to add integers (7.NS.1). “Find each sum -13 + 7.”

• Unit 2, Lesson 9, Session 1, Model It, Problem 5, students develop 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 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-(-2) is the same as -5+2?”

• Unit 4, Lesson 19, Session 2, Apply It, Problem 7, students engage in application 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.” In the teacher notes, it suggests to encourage students to use a table, number line or other visual model to support their thinking and in particular for teachers, “Students should recognize that the phrase no greater than is represented by a less than or equal to symbol. A sum that is not greater than 50 could either equal 50 or be any value less than 50.”

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

• Unit 3, Lesson 13, Lesson Quiz, Problem 5, students attend to procedural skill and fluency and application as they apply properties of operations to calculate with numbers in any form and convert between forms as appropriate (7.EE.3). “A cooler contains 4L of water. The cooler has marks on it at every 02. L. Water bottles are filled with water from the cooler, and each bottle holds approximately \frac{4}{9}L. After 4 water bottles are filled, between which two marks is the water level in the cooler? Show your work.”

• Unit 4, Lesson 18, Session 4, Apply It, Problem 9, students attend to procedural skill and fluency and conceptual understanding as they compare algebraic solutions and identify the sequence of the operations used in each approach, (7.EE.4). “Damita 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?”

• Unit 6, Lesson 25, Session 2, Fluency and Skills Practice, Problem 7, students engage with conceptual understanding and application 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 a 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.”

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

The Table of Contents and the Lesson Overview both include the Standards for Mathematical Practice for each lesson. In the Student Worktext, the Learning Target also highlights the MPs that are included in the lesson. MP1 and MP2 are identified in every lesson from 1-33.  

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

• Unit 2, Lesson 7, Session 3, Try It, students make sense of a problem in order to make a generalization about triangles. “Jorge wants to draw two triangles that have the same angle measures and are not similar. Carlos says that is not possible to do. Make or draw two triangles that have the same three angle measures but different side lengths. Are the triangles similar?”

• Unit 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 went 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. Each distance includes how far we will drive to reach each city and other stops we will make. 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 deep 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?”

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

• Unit 1, 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 ft². She wants to choose one company to paint 2,880 ft² at her horse and a second company to paing 960 ft² 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 ft³, 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.” In the Discuss It routine, students are prompted with a question and a sentence frame to discuss their reasoning with a partner. Teachers are further provided with guidance to support partners and facilitate whole-class discussion. Additionally, fewer problems in the materials ask students to critique the reasoning of others, or explore and justify their thinking.

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

• Unit 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, provides guidance for teachers to engage students in MP3 as they critique an argument about multiplying integers . “Have students consider this claim: If you know how to find the product of two positive numbers, then you can find the product of related negative numbers by factoring out -1.”  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 made 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, 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.” 

• Unit 7, Lesson 30, Session 2, Develop, Model It, students answer, “How do you know an event is equally likely as not?” Students justify their reasoning with other students. “A bag contains 6 red marbles, 6 green marbles, and 12 blue marbles. Paloma reaches into the bag and selects a marble without looking. a. What is the total number of marbles in the bag? b. What is half the number of marbles in the bag? c. Name an event that is possible. d. Name an event that is unlikely. e. Name an event that is equally likely as not. f. Name an event that is likely. g. Name an event that is certain.” Students are then asked: “How do you know an event is equally likely as not?”

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 generally identify MP4 and MP5 in most lessons and can be found in the routines developed throughout the materials. 

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

• Unit 4, Lesson 19, Session 4, Model It, 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 is showing 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? List 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, Deepen Understanding, 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, Unit Review, Performance Task, students design a probability model. The text presents an online game, Downtown, 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 notes the actions she takes for 350 turns. Find the experimental probability for each action in Downtown 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 it’s full intent in connection to grade-level content.” Many problems include the Math Toolkit with suggested tools for students to use. Examples include:

• Unit 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℉. 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 needs to use some of the bricks to make a chimney and 4 times as many bricks to make an arch. He also saves 15 bricks in case some get crushed. How many cardboard bricks can he use to make the arch?”

• Unit 5, Lesson 23, Session 3, Analyze It, 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. The Analyze It section shows students 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 it’s full intent in connection to grade-level content. Many problems present students with the opportunity to attend to precision within the mathematics and the reasoning of the answer. Examples include: 

• Unit 3, Session 2, Math in Action, students are asked to reflect as they work through the problem. “Why is it important to label each part of your final solution with units?” The students engage in MP 6 while realizing that being precise with units is important in solving the problem.

• Unit 5, Lesson 23, Session 2, Additional Practice, Problem 1, the materials attend to the specialized language of mathematics as students complete problems about making inferences from samples about populations. “Jacob conducts another survey of students in the school in the Example. This time, he surveys a random sample of 30 students. a. In Jacob’s sample, 24 students say they will 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, Additional Practice, 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 attend to the specialized language of mathematics. The materials use precise and accurate mathematical terminology and definitions, and the materials support students in using them. The Collect and Display routine is described as, “A routine in which teachers collect students' informal language and match it up with more precise academic or mathematical language to increase sense-making and academic language development.” Teacher’s guides, student books, and supplemental materials explicitly attend to the specialized language of mathematics. Examples include:

• Unit 1, Lesson 4, Session 2, 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, 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, Vocabulary Review, Problem 1, provides practice with specialized mathematical vocabulary. Students are provided with a word bank of 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, Close, provides teachers with guidance around MP7 to correct a common misconception around subtraction of negative numbers using structure. “If students think that a subtraction problem with negative numbers always results in a negative answer, then have them simplify -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 it’s 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 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, C6.28 cm, D: 3 cm, C9.42 cm, D: 4 cm, C12.56 cm, D: 2.5 cm, C15.70 cm

• Unit 3, Lesson 13, Session 3, Deepen Understanding, provides teachers with guidance to support students in applying 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, 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?” Teachers can 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 for teachers to engage students in MP8 as they notice repeated reasoning to make generalizations about finding the sample space for a compound event. Try It, “Lucia has a four-digit passcode on her phone. You know her code only uses the digits 0 and 1. What is the probability of guessing her passcode on the first try?” Model It shows an organized list to model the sample space. It is organized by numbers with four zeros, three zeroes, two zeros, one zero, and zero zeros. The Differentiation Extend provides guidance for the teacher to help students understand the patterns in the list. “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? Listen For In the line for the three 0s, the 1 starts in the last place and then moves to the left one digit at a time. In the line for two 0s, the 0s start in the first two places. The second 0 moves to the third and then the fourth place. Then the first 0 moves one digit to the right and the process repeats. Ask How does using patterns help you find the sample space for a compound event? Listen For It gives you a way to check that you have found all the possible outcomes. Ask There are 16 possible passcodes. How could you use multiplication to show this number? Listen For There are two possible digits, 0 and 1, in each of the four places in the passcode. So the total number of passcodes is 2×2×2×2 or 16.”