6th Grade - Gateway 2

The materials reviewed for i-Ready Classroom Mathematics Grade 6 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 9, Session 2, “Connect It”, Problem 4, Understand Division with Fractions, students develop conceptual understanding by using and comparing models of dividing fractions by fractions (6.NS.1).  “How can you show the quotient $$\frac{4}{6}÷\frac{1}{6}$$ with a bar model? How is using a bar model similar to showing the quotient with a number line? How is it different?”

• Unit 5, Lesson 19, Session 4, “Try It”, is an example of an opportunity for students to develop conceptual understanding with teacher guidance. “Which of these three expressions are equivalent? $$3(x+2)+2x$$; $$2+4(x+1)+x$$; $$2(3+3x)-2x$$.” The teacher guide provides guidance for teachers to facilitate discussion and student connections to using properties of operations to rewrite expressions (6.EE.3). Students are prompted to explain the steps used to rewrite the expression. “How does each expression change from one step to another? Why is each step necessary?”

• Unit 6, Lesson 25, Session 1, “Model It”, Problem 1, students develop conceptual understanding of absolute value of rational numbers (6.NS.7) by describing distances of objects below and above sea level. “A scientist standing on the deck of a boat uses a drone, and a scuba diver uses a camera to explore a sea cave. The table shows the elevations of four objects relative to sea level. (The table shows the following objects and their elevations: Camera -20 ft., Cave floor -30 ft., Drone 20 ft., Boat deck 5 ft.) a. Use the number line to show the elevations of the objects from the table. Label each object at its elevation. b. Are any objects the same distance from sea level? If so, how far from sea level are they? c. Another object is 3 ft from sea level. Is the object’s elevation positive, negative, or could it be either? Explain.”

The materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade with the use of visual models, real world connections, mathematical discussion prompts, concept extensions, and hands-on activities. Examples include: 

• Unit 2, Lesson10, Session 3, “Apply It”, Problem 9, students develop conceptual understanding of dividing fractions by fractions in real world situations (6.NS.1). “It takes Francisco $$\frac{5}{6}$$ minute to upload a video to his blog. How much of one video can he upload in $$\frac{1}{2}$$ minute? Show your work.” 

• Unit 3, Lesson 12, Session 1, “Model It”, Problem 4, students independently engage in writing while developing conceptual understanding of ratio relationships (6.RP.1). Students explain the difference and demonstrate their understanding of units in ratios. “Explain how the ratios of 5 tacos for every 2 guests and 2 tacos for every 5 guests are different. Include a model in your explanation.” 

In Unit 6, Lesson 23, Session 2, “Model It”: Vertical Number Lines, Problem 3, students develop conceptual understanding of negative numbers and plot numbers and their opposites on a number line (6.NS.5 and 6.NS.6). “a. Use a rational number to label each point on the number line. b. What is the opposite of each number you wrote on the number line? c. Plot points at -1.75 and $$-\frac{3}{4}$$.”

The materials reviewed for i-Ready Classroom Mathematics Grade 6 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 5. Session 2, Apply It, Problem 9 provides the opportunity for students to develop procedural skill and fluency with evaluating numerical expressions with whole number exponents (6.EE.1) with teacher guidance. “On the Luck Five game show, Troy wins $5 if he answers one question correctly. Each time he answers another question correctly without making a mistake, the amount of money he wins is multiplied by 5. Troy answers 6 questions correctly without making a mistake. His winnings are represented by the expression $$5^6$$. How much money does Troy win? Show your work.” In the teacher’s edition, teacher’s are directed to provide guidance to the students if they cannot evaluate the expression correctly. “If student add six 5s and get $30, then ask them to think about the difference between $$5^6$$ and 6(5). Ask if they can rewrite each expression in a different way that uses six 5s. Elicit the fact that $$5^6$$ is $$5×5×5×5×5×5$$, or the product of six 5s, and 6(5) is 5 + 5 + 5 + 5 + 5 + 5, or the sum of six 5s.”

• Unit 2, Lesson 7, Session 3, Practice Problem 1, “A green rope is 60.5 m long. Each meter of the rope has a mass of 0.052 kg. What is the total mass of the green rope? Show your work.” Problem 2, “Find 0.102 × 7.3. Show your work.” (6.NS.3)

• Unit 4, Lesson 18, Session 2, Fluency Skills and Practice contains multiple problems for students to find percent of a number. (6.RP.3c) In Problem 6, students “Find the percent of the number. 75% of 80.”

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 1, Lesson 5, Session 4, Apply It, Problem 4, students evaluate multiple expressions involving whole-number exponents (6.EE.1). “Which expressions have a value of 100 when m=5? Select all that apply. $$2m^2+50$$; $$(2m)^2+50$$; $$(m+5)^2$$; $$m^3÷5⋅4$$; $$4m^2$$; $$(4m)^2$$.”

• Unit 2, Lesson 8, Session 2, Practice, Problem 2, students fluently divide multi-digit numbers using the standard algorithm. (6.NS.2) “Platon’s mom buys a car using a loan. She repays the loan by paying $22,032 in 48 equal monthly payments. How much is each payment? Show your work.” 

• Unit 5, Lesson 21, Session 4, Practice, Problem 5, students solve multi-digit division problems using the standard algorithm. (6.NS.2) “Neva is training for a race. This week, she bikes 5.5 times as far as she runs. Her total distance running and biking this week is 26 mi. How far does Neva run this week? Show your week.”

The materials reviewed for i-Ready Classroom Mathematics Grade 6 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, Math in Action, Session 2, Discuss Models and Strategies, non-routine problem, students find the volume of right rectangular prisms with fractional edge lengths by packing it with unit cubes with fractional edge lengths (6.G.2). “Alberto wants to set up an aquarium as a demonstration of freshwater ecosystems for the science club. Read the information he finds about aquarium ecosystems. Then suggest a tank, a number of guppies, and an amount of gravel for Alberto to use to set up his ecosystem.” 

• Unit 4, Lesson 16, Session 2, Try It, routine problem, students use unit rates (6.RP.3) to solve real-world problems. “Aswini jogs on the track at her school. She uses a watch to track her progress. At this rate, how long will it take her to jog 16 laps?” There is a picture of a watch that says “15 minutes, 6 laps.” 

• Unit 7, Lesson 30, Session 2, Try It, non-routine problem, students use a set of data to answer a statistical question (6.SP.2). “Elizabeth records the number of points her favorite basketball team scores in each game. She predicts that the team will score about 120 points in its next game. Is Elizabeth’s prediction reasonable? Display Data Set: Points Scored in a way that supports your answer.” There is a picture of the data set included.

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

• Unit 2, Lesson 11, Session 4, Apply It, Problem 9, non-routine problem, students identify the dimensions and find volume of a right rectangular prism when given an edge length of a cube as a fraction (6.G.2). “Give the dimensions of a right rectangular prism that can be filled completely with cubes that have edge lengths of $$\frac{1}{2}$$in. Explain how to use the cubes to find the volume of the prism.”

• Unit 5, Math in Action, Session 1, non-routine problem, students use variables to represent two quantities that change in relationship to each other (6.EE.9). “The cheerleader, marching band, football team, and school mascot purchase new uniforms. The packing slip shows the total amount each team pays and provides information about tax and shipping charges. Choose one type of uniform. Write and solve an equation to find the price of one uniform before shipping and tax.” Teacher directions include: “read this problem involving writing and solving an equation. Then look at one student's solution to this problem on the following pages. There are many ways to solve problems. Think about how you might solve the New Uniforms problems (previous) in a different way.” 

• Unit 6, Lesson 26, Session 5, Apply It, Problem 2, routine problem, students draw a graph and write an inequality to model possible values of a given situation (6.EE.5). “Each week, Patrick buys more than 2 pounds of apples. Apples cost $1.37 per pound. Draw a graph that represents the possible amounts of money that Patrick spends on apples in a week. Then write an inequality that represents your graph.Show your work.”

The materials reviewed for i-Ready Classroom Mathematics Grade 6 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 1, Lesson 4, Session 4, Apply It, Problem 4, students develop conceptual understanding by using variables to represent numbers while writing expressions and solving mathematical problems (6.EE.2, 6.EE.6). “In a video game, players start with a score of 100 points. They earn 8 points for each gold coin and 25 points for each gem they find. Isaiah finds 3 gold coins and 2 gems. Write and evaluate an algebraic expression to find Isaiah’s score. Use c for the number of gold coins found and g for the number of gems found. Show your work.”

• Unit 3, Lesson 13, Session 2, Practice, Problem 5, students practice procedural skills and fluency while solving real world problems with ratios and rates (6.RP.3). “A manager of a clothing store always orders 2 small T-shirts and 3 large T-shirts and 3 large T-shirts for every 4 medium T-shirts. The manager plans to order 24 medium T-shirts. How many small T-shirts and large T-shirts should the manager order? Show your work.” 

• Unit 5, Lesson 19, Session 2, Develop, Apply It, students apply the properties of operations to generate equivalent expressions (6.EE.3) by solving, "A company sells fruit cups in packs of 4. The packs currently weigh 20 oz. The company plans to reduce the weight of each cup by n oz. The expression 20-4n represents the new weight, in ounces, of a pack of fruit cups. Rewrite the expression for the new weight as a product of two factors. Show your work." 

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 2, Unit Review, Performance Task, attends to conceptual understanding and application as students apply their understanding of volume (6.G.2) and dividing fractions by fractions (6.NS1.) to solve a real-world problem. “Geraldine supplies number cubes to companies that make board games. Each number cube measures $$\frac{3}{4}$$ inch on each edge. For shipping, the number of cubes can be packed into any of the boxes shown.” Images of three boxes are shown labeled with their dimensions (4 in × 4 in × 4in, 4 in × 3$$\frac{1}{2}$$in × 2 in, $$2\frac{1}{2}$$in × $$6\frac{1}{4}$$in ×$$2\frac{1}{2}$$in). “Geraldine receives an order for 780 number cubes. First, she needs to know the maximum number of cubes she can fit in each box. Then she needs a packing plan for the order. Remember: only whole cubes can be packed. Design a packing plan for Geraldine. Your plan must include the following requirements: the maximum number of cubes that can fit into each box is identified, the fewest number of boxes is used to pack the 780 number cubes, no box is packed with fewer than half the total number of cubes it can hold.”

• Unit 5, Lesson 21, Lesson Quiz, Problem 5, attends to conceptual understanding, procedural skill and fluency, and application as students write and solve equations (6.EE.7) and divide decimals (6.NS.3) in a real-world situation. “Jerel runs 5 days each week. On each of 4 days, he runs 2.3 km. If Jerel runs a total of 14 km, how many kilometers does he run on the fifth day? Show your work.”

• Unit 6, Lesson 26, Session 5, Refine, Apply It, Problem 2, students attend to procedural skill and fluency and conceptual understanding while solving inequalities (6.NS.3). “Each week, Patrick buys more than 2 pounds of apples. Apples cost $1.37 per pound. Draw a graph that represents the possible amounts of money, m, that Patrick spends on apples in a week. Then write an inequality that represents your graph. Show your work.”

The materials reviewed for i-Ready Classroom Mathematics Grade 6 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 11, Session 1, Explore, Try It, students find the volume of a right rectangular prism “Jiro has some small cubes. He puts them together to make a large cube, as shown. What is the volume of each small cube.” 

• Unit 5, Math in Action, Session 2, students analyze the relationship between dependent and independent variables using graphs and tables and write an equation to model a real-world problem . “Track and Field Training. A coach plans workouts for several groups of athletes on the track and field team. Read the coach’s plans for how each group should complete a 400-meter lap around the track. Choose one group and make a table and a graph to analyze the relationship between distance, d, and time t, for that group. Then write an equation that models the relationship.” Data about the speed and distance of each group is included in the problem. In the Reflect section, students discuss how to make sense of the problem. “Use Mathematical Practices - As you work through the problem, discuss these questions with a partner. Make Sense of Problems - Which variable is the dependent variable and which is the independent variable? Explain.”

• Unit 7, Lesson 30, Session 1, Explore, Try It, students display data in plots on a number line and summarize numerical data sets in relation to their context. “The parks department can add one new program to its summer camp. The data shows the ages of children who have signed up. Based on the Data Set: Ages in Years, which age group should get the new program?” 

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 3, Lesson 12, Session 1, Explore, Model It, Problem 2, students understand the concept of a ratio and use ratio language to describe a relationship between two quantities as they solve, “You can also use a ratio to compare two quantities. One way to describe a  relationship between ratios is to use the language for every or for each. a. In Eldora’s lab group, there are 3 test tubes for every 1 student. Complete the model to show this ratio relationship. b. Use your model to complete these sentences that use ratio language. For every 1 student, there are ____test tubes. There are ____test tubes for each ____. There is ____student for every ____test tube.” 

• Unit 3, Lesson 14, Session 1, Try It, students deconstruct data in the problem and then reconstruct data in a table using equivalent ratios. . “Hasina is making green tea lattes.  She steams milk to mix with hot tea. Hasina has 12 fl oz of hot tea. Based on the ratio in the recipe, how much milk does Hasina need to steam?” A ratio of “Green Tea Latte, 4:3, Tea:Milk” is provided for the problem.

• Unit 7, Lesson 29, Session 1, Explore, Model It, Problem 1, students reason quantitatively with numerical data sets in relation to their context . “Keiko is on her school’s track team. She collects data to answer this question. How high did members of the track team jump in yesterday’s high jump event? Complete the dot plot to show Keiko’s data.”

The materials reviewed for i-Ready Classroom Mathematics Grade 6 meet expectations for supporting the intentional development of MP3: “Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.” In the Discuss It routine, students are prompted with a question and a sentence frame to discuss their reasoning with a partner. Teachers are further provided with guidance to support partners and facilitate whole-class discussion. Additionally, fewer problems in the materials ask students to critique the reasoning of others, or explore and justify their thinking.

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

• Unit 1, Math in Action, Session 1, Reflect, Critique Reasoning, students critique a partner’s solution to designing pens for hens that meet certain criteria (i.e. “all pens will be the same size and hold the same number of hens, each pen should be at least 4 feet high, and there should be at least 8 square feet of floor space per hen.”) . “Do the pens your partner described meet the requirements from Juan’s teacher? Explain.”

• Unit 4, Lesson 16, Session 3, Try It, Facilitate Whole Class Discussion, provides guidance for teachers to help students construct viable arguments to defend their problem solving strategies. “Call on students to share selected strategies. Remind listeners to be specific when explaining why they disagree with a speaker's idea.” 

• Unit 5, Lesson 20, Session 3, Apply It, Problem 1, students determine if the reasoning of another makes sense and justify their response. “Greg says that x could represent a value of 3 in the hanger diagram. Do you agree? Explain your reasoning.” The problem is accompanied with a hanger diagram with 3 x’s on one side and 6 1’s on the other side. 

• Unit 7, Math in Action, Session 2, Reflect, Critique Reasoning, students critique a partner’s solution to using measures of center and variability to make conclusions about a data set (6.SP.5c). “What did your partner conclude about the word lengths in the first round of both spelling bees? Is your partner’s conclusion supported by the data sets? Explain.”

• Teacher Toolbox Program Implementation Support, Standards for Mathematical Practice in Every Lesson, SMPs are integrated in the Try-Discuss-Connect routine with SMP 3 identified when partners critique each other’s reasoning. In the Teacher’s Guide, at the front of the book, “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 6 meet expectations for supporting the intentional development of MP4: “Model with mathematics;” and MP5: “Use appropriate tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.” The materials generally identify MP4 and MP5 in most lessons and can be found in the routines developed throughout the materials. 

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

• Unit 1, Lesson 6, Session 2, Connect It, Problem 3, students connect a tree factor model to a given situation. “How many blankets and how many flashlights will be in each kit if Akio uses the GCF as the number of kits? Where do you see these amounts in the equations under the factor trees?”

• Unit 2, Lesson 9, Session 1, Model It, Problem 3, students write a division equation to model a situation . “Lin starts with a board that is $$\frac{3}{2}$$ feet long. She cuts it into pieces that are each $$\frac{1}{4}$$ foot long for her stacking game. a. Complete the model to show how many pieces Lin cuts her board into.” A blank tape diagram that is divided into 3 sections and labeled as $$\frac{3}{2}$$ is included. b. “Write a division equation that represents your model and shows how many pieces Lin cuts her board into. What related multiplication equation does your model represent?”

• Unit 6, Math In Action, Session 2, Solve It, students model a situation using inequalities . A table is included with 4 different ceramic pottery samples along with the least and greatest estimated age in years for each sample. “Find a solution to the Estimating Ages of Artifacts problem. Choose a sample. Write an inequality to represent the possible ages of the sample based on the least age given in the table. Then graph the inequality. Write an inequality to represent the possible ages of the sample based on the greatest age given in the table. List three possible years in which the ceramic could have been made. Give an early estimate, a middle estimate, and a late estimate.” In the Reflect section, students are prompted to discuss with a partner. “Use a Model. How could you show the possible estimated ages of the ceramic sample using a single number line?”

• Unit 7, Lesson 32, Session 2, Differentiation, students are guided by the teacher to demonstrate why the mean can be thought of as a balance point. The students use counters and rulers to label a number line above each value to represent the data. Teachers ask, “What is the mean of this data? Teachers have students move each counter to 4, keeping track of how many units they move each counter.” Teachers then ask, “What do you notice about the total units the counters to the left of 4 had to move and the counters to the right of 4 had to move to get to 4?” 

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 3, Model It, Differentiation Extend provides guidance for teachers to engage students in MP5 as they discuss finding areas of composite figures. Students find an area of a three-dimensional figure by decomposing the figure into triangles and a parallelogram. “Prompt students to compare the advantages and disadvantages of each strategy? Ask: What are the advantages and disadvantages of using decomposition and addition to find the area of the logo? Listen for the decomposition strategy allows you to use simple calculations to find the area, but you may find it difficult to decompose the logo into familiar shapes. Ask: What are the advantages and disadvantages of composing a rectangle around the logo and then subtracting the areas of the right triangles? Listen For: With this strategy, you can find the area of a rectangle, which is a simple calculation. However, more calculations are needed to subtract the areas of the triangles from the total area. Generalize: Encourage students to explain how they might choose an appropriate strategy when solving area problems. Students may state that they use the method that is the most efficient for the given shape, or they may state that they like to use the same strategy to solve all types of area problems.”

• Unit 3, Lesson 13, Session 2, Try It, students can choose from a variety of tools to demonstrate understanding of ratios. “The ratio of picnic tables to garbage cans in a new national park should be 8:3. The park design shows plans for picnic tables in a small campground and a large campground. How many garbage cans should there be in each campground?” The number of picnic tables in each campground is provided, 40 in a small campground and 120 in a large campground. The math toolkit includes: connecting cubes, counters, double number lines and grid paper.

• Unit 6, Lesson 28, Session 2, Connect It, Problem 7 students reflect on the models and strategies they use in the Try It to find distance in the coordinate plane. “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 states, “Have all students focus on the strategies used to solve the Try It. If time allows, have pairs discuss their ideas.”

• Unit 7, Lesson 31, Session 3, Deepen Understanding, provides prompts for students to generalize when they may use a box plot instead of a dot plot to represent a data set. “Prompt students to consider when and how box plots are useful representations of data. Ask: What are some advantages to using a box plot to display a data set?...Why is the median not directly in the middle of the box plot?... What are some disadvantages to using a box plot to display a data set.”

The materials reviewed for i-Ready Classroom Mathematics Grade 6 meet expectations for supporting the intentional development of MP6: “Attend to precision;” and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

There is intentional development of MP6: “Attend to Precision,” to meet it’s full intent in connection to grade-level content. Many problems present students with the opportunity to attend to precision within the mathematics and the reasoning of the answer. Examples include: 

• Unit 2, Session 1, Math in Action, Reflect. Students explain the results of their calculations. “Be Precise: Would it make sense to round the results of your calculations to get your final answer? Why or why not?” The students struggle with solving how many grains of salt it takes to grow one of the salt crystals. The problem is using decimals as the length of the side of a cube. 

• Unit 3, Math in Action Session 2, students use and label graphs to compare ratios . “Josephine and her family set goals for how much water to drink compared to other beverages. Choose two family members. Use ratio tables to determine who will drink more water compared to other beverages. Make a graph comparing the two goals that Josephine can share with her family.” The text includes the amount of water in ounces each family member drinks compared to other beverages. For example, Josephine drinks “10 ounces of water for eerie 5 ounces of other beverages.” The question in the Problem-Solving Tips box reminds students to attend to precision, “How will you label the axes of your graph?”

• Unit 6, Lesson 26, Session 4, Apply It, Problem 7, students attend to precision when identifying how the solutions to an inequality are related to the situation. “Aimee works up to 50 hours a month and earns $12 per hour. She wants to save more than $240 to buy a computer. The inequality 12h > 240, where h is the number of hours Aimee works this month, models this situation. Which values from 0 to 50 are solutions to the inequality? What do the solutions mean in this situation? Explain your reasoning.”

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. 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 3, Lesson 14, Session 3, Discuss It, provides teacher guidance to correct a common misconception when describing paint ratios with the appropriate terms. “Listen for students who think that the quantities in two ratios determine which ratio is greater. For example, they may say that 2:3 is bluer than 1:2 because 2 > 1 and 3 > 2. As students share their strategies, rephrase bluer and repeat the definition of ratio. Elicit discussion on how to determine who has a bluer mixture.”

• Unit 4, Lesson 15, Session 2, Discus It, Teacher’s Edition, Develop Academic Language: “Why? Support students as they craft clear explanations using precise language. How? Remind students that using precise mathematical language and complete sentences makes explanations clearer and easier to understand. Prior to each Discuss It, work with students to develop a list of precise terms from Model It, such as rate, per minute, equivalent ratios, and ratio relationships. During Discuss It, Collect and Display authentic examples of clear explanations.”

• Unit 7, Lesson 31, Overview, “Language Objectives: Explain in wiring why the median can be used as a measure of center, Summarize a data set using lesson vocabulary, including lower quartile (Q1), median (Q2), and upper quartile (Q3), Describe the variability of a data set by explaining how box plots and the IQR represent a data distribution in whole-class discussion, Demonstrate understanding of word problems by explaining how the median and IQR connect to the problem context.”

The materials reviewed for i-Ready Classroom Mathematics Grade 6 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 1, Lesson 6, Session 3, Extend, students find the least common multiple. Teachers are asked to do the following routine for MP 7: Deepen Understanding--When discussing using a list of multiples to find an LCM, share with students that another method for finding the LCM of 6 and 8 is to list multiples of 8 and then check for the first number in the list that is divisible by 6. Prompt students to think about the advantages and disadvantages of this method. Ask: Why does this method work? Listen for: If a multiple of 8 is also a multiple of 6, it is divisible by 6. Ask: What is an advantage to finding the first multiple of a number that can be divided by the other number? What is a disadvantage of this method? Listen for: An advantage is that you only have to list the multiples of one of the numbers and you can stop listing multiples as soon as you find one that is divisible by the other number. A disadvantage is that you have to think about several division problems, which may be harder to do than just listing multiples would be.” 

• Unit 2, Lesson 7, Session 1, Connect It, Problems 2, students use structure of place value to add and subtract decimals . Problem 2, “Place value can help you add or subtract decimals. You add 25.393 and 24.138 to find Mateo’s time. You can subtract 24.138 from 25.292 to find how much faster Mateo swims the first lap than the second lap. a. How could it help you to line up the decimals on their decimal points? b. What do you need to do before you can subtract the digits in the thousandths place in this problem? Explain. c. Complete the equation: 9 hundredths + 3 thousandths = 8 hundredths + ___ thousandths. d. How much faster is Mateo’s time for the first lap than the second lap. How did you find your answer?”

• Unit 4, Lesson 17, Session 1, Model It, Problem 4 students use structure of fractions to develop understanding of percents . “How is using a model to show a percent similar to using a model to show a fraction? Use either 50% or 10% as an example in your explanation.”

• Unit 5, Lesson 22, Session 2,Develop, Try It,  students use variables to represent two quantities in a real-world problem that changes in relationship: “An animal reserve is home to 8 meerkats. It costs the reserve $1.50 per day to feed each meerkat. Write an equation with two variables that can be used to determine the total cost of feeding the reserve’s meerkats for any number of days.” Teachers should be asking questions that “prompt students to think about how the Try It problem describes the relationship between quantities and the two variables.” 

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

• Unit 2, Lesson 9, Session 2, Start, students use repeated reasoning to make generalizations about halves and fourths in a given number. The table lists $$\frac{1}{2}$$, 1, $$1\frac{1}{2}$$, and 2, then identifies how many halves and fourths are in each of those values. Students use the Same and Different routine to compare and contrast the number of halves and fourths using a table. The materials list possible solutions as, “There are a whole number of halves and fourths in all four numbers. There are twice as many fourths as halves in each number. There are different numbers of halves and fourths in the numbers.”

• Unit 3, Beginning of Unit, Math Background, Insights on Finding Equivalent Ratios by Multiplying and Dividing, provides teachers with a background of how students can use repeated reasoning to discover multiplicative relationships. “As students continue to use repeated addition to find equivalent ratios, they may begin to notice that each equivalent ratio is related to the original ratio by multiplication. This realization points to another way of finding equivalent ratios: Multiply both quantities in the ratio by the same nonzero number. Once they have discovered this multiplicative relationship, they can use their prior knowledge of multiplicative comparisons to solve ratio problems.” 

• Unit 4, Lesson 16, Session 2, Develop, Try It, students use ratio and rate reasoning to solve real-world and mathematical problems by solving: “Ashwini jogs on the track at her school. She uses a watch to track her progress. It takes her 15 minutes to jog 6 laps. At this rate, how long will it take her to jog 16 laps?” Teachers should “prompt students to look for the relationships between quantities in a ratio and use fractions and division to find unit rates.” 

• Unit 6, Math in Action, Session 2, Reflect, prompts students to use repeated reasoning to make a connection between elevation and negative numbers. “How is the depth of an artifact related to its elevation?” Students are provided with depths of an artifact, in meters, and asked to determine their elevations.