7th Grade - Gateway 2

The materials reviewed for Reveal Math 2025 Grade 7 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

Materials provide opportunities for students to develop conceptual understanding throughout the grade level. Unit Overviews outline the three parts of rigor - conceptual understanding, procedural skill & fluency, and application. The Be Curious activities, which occur during the Launch, focus on sense making with different routines, such as the Notice and Wonder^TM. During the Explore & Develop (Activity-Based and Guided Exploration), instruction links the sense-making activity to conceptual understanding, ensuring students understand the “why” behind operations and other important mathematical skills. Additionally, the eToolkit provides eTools to help students develop a conceptual understanding of math concepts.” Examples include:

• The Unit Overview outlines the three parts of rigor–conceptual understanding, procedural skill & fluency, and application. In the Unit Overview for Chapter 3: Proportional Relationships, “Students explain proportional relationships using tables, graphs, and equations, understand the unit rate informally as a measure of steepness of the related line, called the slope, and distinguish proportional relationships from other relationships. (7.RP.2)

• Unit 5: Sampling and Statistics, Lesson 5-2: Identify Unbiased and Biased Samples, Launch, Session 1, Be Curious: Notice & Wonder, students discuss similarities and differences between the images shown about a survey, “How are they the same? and How are they different?” Students can also be asked the following questions: “What is the purpose of each survey, What type of responses might result from each survey, and Would each survey method favor a group of students? Why or why not?” Students are given two examples of how two students will conduct their surveys. (7.SP.1)

• Unit 6: Solve Problems Involving Operations with Integers and Rational Numbers, Lesson 6-2: Add Integers and Rational Numbers, Explore, Session 1, Guided Exploration, Guardian of the Reef, students explore the addition of positive and negative integers through the context of an underwater statue. The problem states, “The Guardian of the Reef is a bronze casting that stands as a symbol of environmental stewardship in the water off the Grand Cayman Island. The bottom of the statue sits on the ocean floor at 60 feet below the surface. How far below the water’s surface is the top of the crown of the Guardian of the Reef.” Students are asked the following questions, “How can you determine the integer that represents the depth of the base of the statue? What integer represents the depth beneath the top of the statue’s crown? How can you tell?, “What number line could help you understand and evaluate the expression -60 + 17?” Students are shown a picture which gives the height of the reef. Students are given two ways to solve this problem and the first way is to use a number line and students are provided with a number line that shows the expression -60 + 17. (7.NS.1)

• Unit 7: Work with Linear Expressions, Lesson 7-1: Combine Like Terms, Explore, Session 1, Guided Exploration, Electric Vehicles, students write an expression to represent the perimeter of a figure. Then they combine like terms to write the expression in simplest form. The problem states, “The motors of electric vehicles must be created exactly to the specifications presented by the engineers. The dimensions of the motor’s base is determined by the type of vehicle in which it is installed. What expression in simplest form represents the required perimeter of the rectangular base of the motor?” Students can be asked the following questions, “How may representing the expression with algebra tiles help you to see like terms? Why can you move the x-tiles to simplify the expression? The -1-tiles?, How could you use algebra tiles to represent each term of (2x-3 ) + (2x - 3) + x + x?” Students are given pictures of algebra tiles to represent the given problem. (7.EE.1 and 7.EE.2)

• Unit 9: Probability, Lesson 9-1: Understanding Probability, Launch, Session 1, Be Curious: Notice & Wonder, students are given a five-day forecast with a high and low temperature and a certain percentage chance of rain for each day. Students are asked the following discussion questions, “What do you notice?, What do you wonder?, What predictions can you make about the weather during the week?, What type of weather is least likely to occur during the week?, How might you use the forecast information to plan an outdoor event this week?” (7.SP.5)

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade level. Examples include:

• Unit 2: Solve Problems Involving Geometry, Lesson 2-3: Analyze Attributes of Geometric Figures, Launch, Session 1, Be Curious: Notice & Wonder, students explore relationships between a square and a rhombus. It states, “What questions can you ask? Why might you use models like these to make figures? What characteristics can you identify from each figure? In Pause and Reflect, students share their thinking about the characteristics of the figures. How can you change the characteristics of each figure? Establish Mathematics Goals to Focus Learning Let’s think about how the two quadrilaterals are related. What seems to be true about the side lengths in each figure? How might we compare a triangle with the quadrilaterals? What does that suggest about quadrilaterals?” (7.G.2)

• Unit 5: Sampling and Statistics, Unit Review, Questions 18 and 19, students determine if a sampling method is biased or unbiased and whether the inferences made are valid. The problem states, “An amusement park director asks every 10th person that enters the park to vote on their favorite attraction. The results are shown below. Based on the results, the director infers that the most popular attraction is the rollercoasters. 18. Is the sampling method biased or unbiased? Explain. 19. Is the inference made by the director valid? Explain.” Students are given a table of information with the attractions and the percent of votes. (7.SP.1)

• Unit 6: Solve Problems Involving Operations with Integers and Rational Numbers, Lesson 6-2: Add Integers and Rational Numbers, Summarize & Apply, Apply: Find the Charge of the Ion, students use their knowledge of positive and negative integers to answer a question about a nitrogen atom and ion. The instructions state, “In an atom, the number of positively charged protons is equal to the number of negatively charged electrons. However, atoms that do not have exactly 2, 10, or 18 electrons tend to gain or lose electrons to fill their outer shell, creating a charged atom called an ion. A nitrogen atom has 7 protons, so it only has 2 shells. Question: What will be the charge of a nitrogen ion?” Students are provided a picture of ion with the headings: 1st shell: can hold up to 2 electrons, 2nd shell: can hold up to 8 electrons, 3rd shell: can hold up to 8 electrons. (7.NS.1)

• Unit 7: Work with Linear Expressions, Lesson 7-2: Expand Linear Expressions, Practice, Question 1, students use properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. The problem states, “A basketball court has the dimensions shown. A border with a width of x feet is built surrounding the basketball court. What is the perimeter of the basketball court and border? Write the expression in simplest form.” Students are given a picture of a basketball court which dimensions labeled 92 ft and 49 ft and with border of x labeled on each dimension. (7.EE.1 and 7.EE.2)

• Unit 9: Probability, Lesson 9-1: Understand Probability, Lesson Quiz, Error Analysis, Question 14, students use their understanding that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. The problem states, “A classmate states that it is equally likely to roll a number greater than 4 on a six-sided number cube with sides labeled 1-6. Explain two ways your classmate can change the statement to make it true.” (7.SP.5)

The materials reviewed for Reveal Math 2025 Grade 7 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

Materials provide opportunities for students to develop procedural skill and fluency throughout the grade level. Reveal Math provides students with multiple opportunities to revisit concepts and develop these areas of fluency within each unit. Implementation Guide (page 58) “Number Routines provide students with daily opportunities to develop number sense, deepening their understanding of number relationships. In addition, every unit reviews a computational strategy previously learned to revisit concepts and strategies adding to students’ flexibility when choosing methods.” Examples include:

• Unit 5: Patterns of Association, Fluency Practice, the materials state, “Fluency practice helps students develop procedural fluency, that is, the “ability to apply procedures accurately, efficiently, and flexibly.” Because there is no expectation of speed, students should not be timed when completing the practice activity.” According to the Build Fluency section, “Objective Students build fluency with relating unit rate with slope. As students work to develop fluency with unit rates, have them reflect on and share with classmates the strategies they find the most useful.” Fluency Talk states, “How would you describe the relationship between the slope of a linear graph and the unit rate for the relationship it shows?” 

• Unit 6: Solve Problems Involving Operations with Integers and Rational Numbers, Lesson 6-1: Terminating and Nonterminating Decimals, Session 2, Guided Exploration, students develop procedural skill and fluency as they convert rational numbers to decimals. The materials state, “Tyrone is trying to make a pie chart with 3 equal sections, but he cannot find a decimal that equally represents the sections.” Teacher guide states, “Students explore how to evaluate a division expression that results in a repeating decimal. Have students work in small groups or with a partner to evaluate the division expression 1\div3.” This activity provides an opportunity for students to develop procedural skill and fluency of 7.NS.2d (Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats).

• Unit 8: Solve Problems Using Equations and Inequalities, Lesson 8-1: Solve Equations: px + q = r, Explore, Session 1, Activity-Based Exploration, “Students explore solving equations in the form px + q = r by applying properties of equality and by reasoning about real-world connections to their strategy and solution. Group students in pairs or small groups. Write the equation px + q = r as well as an example of an equation of this form. Have students generate several more equations of this form before having them read and respond to the Introductory Question in their Activity Exploration Journal. How can you solve equations in the form px +q = r? Have students read the scenario on the Teaching Resource page. The students will write an algebraic solution by sorting the cards into a logical order. Then have students choose two description cards that correctly interpret their solution to the given scenario. Students will not need to use all of the cards on the resource page. Provide a set of algebra tiles for each pair to model their solution process.” Students develop procedural fluency and conceptual understanding of 7.EE.4a (Solve word problems leading to equations of the form px + q = r and p(x + q) = r, p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.)

Materials provide opportunities for students to independently demonstrate procedural skill and fluency throughout the grade level. Examples include:

• Unit 2: Solve Problems Involving Geometry, Lesson 2-1: Solve Problems Involving Scale Drawings, Number Routines, In My Head, students decide which tool–mental math, paper and pencil, or calculator–is most appropriate for them to evaluate the expressions given. The materials state, “In My Head? empowers students to think flexibly about computing and evaluating on paper or in their head. Students determine which of the given expressions they could do mentally and share how they would do so. Students also talk about why certain problems are better done on paper or even with a calculator.” This activity provides an opportunity for students to develop procedural skill and fluency of 7.G.1 (Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale).

• Benchmark Assessment, Item 11, students develop procedural skill and fluency with finding unknown angle measures. The materials state, “Angles ABD and DBC are complementary. What is the measure of angle DBC?” This activity provides an opportunity for students to develop procedural skill and fluency of 7.G.5 (Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure). 

• Unit 8: Solve Problems Using Equations and Inequalities, Lesson 8-3: Solve Equations p(x + q)=r, Differentiate, Reinforce Understanding, Independent Work, Exercises 5-6, students solve two-step equations in the form p(x+q)=r. The materials state, “For exercises 5 and 6, solve each equation using any method. Show your work. 5. 1.5(x+3)=-6 6. 7(x-11)=14.” This activity provides an opportunity for students to develop procedural skill and fluency of 7.EE.4a (Solve word problems leading to equations of the form px + q = r and p(x + q) = r, p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.)

The materials reviewed for Reveal Math 2025 Grade 7 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

There are multiple routine and non-routine application problems throughout the grade level, including opportunities for students to work with teacher support and independently. The materials state, “While single and multi-step application problems are included across various portions of lessons, independent application opportunities are most often found within the Apply section. Many Apply problems provide multiple options, helping to build student agency through choice.” Materials provide opportunities for students to engage with routine application problems throughout the grade level. Examples include:

• Unit 7: Work with Linear Expressions, Lesson 7-2, Expand Linear Expressions, Session 1, Practice, Item 7 states, “A contractor is building a raised deck with the dimensions shown on the back of a house. The contractor is going to add a railing along the two shorter sides and one of the longer sides of the deck. What expression represents the length of railing needed to go around the three sides of the deck? Write the expression in simplest form.” Students are shown a rectangle with a width of (5x+1)ft and a height of (3x+7)ft. (7.EE.1)

• Unit 6: Solve Problems Involving Operations with Integers and Rational Numbers, Lesson 6-3: Understand Additive Inverse, Differentiate, Reinforce Understanding, Independent Work, students apply and extend previous understandings of addition and subtraction to add and subtract rational numbers. The materials state, “A number and its additive inverse have a sum of zero. An additive inverse can help you determine the range between a positive and negative number. 1. A football team loses 6.5 yards on the first down. On second down, it gains 6.5 yards, How many yards is the football team from their starting yard line? 2. On her first turn, Michaela moves forward 5 spaces. On her second turn, she moves back 5 spaces. What position will she be in after her second turn? 3. The Earth’s highest elevation is the summit of Mt. Everest, measuring 8,848 meters above sea level. The Earth’s lowest land elevation is at the Dead Sea, measuring 420 meters below sea level. What is the range of elevations?” (7.NS.1)

• Unit 8: Solve Problems Using Equations and Inequalities, Lesson 8-2: Write Two-Step Equations, Summarize & Apply, Apply: STEAM Fun, students solve multi-step real-life and mathematical problems posted with positive and negative rational numbers. The materials state, “A school district contracted 7 charter buses to take students to a STEAM competition. There are 350 students signed up for the events, as well as 29 chaperones. The table shows how many seats are still available on each bus. Question 1, Assuming the charter buses have the same number of seats, how many seats are on each bus? Question 2, A different charter bus service rents buses with a capacity of 47 passengers. The equation 47x - 44 = 379 is used to represent the scenario. What does each term of the equation represent in the scenario? What does the variable represent?” (7.EE.3)

Within the Implementation Guide, Focus, Coherence, Rigor, Application, “Students encounter real- world problems throughout each lesson. The On My Own exercises include rich, application-based question types, including Error Analysis and Extend Thinking. Lesson differentiation provides opportunities for application through the STEM Adventures. The unit performance task and the Mathematical Modeling Project, both found in the Student Edition offer additional opportunities for students to apply their knowledge of math concepts to solve non-routine application problems.” Examples of non-routine application problems include:

• Unit 2: Solve Problems Involving Geometry, Mathematical Modeling, Urban Planning, Project One, students solve problems involving scale drawings of geometric figures. The materials state, “Your town is holding a contest to design a new park. The park will be in the shape of the model shown below and will have an approximate area of \frac{1}{2} acre (or about 20,000 square feet). Use what you know about scale drawings and geometric figures to design your dream park in the model below. Think about the different types of park features and how to attract the most guests.” (7.G.1)

• Unit 3: Proportional Relationships, Lesson 3-6: Use Proportional Reasoning to Solve Multi-Step Ratio Problems, Summarize & Apply, Apply, Creating a Time-Lapse Video, students use proportional relationship to solve multi-step ratio and percent problems. “A group of students wants to create a 2-minute-long time-lapse video of a plant growing from a seed over a two-week period. They will set up the timer on the camera to take photographs at a set interval of time. They determine that they will need 30 photographs for every second of video. Question: How can the students set up the camera to capture the photographs they will need to create the video?” (7.RP.3)

• Unit 5: Sampling and Statistics, Mathematical Modeling, Pesticide Sampling, students design a study to collect soil samples from the middle school grounds, which will be analyzed for banned pesticides. The materials state, “Environmental engineers can tell when a banned pesticide has been used by testing the water or soil surfaces for residue chemicals from the banned pesticides. You and your classmates want to determine whether any banned pesticides have been used in the area of your middle school. You find out that a local university will analyze any samples you collect for pesticides. Design a study to determine the effect of banned pesticides on a middle school campus. Your study should include the following: the chemical or chemicals you will be testing for, how you plan to conduct your sample, how your results will be used to make inferences, how actions you plan on taking based on the inferences you make.” (7.SP.2)

The materials reviewed for Reveal Math 2025 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.

All three aspects of rigor are present independently throughout the grade. Examples where instructional materials attend to conceptual understanding, procedural skill and fluency, or application include:

• Unit 3: Proportional Relationships, Lesson 3-5: Describe Proportional Relationships, Summarize & Apply, Apply: Turning Up the Heat, students apply their understanding of proportional relationships in a real-world scenario. The materials state, “A gas is being heated in a 500-cubic centimeter glass container. The diagram shows the volume of the gas at two different temperatures. Question 1 Is the gas likely to escape from the container when it heats 375°? Justify your reasoning. Question 2 At what temperature will the gas begin to leak out from the top of the container? Choose a question to answer. Then answer it in the space below.” (7.RP.2)

• Unit 5: Sampling and Statistics, Lesson 5-2: Identify Unbiased and Biased Samples, Explore, Session 1, Activity-Based Exploration, Unbiased and Biased Samples, students develop conceptual understanding of random sampling, representative samples, and supporting valid inferences. The materials state, “Hands-On: Provide students with access to a large tub of interlocking building bricks. Students work in groups of 3 or 4. Give groups time to brainstorm about various ways that they could obtain samples from the tub. Structure the class so that initially, one group has access to the tub at a time. Make sure students understand that each member of the group must obtain a sample using the method agreed upon by the group. Have students use the table in the Teaching Resource to guide them in analyzing their sampling strategies. Remind students to return their bricks to the tub before obtaining their next sample, so that representative samples remain available.” (7.SP.1)

• Unit 6: Solve Problems Involving Operations with Integers and Rational Numbers, Lesson 6-5: Multiply Integers and Rational Numbers, Lesson Quiz, Question 3, students practice procedural skill and fluency as they multiply and divide rational numbers. Question 3 states, “A school reduces the amount of paper by 75 pounds per week by switching to electronic documents. Write an expression that represents the reduction in paper usage after 6 weeks. How much does the school reduce its paper usage in 6 weeks?” (7.NS.2c)

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single unit of study throughout the materials. Implementation Guide, Lesson Walk-Through, Rigor, state “Every lesson describes the main rigor focus of each lesson based on the goals and expectations of the standards.” The Apply section states, “The Apply offers students a non-routine problem to solve. Many Apply problems provide multiple options, helping to build student agency though choice. The Teacher Edition offers scaffolded prompts that the teacher can ask students who may need support getting started.” Practice & Reflect states,“Practice & Reflect provides students with practice that address all elements of rigor.” Many lessons include more than one aspect of rigor. Examples include:

• Unit 2: Solve Problems Involving Geometry, Performance Task, students build conceptual understanding and apply their understanding to solve problems using scale drawings of geometric figures, use formulas for the area and circumference of a circle, and solve real-world and mathematical problems. The materials state, “Priya works as an urban planner for a town. She is helping to design a new park and recreation area. Part A The park will be built on a rectangular parcel of land that is 1,00 feet long and 300 feet wide. Priya creates a scale drawing of the park, with a scale of 1 inch = 40 feet. What are the dimensions of her scale drawing? Part B A picnic pavilion on the scale drawing has the dimensions shown. Explain how to use the scale to find the area of the actual pavilion. Then find the area. Part C A circular fountain on the scale drawing has a diameter of 1 inch. What is the actual circumference of the fountain? How much space will the fountain cover?” (7.G.1, 7.G.4, 7.G.6)

• Unit 3: Proportional Relationships, Lesson 3-4: Represent Proportional Relationships with Equations, Explore, Session 1, Guided Exploration, Clay Animation, students build conceptual understanding, use application, and develop procedural fluency to recognize and represent proportional relationships between quantities. The materials state, “Clay Animation is a form of animation where animators sculpt characters from clay-like materials, arrange the items on a set, and photograph. The animators move the figures by hand for the next shot. To produce 5 minutes of film, animators take 3,600 photos. Each minute of a clay animation requires 720 photos. How can you determine the number of photos needed for any number of minutes of a clay animation film? One Way Make a table of values. Another way: Use an equation. Let’s Explore More: a. How can you use the equation to determine the number of photos an animator needs for a 30-second-long animation? b. A different kind of animation requires 540 photos for one minute of animation. What equation could you use to determine the number of photos an animator needs for a 7-minute-long animation?” (7.RP.2)

• Unit 8: Solve Problems Using Equations and Inequalities, Lesson 8-6: Write and Solve Two-Step Inequalities, Session 2, Guided Exploration, Emperor Penguins, students build conceptual understanding and develop procedural fluency as they explore solving inequalities that require a multiplication step. The materials state, “A team of 4 scientists has a grant to study a colony of emperor penguins. The grant provides funding for each scientist to study at most 50 penguins. There are currently 140 adult penguins in the study and as chicks hatch, they are added to the study. How many chicks can be added to the study under this grant?” Let’s Explore More, “a. What do you notice about the following 4 expressions? \frac{x+140}{4}, (x+140)\div4, \frac{1}{4}(x+140), \frac{x}{4}+\frac{140}{4}” (7.EE.3)

The materials reviewed for Reveal Math 2025 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. 

Within the Implementation Guide, Math Practices states, “To think like mathematicians, students must build thinking habits that help them develop a problem-solving frame of mind. Reveal Math helps students build proficiency with these important thinking habits and problem-solving skills through the Math is... Prompts found in every lesson. These prompts model the kinds of questions students can ask themselves to become proficient problem solvers and doers of math.” The Standards for Mathematical Practice are identified for teachers in the Lesson Overviews, and within the lesson margins labeled in orange as “Math Practices and Processes” or “MPP”. Although most lessons offer opportunities for students to engage with more than one Standard for Mathematical Practice, this guidance identifies a focus MP within each lesson. The notes also provide lesson-specific information, ideas, and questions teachers can use to deepen students’ engagement with the focus MP. 

MP1 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students make sense of problems and persevere in solving them as they work with support of the teacher and independently throughout the modules. Examples include:

• Unit 4: Solve Problems Involving Percentages, Unit Overview, Math Practices, Make Sense of Problems and Persevere in Solving Them. The materials state, “The ability to make sense of problem situations and persevere in solving them is an essential skill, not only in middle and high school mathematics but also in academic and real-world situations. Helping students analyze problems and identify a solution path will help them succeed in higher-level mathematics. Encourage students to make connections between problem situations and similar problems they have solved before. Focus students’ attention on identifying the important information that is provided and the information they need to find. Encourage students to persevere when they get stuck by discussing the problem and possible entry points. Provide consistent opportunities for students to focus on making sense of problems. Some suggestions include: Have students highlight, underline, or write bullets with the key information in the problem. Have students mark up problems involving percentages by circling the part, underlining the percent, and boxing the whole (or using similar markings) to help them set up the percent equation and identify the missing information. Have students discuss problems with a partner before solving to help them devise a solution method. Engage students in conversation about their solution plans.”

• Unit 4: Solve Problems Involving Percentages, Lesson 4-6: Solve Percent Error Problems, Explore, Session 1, Guided Exploration, Sunflower Seeds, students identify the information given in the problem, and discuss how they can use that information to begin solving…” The materials state, “A vendor at a farmer’s market claims that his sunflowers have 1,500 seeds. Imani buys a sunflower from the vendor and counts the seeds to see how many seeds are actually on the flower. She counts 1,821 seeds on the flower she purchased. By what percent does the vendor’s claim differ from the number of seeds Imani counted? Step 1: Find the difference between the quantities. Step 2: Find the percent represented by the error. Let’s Explore More: a. Give an example of a situation where a small percent error could be a major issue. Then give an example of a situation where a large percent error is not a major issue. Explain why this is important. b. Is the statement, “A small error results in a small percent error” sometimes, always, or never true? Explain your reasoning.” 

• Unit 9: Probability, Lesson 9-6: Simulate Chance Events, Develop, Session 2, Activity-Based Exploration, students analyze the information in the problem and determine if their answers make sense. The materials state, “1. Present: Provide the following scenario and flawed response: Players at a carnival game win about 20% of the time. To simulate the probability that the next six players will win, a spinner is divided into 5 equal sections in which 1 of the sections represents a win. The spinner is then spun 6 times and the number of wins is recorded. 2. Prompt: Prompt students to identify the error in the statement and think of a way to change the statement to make it mathematically correct. 3. Share: Pairs share out their draft improved response. 4. Refine: Students refine their own draft response, as needed. Facilitate a whole-class discussion regarding the students’ responses to the Concluding Questions. Using evidence of student thinking that you gathered while students were completing the Math Language Routine, sequence students’ sharing of findings to highlight different approaches and thinking strategies used to respond to the Concluding Questions. What factors are important to consider when designing a simulation model? Why is experimental probability used with simulation? MPP: Students should recognize that when they design a simulation, they are designing an experiment. Each time the simulation is performed is a trial.”

MP2 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students reason abstractly and quantitatively as they work with support of the teacher and independently throughout the modules. Examples include:

• Unit 3: Proportional Relationships, Lesson 3-1: Connect Ratios, Rates, and Proportions, Explore, Session 1, Guided Exploration, Salt-Water Aquarium, students understand the relationships between problem scenarios and mathematical representations as they compute unit rates associated with ratios of fractions. The material states, “Jacob is setting up a salt-water aquarium. The aquarium needs to have a salinity level of 35 grams of salt per kilogram of water (1,000 cm^3= 1 kg). How much salt will Jacob need to add to the water for the salinity level to be correct? One Way, Use a table of equivalent ratios. Another Way, Use an equation.” Let’s Explore More “a. How is finding an unknown in a proportion similar to determining equivalent ratios? b. Ayana is setting up a salt-water aquarium that holds 120 kilograms of water. How much salt will she need if she maintains the same salinity level?” 

• Unit 4: Solve Problems Involving Percentages, Lesson 4-1: Connect Percentages and Proportional Reasoning, Explore, Session 1, Guided Exploration, Protecting the Goal, students represent situations symbolically as they use proportional relationships to solve multistep ratio and percent problems. The materials state, “The game stats for the goalkeeper for the local soccer team are shown. What percent of the shots taken did the goalkeeper save? You can use proportional reasoning to solve problems involving percentages. Step 1: Draw a tape diagram to represent the problem. Step 2: Use the tape diagram to write a proportion. Step 3: Solve the proportion. Let’s Explore More: a. What proportion can be used to determine the percent of goals scored? MPP: Ask students to share their thinking about tools they can use to represent the relationship.” 

• Unit 8: Solve Problems Using Equations and Inequalities, Unit Overview, Math Practices, Reason Abstractly and Quantitatively states, “Writing and solving algebraic equations and inequalities is a foundational skill for higher-level mathematics. Helping students to reason abstractly and quantitatively in order to represent and solve mathematical and real-world situations with equations and inequalities will provide them with the skill set they need to be successful in high school mathematics and beyond. Encourage students to think of problems they have solved in the past when making sense of new problems. This can help them identify appropriate solution methods. Focus students’ attention on the relationships between quantities in a problem and how they can use the relationships to determine an appropriate solution method, including identifying the operations and symbols needed to represent the situation. Provide consistent opportunities for students to focus on reasoning. Some suggestions include the following: Students work with partners to craft a scenario that can be modeled with a linear equation or inequality. Student-groups exchange scenarios and write an equation or inequality to model each other’s scenarios. Give student-pairs the graph of the solution of an inequality. Have student-pairs write the inequality that the graph represents. Then have students write one- and two-step inequalities with that solution set.”

The materials reviewed for Reveal Math 2025 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.

Within the Implementation Guide, Math Practices states, “To think like mathematicians, students must build thinking habits that help them develop a problem-solving frame of mind. Reveal Math helps students build proficiency with these important thinking habits and problem-solving skills through the Math is... Prompts found in every lesson. These prompts model the kinds of questions students can ask themselves to become proficient problem solvers and doers of math.”

Students construct viable arguments and critique the reasoning of others in connection to grade-level content as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 2: Solve Problems Involving Geometry, Lesson 2-7: Solve Problems Involving Volume, Explore, Session 1, Guided Explanation, Fish Tank, students construct viable arguments as they explore volume of an irregular right prism in a real-world situation. The materials state, “Sani bought a pentagonal tank at a yard sale that he plans to use for saltwater fish. To get an idea of how much salt to purchase for the tank, Sani needs to determine the volume of the tank. How much water does the tank hold? Math is… Explaining, What argument can you use to explain the solution? Have students discuss their arguments explaining the solution with a partner, asking and answering questions to ensure their arguments are sound.” 

• Unit 4: Solve Problems Involving Percentages, Performance Task, students construct viable arguments as they use proportional reasoning to solve multi-step percent problems. The materials state, “DeShawn works at a company that creates reusable packaging. Part A: DeShawn sent a survey to 560 companies asking if they would be interested in reusable packaging for their products. 255 companies expressed interest. What percent of the companies surveyed expressed interest in reusable packaging? What factor (s) might prevent a company from using the reusable packaging? Part B: DeShawn sent surveys to the residents of two different cities to find out their thoughts on reusable packaging. In Johnstown, 88% of 640 people responded that they would like to see companies use reusable packaging. In Springfield, 646 of 760 were in favor of reusable packaging. In which city did residents have a greater interest in reusable packaging? Explain. Part C: DeShawn’s company has a goal of increasing the number of companies using their packaging by 10% over a five-year period. Five years ago, 242 companies used their packaging. If DeShawn’s company reached its goal, how many companies would be using their packaging now? Explain.” 

• Unit 5: Sampling and Statistics, Unit Overview, Math Practices, Construct Viable Arguments and Critique the Reasoning of Others states, “Proficiency in sampling and statistics not only requires students to be able to analyze data, but also to be able to recognize flawed inferences. Helping students develop the ability to justify their reasoning and explain why others’ reasoning is flawed will provide them with skills for inside and outside the classroom. Encourage students to look at survey questions, sampling methods, and analysis critically. Students should be able to explain how each step contributes to the validity or invalidity of inferences about the population. Guide students to use the key terms from the unit when formulating arguments. Using correct mathematical language helps students craft more effective arguments and justifications for their reasoning. Provide consistent opportunities for students to focus on constructing viable arguments and critiquing the reasoning of others. Some suggestions include: Students generate statistical questions about a population and exchange with a partner. Partners design a sampling method for each question and justify their method based on whether it would produce a representative sample. Give partners data from surveys and have them make inferences about the population from the data. Students can make written or verbal statements about their inferences with justifications. Partners can critique each other’s reasoning.”

• Unit 9: Probability, Lesson 9-3: Theoretical Probability of Simple Events, Develop, Session 2, Activity-Based Exploration, Activity Debrief, Stronger and Clearer Each Time: Successive Pair Share, students construct viable arguments and critique the reasoning of others as they explore uniform probability models. The materials state, “1. Think Time: Give students 5-10 minutes to review their response to the Concluding Question from the previous session and to think about what they will say to their first partner to explain the summary of what they have learned. 2. Structured Pairing: Using a successive pairing structure, students explain their response to at least two different partners. Each time, the students speaking focuses on explaining their reasoning clearly and precisely. The student listening asks clarifying questions to help their partner to be clearer and more precise in their communication. 3. Post-Write: Students revisit and revise as needed their response to the Concluding Questions. Math is… Justifying, Remind students that they can use mathematical terms, drawings, and properties as justifications.”

The materials reviewed for Reveal Math 2025 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.

Within the Implementation Guide, Math Practices states, “To think like mathematicians, students must build thinking habits that help them develop a problem-solving frame of mind. Reveal Math helps students build proficiency with these important thinking habits and problem-solving skills through the Math is... Prompts found in every lesson. These prompts model the kinds of questions students can ask themselves to become proficient problem solvers and doers of math.”

MP4 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students model with mathematics as they work with teacher support and independently throughout the modules. Examples include:

• Unit 6: Solve Problems Involving Operations with Integers and Rational Numbers, Lesson 6-6: Divide Integers and Rational Numbers, Explore, Session 1, Guided Exploration, How Much Money, students explore the meaning of division with negative numbers by modeling the situation with an appropriate representation and using an appropriate strategy. The materials state, “DeShawn wants to buy the video game system shown. He asks his parents to loan him the money and makes a plan to pay them back by making the same payment each month for 12 months. How much money will DeShawn owe his parents each month? One Way Use a number line. Another Way Use division. MPP: Have students reread the problem focusing on quantities and descriptive words that can help them understand the meaning of the quantities in the context of the situation. Then have them describe to a partner how they would represent the problem mathematically.” 

• Unit 7: Work with Linear Expressions, Lesson 7-3: Add Linear Expressions, Explore, Session 1, Guided Exploration, Mosaic, students describe what they do with algebra tiles and how it relates to the situation as they simplify when adding two linear expressions. In the first method, algebra tiles are used to demonstrate the process. In the second method, the expression is rewritten by using the additive inverse, “An artist is designing a mosaic pattern using glass tiles and stone tiles set in rows. The number of each type of tile will be based on the row number. An expression to find the number of each type of tile needed based on the row number x is shown. What expression in simplest form represents the total number of tiles in a row? One Way Use algebra tiles. Another Way: Use properties to simplify the expression. MPP: Have students describe how to add linear expressions using algebra tiles to a partner. Listen for understanding that like tiles can be grouped and that pairs of opposite tiles can be removed.”

• Unit 9: Probability, Lesson 9-2: Experimental Probability of Simple Events, Develop, Session 2, Guided Exploration, Probability of a Pearl, students calculate experimental probabilities and check to see whether their answer makes sense. The materials state, “Talia wants to calculate the probability of finding a pearl in an oyster. She collects a bucket of 103 oysters, opens each one, and finds that none of them have a pearl. Based on the oysters that Talia opened, what is the experimental probability of finding a pearl in an oyster? MPP: Have students brainstorm potential questions with a partner and describe how these questions help them to better define the problem.” 

• Implementation Guide, Unit Walk-Through, Mathematical Modeling, “As part of the STEM focus, each unit ends with a Mathematical Modeling project that offers students the opportunity to apply the math concepts they have learned. Each Unit contains two options from which students can choose, promoting engagement and student agency. These STEM-focused projects also encourage students to make decisions about how to approach the project, what mathematics to use, and how to present their project findings.”

MP5 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students choose tools strategically as they work with teacher support and independently throughout the modules. Examples include:

• Unit 2: Solve Problems Involving Geometry, Lesson 2-8: Solve Problems Involving Circumference of Circles, Explore, Session 1, Guided Exploration, Exercising Horses, students choose appropriate tools and/or strategies that will help develop their mathematical knowledge as they explore the relationships among radius, diameter, and circumference. The materials state, “Horse trainers use lunge lines to exercise horses in circular training pens. In one pen, a trainer uses a 30-foot-long line and the horse walks 188.4 feet in one lap. In a different pen, the trainer uses a 20-foot-long line and the horse walks 125.6 feet in one lap. How do the lengths of the lunge lines and one lap around each pen relate? MPP: Discuss with students the different technology tools they are familiar with, and how they could use one to find the relationship between circumference, radius, and diameter.”

• Unit 8: Solve Problems Using Equations and Inequalities, Lesson 8-2: Write Two-Step Equations, Explore, Session 1, Guided Exploration, Donating Hair, students use technological tools, as appropriate, to explore and deepen their mathematical understanding. The materials state, “To donate hair to a non-profit organization, a ponytail must be 300 millimeters in length. On average, hair grows at a rate of about 0.4 millimeter per day. Aiden’s ponytail is currently 110 millimeters long. About how many days will it take Aiden’s hair to grow to be long enough to donate? MPP: Have students explain how they would use technology to solve the equation.” 

• Unit 9: Probability, Unit Overview, Math Practices, Use Appropriate Tools Strategically states, “Throughout the unit, students explore and apply tools to generate and represent outcomes for probability experiments. They use spinners, number cubes, coins, and random-number generators to generate outcomes for experiments and tree diagrams, tables, and organized lists to represent sample spaces. Proficiency in choosing tools is a skill that is key to success in the student of probability, and applicable to all areas of mathematics. As students use various tools throughout the unit, engage them in conversation about what insight they expect to gain from the tool and any limitations the tool might have. Encourage students to use tree diagrams, tables, and organized lists to represent the sample space of compound events so that they can gain proficiency with all three tools. Provide consistent opportunities for students to focus on structure. Some suggestions include: Students work with partners to compare the benefits and limitations of using tree diagrams, tables, and organized lists to represent the sample space of compound events. Student-groups discuss situations for which a random-number generator is a useful tool to generate outcomes and situations for which it is not useful. Challenge students to think about how the likelihood of each possible outcome affects how a random number generator should be used.”

The materials reviewed for Reveal Math 2025 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. 

Within the Implementation Guide, Math Practices states, “To think like mathematicians, students must build thinking habits that help them develop a problem-solving frame of mind. Reveal Math helps students build proficiency with these important thinking habits and problem-solving skills through the Math is... Prompts found in every lesson. These prompts model the kinds of questions students can ask themselves to become proficient problem solvers and doers of math.”

Students attend to precision in mathematics, in connection to grade-level content, as they work with teacher support and independently throughout the units. Examples include:

• Unit 2: Solve Problems Involving Geometry, Lesson 2-6: Solve Problems Involving Area and Surface Area, Explore, Session 1, Guided Exploration, Two-Tiered Roof, students attend to precision as they calculate the surface area of a roof composed of trapezoids and triangles. The materials state, “A contractor is replacing the 2-tiered pagoda-style roof on an octagonal gazebo that was damaged in a storm. How much roofing material does the contractor need to order? Step One: The top tier is made up of 8 congruent triangles. Use the formula for the area of a triangle to find the area of one triangle. Step Two: The bottom tier is made up of 8 congruent trapezoids. Decompose each trapezoid into a rectangle and two triangles.”

• Unit 7: Work with Linear Expressions, Lesson 7-4: Subtract Linear Expressions, Explore, Session 1, Guided Exploration, Budgeting, students attend to precision as they write expressions in simplest form. The materials state, “Isaiah has $150 in his checking account. This week, he has x dollars to deposit. He puts 5% of his deposit into his savings account and the rest in his checking account. What expression in simplest form represents the amount in Isaiah’s checking account after paying his $50 phone bill. Step 1: Write an expression to represent the situation. Step 2: Use the additive inverse and then simplify the expression. MPP: Encourage students to discuss the quantities given in the problem, and how the quantities should be represented.”

• Unit 8: Solve Problems Using Equations and Inequalities, Lesson 8-4: Write and Solve One-Step Addition and Subtraction Inequalities, Explore, Session 1, Guided Exploration, Time to Change the Oil, students attend to precision as they write an inequality to represent a real-world problem and graph its’ solution set. The materials state, “A customer brings their car to a service center to get an oil change and to have the brake pads replaced. It typically takes the mechanic 45 minutes to complete an oil change. How long does the mechanic have to replace the brake pads in order to have the car ready in no more than 2 hours? Step 1: Define a variable, Step 2: Write an inequality to represent the situation, Step 3: Solve the inequality and graph the solution set.”

Students attend to the specialized language of mathematics, in connection to grade-level content, as they work with teacher support and independently throughout the units. Examples include:

• Unit 4: Solve Problems Involving Percentages, Lesson 4-2: Understand the Percent Equation, Explore, Session 1, Activity-Based Exploration, Dining Out, students use grade-level appropriate vocabulary and formulate clear explanations as they calculate different tips. The materials state, “Students work in pairs or small groups. Tell students that they will be ordering off the Restaurant Menu Teaching Resource, and group members’ order will be combined into one check. Write: 10%, 15%, and 20% on the board and tell students that their job is to calculate the tip amount and total bill for each of these tip percentages. After finding the total bill for the 3 percentages, challenge students to come up with a one-step algebraic expression that will give the total bill amount for any given food purchase, f, with an 18% tip… MPP: Encourage students to use words and phrases such as percent, percent equations, and total cost as they discuss their understanding of finding percent tip.”

• Unit 6: Solve Problems Involving Operations with Integers and Rational Numbers, Lesson 6-1: Terminating and Nonterminating Decimals, Explore, Session 1, Activity-Based Exploration, students convert fractions to decimals and explain what it means to have terminating decimals. The materials state, “...Distribute the materials and briefly demonstrate how to use the paperclip with the spinner. Explain that they will be moving across the game board, one column at a time. Then briefly explain the rules. Review the game board with the students, making sure that they understand that “R1” means a remainder of 1. MPP: Have students think about the word terminate. Ask students what it means to terminate something.”

• Unit 7: Work with Linear Expressions, Unit Overview, Math Practices, Attend to Precision, students use specific mathematical language when providing written responses throughout the unit. The materials state, “Working with linear expressions is a foundational concept for middle and high school mathematics. To successfully work with linear expression, students need to be fluent with vocabulary terms and precise with their computations. Encourage students to use precise mathematical language when they explain their work or their reasoning, including key vocabulary from the unit such as simplest form, like terms, and additive inverse. Remind students to check their computations carefully, particularly when distributing the negative term when subtracting expressions or applying the Distributive Property to expand or factor expressions. Provide consistent opportunities for students to attend to precision. Some suggestions include: Have students exchange practice problems and check each other’s work for computational accuracy. Model using mathematical language by rephrasing students’ ideas using key vocabulary from the unit. Have students revise written responses to use precise mathematical language.”

The materials reviewed for Reveal Math 2025 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.

Within the Implementation Guide, Math Practices states, “To think like mathematicians, students must build thinking habits that help them develop a problem-solving frame of mind. Reveal Math helps students build proficiency with these important thinking habits and problem-solving skills through the Math is... Prompts found in every lesson. These prompts model the kinds of questions students can ask themselves to become proficient problem solvers and doers of math.”

MP7 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students look for and make use of structure as they work with teacher support and independently throughout the modules. Examples include:

• Unit 3: Proportional Reasoning, Unit Overview, Math Practices, Look for and Make Use of Structure states, “Analyzing and understanding the structure of proportional relationships is an important big idea in middle school and a foundational concept for high school mathematics helping students see the structure of proportional relationships will increase the likelihood of them being successful with high-level mathematics. Encourage students to see patterns when looking at a series of ratios before they carry out any operations to determine constant ratios. Have students predict whether the ratios represent a proportional relationship and justify their prediction based on patterns they notice. Focus students’ attention on the representations used in this unit. When students see ratios in tables and graphs as ordered pairs, they are more likely to recognize patterns that will help them recognize proportional relationships. Provide consistent opportunities for students to focus on structure. Some suggestions include: Students work with partners to create tables of values, some of which represent proportional relationships, white others do not. Partner-groups can exchange their tables of values with other partner-groups who determine which tables represent proportional relationships. Partner-groups can opt to plot the table of values as ordered pairs on a coordinate grid or determine whether there is a constant of proportionality. Give student-groups a constant of proportionality value, and have them create a data set that aligns to the constant of proportionality. Student-groups share their data sets and explain how and why the data set represents a proportional relationship.”

• Unit 4: Solve Problems Involving Percentages, Solve Markup and Markdown Problems, Develop, Session 2, Guided Exploration, Buy One Get One Sales, students decompose a complicated problem into smaller (more simple) problems as they find the sales price for a pair of sunglasses. The problem states, “Priya wants to buy two pairs of sunglasses for $89 each. She has a coupon code for 35% off the second pair if she pays full price for the first pair. How much will Priya pay for the second pair of sunglasses? Use a tape diagram to represent the problem. One Way Calculate the 35% markdown and subtract it from the original price. Another Way Multiply by the percent of the original price after markdown. MPP: Ask students to share their thinking about how they can break down the problem into pieces that they can solve one step at a time.”

• Unit 7: Work with Linear Expressions, Lesson 7-5: Factor Linear Expressions, Explore, Session 1, Activity-Based Exploration, students look for patterns as they write expressions in factored form. The materials state, “Have students work in pairs or small groups to analyze the structures of the related area diagrams, factored expressions, and expanded expressions on the Teaching Resource. Encourage students to group their algebra tiles to represent the factored form of each expression, as well. MPP: Encourage students to rewrite the expression using the GCF. Then have them think about the mathematical properties they have learned and which might be useful in this situation.”

MP8 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students look for and express regularity in repeated reasoning as they work with teacher support and independently throughout the modules. Examples include:

• Unit 2: Solve Problems Involving Geometry, Lesson 2-2: Use Side Lengths and Angle Measures to Draw and Analyze Triangles, Develop, Session 2, Guided Exploration, Triangle Designs, students evaluate the reasonableness of a design that could be constructed using repeated triangles. The material states, “Carlos has an assignment to create a design with one shape only. His shape is the right triangle shown. He can change only the side lengths and the color of the triangle, not the angle measures. What might his design look like? Let’s Explore More: a. One friend insists that two angle measures and one side length will form a unique triangle. Do you agree with this statement? Explain. MPP: Invite students to share their designs and to respond to questions from their classmates. As they do, remind them to justify their reasoning.”

• Unit 6: Solve Problems Involving Operations with Integers and Rational Numbers, Unit Overview, Math Practices, Look For and Express Regularity in Repeated Reasoning states, “Mathematically proficient students use regularity in repeated reasoning to help them make generalizations. Guiding students to look for and express regularity in repeated reasoning will help them succeed with some key concepts in this lesson. By noticing that they are repeating the same calculations when converting a fraction to a decimal, students will recognize the repeating unit in a nonterminating decimal. By recognizing the pattern of signs for the products and quotients of integers, students will find the general methods for multiplication and division of rational numbers. By recognizing that the sums of additive inverses are zero, students will improve their fluency and proficiency in operations with integers. Provide consistent opportunities for students to reflect on their work and notice regularity that can lead to generalizations. Some suggestions include: Give students sets of fractions and have them convert the fractions to decimals. Students classify the decimals as terminating or non terminating and justify their classification. Give students multiplication and division problems with integers that have different signs. Students classify the problems based on whether the product/quotient is positive or negative and justify their classifications based on generalizations about signs of products and quotients.”

• Unit 8: Solve Problems Using Equations and Inequalities, Lesson 8-5: Write and Solve One-Step Multiplication and Division Inequalities, Develop, Session 2, Activity-Based Exploration, students evaluate the reasonableness of their answers and thinking as they determine how the four operations impact the values and symbol in an inequality. Students are encouraged to think about the steps they take in solving an inequality, and to notice any patterns in what repeats during the process. The materials state, “Facilitate a whole-class discussion of the two activities. Using the evidence of student thinking you have gathered while students were completing Math Language Routine, sequence students’ sharing of findings to highlight different approaches and thinking strategies used to respond to the Concluding Questions. MPP:Encourage students to think about the steps they take in solving an inequality, and to notice any patterns in what repeats during the process.”