6th Grade - Gateway 2

The materials reviewed for Reveal Math 2025 Grade 6 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 7: Integers, Rational Numbers, and the Coordinate Plane notes “Students understand the locations of integers and rational numbers and their opposites on a number line and understand the absolute value of a number as its distance from zero on a number line.” (6.NS.6 and 6.NS.7)

• Unit 3: Ratios & Rates, Lesson 3-4: Determine Equivalent Ratios Using Graphs, Session 1, Guided Exploration, students make tables of equivalent ratios relating quantities with whole- number measurements. The problem states, “The organizers of a soccer league ordered 6 of the large bags shown to hold soccer balls. How many soccer balls will the bags hold?” There is a table and a graph provided to demonstrate a ratio of 1 bag to 6 soccer balls. Discussion questions are “What tools or models represent this relationship? How are the table and graph related? Why do you think the graph is a straight line?” (6.RP.3a)

• Unit 4: Understand and Use Percentages, Lesson 4-2: Relate Fractions, Decimals, and Percents, Be Curious: Notice & Wonder, students use a protocol to discover the relationship between decimals, fractions, and percents. “Which doesn’t belong? 12:80, \frac{80}{100}, 80%, 4 out of 5.” Pose Purposeful Questions, “How can you rewrite these into the same form? How can you actually compare the different values? What would need to happen to compare these values?” (6.RP.3c)

• Unit 5: Solve Area, Surface Area, and Volume Problems, Lesson Lesson 5-1: Determine the Area of Parallelograms and Rhombuses, Session 2, Guided Exploration, Tiling a Backsplash, students explore ways to find the area of a rhombus by decomposing and rearranging it. The problem reads, “Facilitate Mathematical Discourse. Can you think of another way to decompose the rhombus and compose the pieces into a rectangle? Will the area of your rectangle be different from the area of the rectangle shown? How can you decompose and rearrange the parts to justify the formula for area?” Let’s Explore More states, “Is area always represented in square units (u^2)? Explain.” (6.G.1)

• Unit 7: Integers, Rational Numbers, and the Coordinate Plane, Lesson 7-1: Explore Integers and Their Opposites, Session 1, Be Curious: Notice & Wonder uses a real-life example of sea level for students to understand that positive and negative numbers are used together to describe quantities having opposite directions or values; use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation. the problem shows a picture of Death Valley National Park with Telescope Peak labeled at 3,368 meters and Badwater Basin labeled below the line of Sea Level is provided. Students are asked, “What do you notice? What do you wonder?” Pose Purposeful Questions state, “Which features of Death Valley National Park are shown? Which feature has an elevation nearest to sea level? How can you represent the location of each of the features?” (6.NS.5)

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

• Unit 3: Ratios and Rates, Lesson 3-2: Understand Rates and Unit Rates, Session 1, Activity-Based Exploration, students “explore problems involving rates and unit rates” in the “Hands-On'' section. It states, “Students use the Blank Number Lines Teaching Resource to create a double number line to solve the problem. Students will use what they know about the relationship between quantities to label a double number line and use it to answer the question.” Activity Exploration Journal states, “Yelina needs to ride the metro train into the city each day for one week. She needs to decide whether she wants to purchase seven one-day passes or one seven-day pass. The one-day pass costs $6.25 per pass. The seven-day pass costs $35 per pass. Which pass do you recommend Yelina choose? Why?” (6.RP.3b)

• Unit 4: Understand and Use Percentages, Lesson 4-1: Understand Percent, Session 1, Be Curious: Notice & Wonder, students compare the total number of each color card in an array of 20 colored cards to find a percent of a quantity as a rate per 100. The problem states, “Pose Purposeful Questions, How does the number of each color compare to the others? How does the number of each color compare to the whole? How can you express the ratio of the color to the whole consistently? Pause & Reflect Students consider how the quantities of cards can be expressed. They think about the relationship between individual colors and the total number of cards. How can you express the ratio of pink cards to the total number of cards?” (6.RP.3c)

• Unit 6: Numeric and Algebraic Expressions, Lesson 6-8: Generate Equivalent Expressions, Session 1, Activity-Based Exploration, students create algebraic expressions by adding, rearranging, and deleting algebra tiles to model expressions. Students use digital algebra tiles to answer “In how many more ways can you represent 12x+138+4x+22 so that its value does not change?”(6.EE.3)

• Unit 7: Integers, Rational Numbers, and the Coordinate Plane, Lesson 7-5: Represent Rational Numbers on the Coordinate Plane, Session 1, Activity-Based Exploration “Students explore how rational numbers can be represented in the four regions on the coordinate plane.” The hands-on activity states, “Students label the coordinates of the locations shown on the map. Then they will fold the map to see if any points match up. Students can notice and wonder about the coordinates of the reflected points.” Then teachers ask, “What do you notice about the way the regions are numbered? What do you notice about the numbers in the ordered pairs in the first region? In the second region? In the third region? In the fourth region?” Finally, students journal the answer to “What does it mean if two ordered pairs differ only by their signs?” (6.NS.6b)

• Unit 8: Equations & Inequalities, Lesson 8-5: Understand Inequalities and Their Solutions, Session 1, Activity-Based Exploration, students graph inequalities on a number line to understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? “How can you represent an inequality on a number line? You will explore how to represent the solution to each of the following inequalities on a number line. x>-4, x\lesseq4, x<6, x\geq6. How can you represent the solution of an inequality?” (6.EE.5)

The materials reviewed for Reveal Math 2025 Grade 6 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 2: Understanding the World Around Us Through Statistics, Lesson 2-6: Divide Multi Digit Numbers Using an Algorithm, Session 2, Guided Exploration, students explore using an algorithm to divide multi-digit whole numbers that do not divide easily. The materials state, “In the U.S., public schools calculate the amount spent on each student in a year. The typical amount spent for each student each year is shown. How much do schools typically spend on each student each month? You can abbreviate the values when using an algorithm to make calculations simpler. The equation 12630\div12=a represents the problem. Divide the numbers in each place value position from left to right. Start at the highest place value. You can extend the dividend to decimal places to address a remainder.” Students develop procedural fluency of 6.NS.2 (Fluently divide multi-digit numbers using the standard algorithm).

• Unit 3: Ratios & Rates, Lesson 3-4: Determine Equivalent Ratios Using Graphs, Session 1, Exit Ticket, students explore the concept of equivalent ratios through ratio tables and graphs. The materials state, “The table shows the cost of notebooks at the school store. 1. Use the table to plot the points on the coordinate plane. 2. What does the point at (3,15) on the coordinate plane represent?” A table with two rows is included. The first row is labeled “Number of Notebooks” and the second row is labeled “Cost ($).” Students develop procedural fluency and conceptual understanding of 6.RP.3a (Make tables of equivalent ratios relating quantities with whole- number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios).

• Unit 6: Numerical and Algebraic Expressions, Lesson 6-3: Explore Numerical Expressions with Exponents, Session 2, Summarize & Apply states that, “You can use exponents to represent a repeated multiplication expression. An expression written with an exponent is a power. You can evaluate a power using multiplication. Apply: Biology James is studying the growth rate of a specific type of bacteria. He places three cells in a Petri dish and records the number of bacteria every five hours. The table shows his results. Question 1: What powers represent the number of bacteria recorded every 5 hours? Question 2: What power would represent the number of bacteria after 25 hours? Choose a question to answer. Then answer it in the space below.” Students develop procedural fluency of 6.EE.1 (Write and evaluate numerical expressions using whole number exponents).

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

• Unit 4: Understand and Use Percentages, Fluency Practice, students practice dividing decimals using the algorithm. The materials state, “Divide decimals using an algorithm. Multiply the divisor by a power of 10 so the divisor is a whole number. Divide. Annex zeros, if necessary. Place the decimal in the quotient directly above the decimal in the dividend. 9\div2.43=3.75.” After completing practice items, students are asked, “How would you explain to a classmate when to annex zeros?” This activity provides an opportunity for students to develop procedural skill and fluency of 6.NS.2 (Fluently divide multi-digit numbers using the standard algorithm.)

• Unit 5: Solve Area, Surface Area, and Volume Problems, Fluency Practice, gives the students the following instructions: “Add or subtract decimals. Align the decimal points. Annex zeros, if needed. Add or subtract. Multiply decimals. Multiply. Place the decimal point so the product has the same number of decimal places as the addends. Divide decimals. Multiply so that the divisor is a whole number. Divide. Place the decimal in the quotient directly above the decimal in the dividend. Evaluate each expression. 1. 6.2+9.3, 2. 8.92-2.47, 3. 1.5\times2.6.” This activity provides an opportunity for students to develop procedural skill and fluency of 6.NS.3 (Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.) 

• Benchmark Assessment 2, Item 31, students determine the error in the claim made by the fictitious student and then correct the error. The problem states, “For the school bake sale, members of the cheerleading squad bake the number of cookies shown in the table. A student claims that Robert and Melanie bake the same number of cookies and Alicia and Joe bake the same number of cookies. Find the error made by the student and correct it. Sample answer: The student may have multiplied 3\times4 instead. Alicia baked 3\times4 = 12 cookies, but Joe baked 3^4=3\times3\times3\times3= 81 cookies." This activity provides an opportunity for students to independently develop procedural skill and fluency of 6.EE.1 (Write and evaluate numerical expressions using whole number exponents).

The materials reviewed for Reveal Math 2025 Grade 6 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 4: Understand and Use Percentages, Lesson 4-3: Estimate the Percent of a Number, Session 2, Lesson Quiz, Item 1, students use rate and reasoning to solve real-world and mathematical problems. The item states, “Just Shirts has 90 shirts on sale. Of those 90 shirts, 55 shirts are polo shirts. Which is the best estimate for the percent of the shirts sold that are polo shirts? A. 50% B. 55% C. 60% D. 75%” (6.RP.3c)

• Unit 5: Solve Area, Surface Area, and Volume Problems, Lesson 5-2: Determine the Area of Triangles, Session 1, Practice, Item 5, students find the area of triangles and apply these techniques to real-world scenarios. Item 5 states, “STEM Connection: A pipestem triangle is a piece of lab equipment that holds a beaker while it is being heated. What is the area inside the pipestem triangle shown?” (6.G.1)

• Unit 6: Numerical and Algebraic Expressions, Lesson 6-1: Division Expressions with Fractions and Whole Numbers, Session 2, Extend Thinking, students interpret and make comparisons between fraction division problems. The materials state, “For exercises 1 and 2, find the quotient by drawing a model. 1. \frac{1}{3}\div4 and 2. \frac{1}{6}\div2 For exercises 3-5, answer the questions. 3. What do you notice about the quotient \frac{1}{3}\div4 and the quotient \frac{1}{6}\div2? 4. How does \frac{1}{3} compare to \frac{1}{6}? 5. How does 4 compare to 2?” (6.NS.1)

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 3: Ratios and Rates, Lesson 3-2: Understand Rates and Unit Rates, Session 2, Summarize & Apply, students solve unit rate problems and use ratio and rate reasoning to solve real world problems. The materials state, “Yelina’s mother’s current budget for commuting to and from her job is shown. She is considering purchasing a monthly public transportation pass that costs $145. Her walk from the train station to her office would be about 25 minutes. Question: What is your recommendation for Yelina’s mother for her daily commute?” (6.RP.2)

• Unit 5: Solve Area, Surface Area, and Volume Problems, Mathematical Modeling, students find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. The materials state, “Your school has adopted a one-acre (43,560 square feet), parallelogram-shaped plot of land for reforestation. A local conservation group sent the table below of recommended trees for the climate of the area. You and your classmates will decide on a planting plan for the site. The conservationists recommend selecting no more than 3 types of trees and to include a mixture of trees with different growth rates. Your plan should include the number of each type of tree and a diagram showing where each type of tree will be planted. Be sure to consider the area of the plot of land when making your decisions.” Six tree options are pictured and include captions with spacing requirements and growth rate of each type of tree. (6.G.1)

• Unit 7: Integers, Rational Numbers, and the Coordinate Plane, Lesson 7-6: Determine Distance on the Coordinate Plane, Session 1, Activity-Exploration, students explore determining the distance between two points on the coordinate plane. The materials state, “Students plan the locations of different features by plotting points on a coordinate plane map of a town (Coordinate Plane 2 Teaching Resource). Students have freedom as to where they place these features, as long as they follow the criteria listed in the Activity Exploration Journal.” (6.NS.8)

The materials reviewed for Reveal Math 2025 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.

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 2: Understanding the World Around Us Through Statistics, Lesson 2-6: Divide Multi-Digit Numbers Using an Algorithm, Session 2, Lesson Quiz, Item 4, students develop procedural fluency with the standard algorithm for division. Item 4 states, “A deli owner buys 12 pounds of smoked salmon for $333 from the fish market. What is the cost of smoked salmon per pound?” (6.NS.2)

• Unit 8: Equations and Inequalities, Lesson 8-1: Understand Equations and Their Solutions, Session 1, Exit Ticket, Item 1, students engage in conceptual understanding as they use variables to represent numbers and write equations when solving real-world problems. Item 1 states, “Each stack of newspapers is 3\frac{1}{2} inches high. Write an equation you can use to find how many 3\frac{1}{2} inch stacks s of newspapers make a stack 49 inches high.” (6.EE.6)

• Unit 10: Math Is…, Lesson 10-1: Math is Everywhere, Session 2, Summarize & Apply, students apply ratio and rate reasoning in a real-world scenario. The materials state, “Did you know that leaving the water running while brushing your teeth wastes an average of four gallons versus only turning the water on to rinse? How much water would you save each year if every person in your house ran water only to rinse? How does your savings compare to a household of 3, 4, 5, and 8 people?” (6.RP.3)

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. The materials 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 offers, “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: Understanding the World Around Us Through Statistics, Mathematical Modeling, students build conceptual understanding, use application, and develop procedural fluency as they analyze the results of three water analysis tests to determine which contaminant, if any, violates the maximum contaminant level by using mean and fluently dividing numbers using the standard algorithm. The materials state, “Project One The table shows the average amount of water consumed each day in an indoor household in the United States by appliance or device. A local family has made the commitment to reduce their water consumption by \frac{1}{5}. What recommendations can you make to the family? Which appliance or device would you recommend they use less frequently? Develop a plan that could get them to their goal.” (6.NS.2, 6.SP.5)

• Unit 3: Ratios and Rates, Lesson 3-1: Understand Ratios, Session 2, Guided Exploration, students build conceptual understanding and application as they use ratio and rate reasoning to solve real-world and mathematical problems. The materials state, “Making Salad Dressing, Tahira is making a salad dressing using the recipe shown. She plans to use 4 tablespoons of vinegar. How can she determine the amount of olive oil she will need to keep the taste of the salad dressing the same? One Way Use a tape diagram. The vinegar and olive oil are both parts of the salad dressing. Multiply the quantities of vinegar and olive oil by 4 to maintain the same ratio. Another Way, Use a double number line. A double number line shows the ratio relationships on two number lines. With 4 tablespoons of vinegar, 12 tablespoons of olive oil are needed. Let’s Explore More, a. How can Tahira determine the amount of salad dressing she will have if she uses 6 tablespoons of vinegar? b. Tahira added 4 teaspoons of mustard to the 4 tablespoons of vinegar. How will her salad dressing taste?” (6.RP.3) 

• Unit 7: Integers, Rational Numbers, and the Coordinate Plane, Lesson 7-6: Determine Distance on the Coordinate Plane, Session 2, Summarize & Apply, students build procedural fluency and application as they solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. The materials state, “You can use the coordinates and absolute values to find distances between points with the same first coordinate or the same second coordinate. When two points are in the same quadrant, subtract the absolute values of the coordinates to determine the distance. When the points are in different quadrants, add the absolute values of the coordinates to determine the distance. Apply: Deon’s Errands The table shows the locations for several places in town. The grid shows a map of the town where each unit labeled on the grid represents one mile. Deon needs to go to the dry cleaner, which is \frac{3}{4} mile west and 1\frac{1}{4} miles north of the library. Question: What are the coordinates of the dry cleaner?” The item includes a table with location coordinates. (6.NS.8)

The materials reviewed for Reveal Math 2025 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. 

Within the Implementation Guide, Math Practices, the materials state, “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 3: Ratios and Rates, Lesson 3-5: Compare Ratio Relationships, Session 1, Activity- Based Exploration, students use a variety of strategies to compare ratios. The materials state, “Two friends create shades of purple paint. Reginald uses 3 parts red and 5 parts blue to make his shade of purple. Anwar uses 2 parts red and 3 parts blue to make his shade of purple. Whose mixture contains the most blue paint? Use the space below to record your findings with descriptions, drawings, and representations. Be prepared to share your findings. Concluding Question 1. What methods can you use to compare two ratios that are not equivalent?” 

• Unit 6: Numerical and Algebraic Expressions, Unit Overview, Math Practices states,“Making sense of quantities and their relationships in problem situations is a key skill in working with numeric and algebraic expressions. Students will use sense-making and perseverance throughout the unit. For example, to evaluate numerical expressions with exponents, students use the order of operations to plan a solution pathway, to identify if two expressions are equivalent, students look for entry points such as identifying which, if either, expression can be simplified or analyzing the parts of each expression. When simplifying algebraic expressions, students may check that the original and simplified expressions are equivalent by substituting a value for the variable in each.” The Unit Overview discusses how the mathematical content in this unit is enriched by MP.1: make sense of problems and persevere in solving them. 

• Unit 9: Relationships Between Two Variables, Lesson 9-4: Apply Two-Variable Relationships to Solve Problems, Session 1, Guided Exploration, students analyze, make sense, and compare quantities in two-variable relationships. The materials state, “An assembly line at an automobile manufacturing plant is run in 8-hour shifts and produces 2.5 cars per hour. The manufacturer has received the order shown. Should the manufacturer accept the order?” The task includes a picture of a purchase order for 1,000 black cars in a production time of 3 weeks. 

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 2: Understanding the World Around Us Through Statistics, Lesson 2-4: Represent and Describe Data in a Box Plot, Session 2, Guided Exploration, students represent situations symbolically as they compare data using two box plots. The materials state, “The sixth graders and seventh graders want to compare their finish times in the 5-kilometer race. How do the finish times of the students in the two grades compare?” The task includes two tables with data. The teacher materials prompt teachers to “Have students work with a partner to determine how they can represent the relationship between quantities. Students may mention that they can use and compare box plots to interpret the relationship.” 

• Unit 4: Understand and Use Percentages, Lesson 4-4: Find and Compare with Percentages, Session 2, Lesson Quiz, Items 9-10, students understand the relationships between problem scenarios and mathematical representations as they use ratio and rate reasoning to solve real-world and mathematical problems. The materials state, “A group of sixth-grade students were asked to choose a vegetable snack. The table shows the number of students that chose each type of vegetable. 9. How can you determine what percent of students chose celery? 10. What percent of students chose broccoli?” The items include a table of different vegetables and the number of students. 

• Unit 8: Equations and Inequalities, Unit Overview, Math Practices states, “Reason Abstractly and Quantitatively 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 about 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. For example, have students name the operation used in the given equation, identify the inverse operation, and name the Property of Equality they will use to solve the equation.”

The materials reviewed for Reveal Math 2025 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.

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: Understanding the World Around Us Through Statistics, Lesson 2-9: Describe Data by Mean Absolute Deviation, Session 1, Activity-Based Exploration, students construct viable arguments and critique the reasoning of others as they describe variability of a data set. The materials state, “Have students work with a partner to discuss what they can listen for as others share their arguments. These can include listening for viable arguments and critiquing the reasoning of others. In this problem situation, students can describe the circumstances in which data sets might have the same mean value but have deviations that are significantly different. Have students complete the Concluding Question in their Activity Exploration Journal. How does the average amount that each data value deviates from the mean describe the spread of a data set?”

• Unit 5: Solve Area, Surface Area, and Volume Problems, Lesson 5-8: Determine the Surface Area of Pyramids, Session 1, Guided Exploration, students construct viable arguments as they explore how to determine the surface area of a pyramid. The materials state, “Let’s Explore More a. Why are the heights of the triangles in a square pyramid congruent? b. The triangular faces in a square pyramid are congruent. Is this true for all pyramids

• Unit 6: Numerical and Algebraic Expressions, Lesson 6-3: Explore Numerical Expressions with Exponents, Session 1, Guided Exploration, students critique the reasoning of others as they explore writing addition and multiplication expressions in different ways. The materials state, “Let’s Explore More: a. How does the value of 4\times5 compare to the value of 5^4? b. Can you write the expression \frac{2}{5}+\frac{2}{5}+\frac{2}{5} as a power?” Teacher Guidance states, “Have students think about how they know they have understood the argument presented. Help students listen intently to the argument. Encourage them to write down any parts of the argument they would like clarified.” 

• Unit 7: Integers, Rational Numbers, and the Coordinate Plane, Lesson 7-2: Represent Rational Numbers and Their Opposites on the Number Line, Session 1, Activity-Based Exploration, students construct arguments and critique the reasoning of others as they explore rational numbers and their opposites by plotting numbers on a number line. The materials state, “Introductory Question: How can you plot a non-integer number on a number line? Concluding Question 1. How can you determine whether the opposite of a fraction or a decimal number will be positive or negative?” The Teacher Guidance states, “Students can share out their ideas about the relationships between fraction and decimal numbers and their opposites. Ask students to think about the questions they have about their classmates’ ideas. Have students share anything they want to add on or critique their classmates’ ideas.”

The materials reviewed for Reveal Math 2025 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.

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 5: Solve Area, Surface Area, and Volume Problems, Lesson 5-1: Determine the Area of Parallelograms and Rhombuses, Session 1, Activity-Based Exploration students check to see whether an answer makes sense and change the model when necessary to find the area of special quadrilaterals. Teacher Guidance states, “Hands-On: Students determine which figure on the Area of Quadrilaterals Teaching Resource has the greatest area. Students can cut the paper and move the pieces around to form squares or rectangles if needed. Have volunteers share their responses while others listen respectfully. Guide students to see the value in pausing to check their work and determine whether their answer is reasonable. Encourage students to take the time to check their work throughout the unit.”

• Unit 8: Equations and Inequalities, Lesson 8-3: Write and Solve Equations Using Multiplication or Division, Session 2, Summarize & Apply, students model the situation with an appropriate representation to write and solve one step equations. The materials state, “The student council buys snacks in bulk to sell during lunch. One box of snacks contains the types and quantities of snacks shown below. The first month, the student council sold 115 bags of snacks, including 50 bags of cheese crackers. Question: How many boxes of snacks would you recommend the student council order for next month?” Teacher guidance, “Elicit Evidence of Student Understanding How does creating models such as tape diagrams help you write multiplication and division equations involving variables? How does understanding of the Multiplication Property of Equality and Division Property of Equality make you a more efficient problem solver?” 

• Unit 9: Relationships Between Two Variables, Lesson 9-3: Write Equations to Represent Relationships Between Two Variables, Session 2, Guided Exploration, students write an equation from a graph, describe what they do with a model and how it relates to the problem situation. The materials state, “Let’s Explore More: a. Which representation do you find most helpful when representing the relationship between two variables: a table, a graph, or an equation? b. How are the graph and equation related?” Teacher Guidance states, “How can you apply mathematics to model the context? Listen to students’ responses and make sure that they know that they are applying mathematics to model the context when they use a table, graph, or an equation.”

• Implementation Guide, Unit Walk-Through, Mathematical Modeling states, “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 4: Understand and Use Percentages, Lesson 4-3: Estimate the Percent of a Number, Session 2, Guided Exploration, students recognize the insight to be gained from different tools and strategies as they estimate the percent of a number using rounding or compatible numbers. The materials state, “What is another tool that would be helpful in approaching this problem? Have volunteers share the tools that would be helpful in approaching this problem. As needed, help students brainstorm other tools, while keeping a list for all to reference. Making a rough sketch of a pie chart, using different colors to shade in hundredths’ decimal grid, or making a labeled table with percents and quantities are some other tools that may help students understand the problem.”

• Unit 6: Numerical and Algebraic Expressions, Lesson 6-2: Division Expressions with Fractions and Mixed Numbers, Session 2, Summarize & Apply, students recognize the insight to be gained through the use of certain tools, models or equations to divide fractions and mixed numbers by fractions and mixed numbers. The materials state, “Summarize: Divide Fractions and Mixed Numbers You can use models or equations to divide fractions or mixed numbers by fractions and mixed numbers. How many \frac{1}{4}s are in 2\frac{3}{8}?” Teacher guide, “Elicit Evidence of Student Understanding How would you explain to a friend a scenario in which you might divide a fraction by a fraction? What is beneficial about using a visual model to find the quotient? Why is writing an equation to find a quotient helpful?” 

• Unit 7: Integers, Rational Numbers, and the Coordinate Plane, Lesson 7-3: Understand Absolute Value of Rational Numbers, Session 1, Guided Exploration, students recognize the insights gained from using a number line as they explore the absolute value of rational numbers. The materials state, “Claire is performing a science experiment with two liquids. The temperatures of the two liquids in degrees Celsius are shown. She needs to use the beaker with the temperature that is closer to 0℃. Which beaker should Claire use? You can graph the temperatures on a number line to compare.” Teacher guidance, “Have students work with a partner to find the distance each point is from 0. Student-pairs may want to recreate the number line using the Blank Open Number Lines Teacher Resource. As students work on the activity and answer the question, encourage them to explain how they found their solutions. How does this tool help solve the problem? Have students discuss how a thermometer is the same and different than a number line. Ask students to consider how each tool can be used to help them determine how far a rational number is from zero.”

The materials reviewed for Reveal Math 2025 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. 

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 3: Ratios and Rates, Lesson 3-7: Ratio Reasoning: Convert Measurements Between Systems, Session 2, Guided Exploration, students attend to precision as they explore the concept of converting Customary measures to metric measures. The materials state, “What units of measure are needed to be precise? Have students discuss why the correct unit must be identified when using measurements. Mask sure they understand that mathematicians use notation, including symbols, to communicate ideas accurately.”

• Unit 5: Solve Area, Surface Area, and Volume Problems, Lesson 5-5: Determine the Volume of Rectangular Prisms, Session 1, Activity-Based Exploration, students attend to precision as they find the volume of a rectangular prism by filling it with \frac{1}{2} inch unit cubes. The materials state, “What units of measure are needed to be precise? Ask for volunteers to share their thoughts on the level of precision. Have students consider how their results would change if they used \frac{1}{4} inch unit cubes, 1-inch unit cubes, or 2-inch unit cubes. Have students complete the Concluding Questions in their Activity Exploration Journal. How do you find the volume of a rectangular prism when it is filled with \frac{1}{2} inch unit cubes? You can also use the formula V=lwh to find volume. How is filling a rectangular prism with \frac{1}{2} inch unit cubes related to this formula?” 

• Unit 7: Integers, Rational Numbers, and the Coordinate Plane, Lesson 7-6: Determine Distance on the Coordinate Plane, Session 1, Guided Exploration, students attend to precision as they find the distance between two points on a coordinate plane that have the same y-coordinate and use appropriate units to label graphs accurately. The problem states, “What unit of measure is needed to be precise? Have students consider how the information given in the problem statement can be used to determine the needed unit of measure. The problem statement mentions that one unit on the coordinate plane is equal to one kilometer. Students should notice that they should label their answer of 4.5 with kilometers, so that the answer makes sense in the context of the problem.” 

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 2: Understanding the World Around Us Through Statistics, Lesson 2-5: Describe Data by Range and Interquartile Range, Session 1, Activity-Based Exploration, students use specific mathematical language to describe a data set by giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation). The materials state, “Ask for volunteers to share the math terms they used during the activity. Listen for students to mention the following terms: spread, variability, difference, first quartile, third quartile, lower extreme, and upper extreme. Have students complete the Concluding Questions in their Activity Exploration Journal. What does the variability of the middle half of the data describe? How would changing values in a data set affect the variability of the whole and middle half of the data set?

• Unit 6: Numerical and Algebraic Expressions, Lesson 6-5: Write and Evaluate Algebraic Expressions, Session 2, Activity Based Exploration, students use specific mathematical language to write and evaluate expressions with unknown quantities. The materials state, “During the whole-class discussion, connect students’ understanding of writing and evaluating algebraic expressions to these new key terms and concepts: An algebraic expression is an expression with at least one variable and one operation. A variable represents an unknown quantity that can change or vary. Variables can be letters or symbols. The Substitution Property of Equality states that is a = b, then b may be substituted for a in any expression containing a. Each part of an algebraic expression that is separated by a plus or minus sign is a term. Terms that contain the same variable, with the same exponent, are called like terms. The numerical factor of each term that contains a variable is a coefficient. A term without a variable is a constant. What mathematical terminology can you use to convey understanding? Have students work with a partner to identify the mathematical language they can use to describe how to write an expression.” 

• Unit 9: Relationships Between Two Variables, Unit Opener: Ignite, Building the Language of Mathematics, students use specific mathematical language while using a graphic organizer throughout the unit to build understanding of and proficiency with key mathematical terms and concepts. The materials state, “Building the Language of Mathematics Relationships Between Two Variables.” A graphic organizer is provided and “Multiple Representations of Relationships Between Two Variables'' is in the middle. There are four boxes around the middle that state, “Situation A conveyor belt moves objects at 100 feet per minute. Equation y = 100x, Graph and Table.” At the bottom of the graphic organizer, it says “In my example relationship, ___ is the dependent variable and ___ is the independent variable.”

The materials reviewed for Reveal Math 2025 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.

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 2: Understanding the World Around Us Through Statistics, Lesson 2-8: Describe Data Using the Mean, Session 2, Activity-Based Exploration, students look for and make use of structure as they describe data using the mean and discover how changing a value in a data set impacts the mean. The materials state, “Arihi has these five bags of marbles. She wants to share them with four friends so that each person, including herself, has the same number of marbles. How many marbles does each person get? Explain your problem-solving strategy in the space below. Concluding Questions 1. How does the mean of a data set summarize the data set? 2. How can adding a value to a data set greater than all other values affect the mean of the data set?” Teacher Guidance states, “Math is… Looking for Structure How does exploring patterns help you solve problems? Have students work with a partner to discuss solving problems by exploring patterns. Ask for volunteers to share their responses with the class. Make sure that students understand the relationship among the data values, representations, and the mean. Recognizing this relationship can help solve the problem.”

• Unit 3: Ratios and Rates, Lesson 3-1: Understand Ratios, Session 1, Activity-Based Exploration, students look for and make use of structure in part-to-part and part-to-whole relationships among quantities in real-world scenarios. The materials state, “You will represent and describe the relationship between two quantities in each scenario shown using a tape diagram and a double number line. 1. Tahira is making a dressing for a salad. The recipe calls for 1 tablespoon of vinegar for every 3 tablespoons of olive oil. Tahira plans on using 4 tablespoons of vinegar. How many tablespoons of olive oil will she need? 2. If two cups of apple juice are needed to make 10 cups of a fruit drink, how many cups of apple juice are needed to make 20 cups of the fruit drink?” Teacher Guidance states, “How can you connect mathematical ideas to representations? Students think about how some problems can be represented in different ways. These ways include tape diagrams and double number lines. Some students may also suggest colored counters or tiles, one color representing one quantity and the other the other quantity. Encourage students to describe how each model they make represents the problem and the advantages of each point of view.”

• Unit 4: Understand and Use Percentages, Unit Overview, Look for and Make Use of Structure, Teacher Guide states, “Analyzing and understanding the structure of percentages is a foundational skill for statistical analysis in middle school and beyond. Helping students see the structure of percent relationships and the structure of percent equations will increase their fluency in percent calculations. Help students connect percentages to ratio reasoning by reinforcing the idea of percent as a ratio out of 100. By solving percent problems as equivalent fractions, students apply a known structure to a new concept. When students represent percent relationships with equations, engage students in discussion how the components of the equation relate to the part, the whole, and the percent in the situation. Help students connect the equation to other representations such as double number lines.”

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 5: Solve Area, Surface Area, and Volume Problems, Lesson 5-7: Determine Surface Area of Prisms, Session 1, Guided Exploration, students look for and express regularity in repeated reasoning to explore the surface area of a rectangular prism using a net and a formula. The problem states, “James is putting a special restorative stain on the entire surface of the wooden chest. One can of stain covers about 35 square feet. How many cans of stain will James need?” Teacher Guidance states, “How can generalizing be helpful in solving this problem? Ask for volunteers to share their responses while others listen respectfully. Students should be able to generalize the connection between the prism and its net.”

• Unit 6: Numerical and Algebraic Expressions, Lesson 6-7: Find Factors and Multiples, Session 1, Guided Exploration, students look for and express regularity in repeated reasoning to explore how to use a greatest common factor to solve a real-world problem. The materials state, “The school store manager plans to sell a combo pack as a back-to-school item. Each combo pack will have an equal number of pencils and an equal number of notebooks. What is the greatest number of combo packs that he can make using all the pencils and notebooks?” Teacher Guide states, “Are there calculations that are being repeated? Have students discuss what calculations they repeated to list all the factors or to create the factor trees.”

• Unit 8: Equations and Inequalities, Lesson 8-4: Write and Represent Inequalities, Session 1, Activity-Based Exploration, students look for and express regularity in repeated reasoning as they determine what part of the process is repeating. The materials state, “Students use a number line to represent several possible solutions for each scenario shown on the Inequalities Table Teaching Resource. For each scenario, students should plot at least 10 points on a number line. Students record the points and circle the inequality that represents the scenario.” The Teacher Guide states, “What about the process is repeating? Ask volunteers to share their responses with the class while others listen respectfully. Students should recognize that each value needs to be compared in the inequality to determine if it is a solution of the inequality.”