6th Grade - Gateway 2

The instructional materials reviewed for Reveal Math Grade 6 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. 

The structure of the lessons provide several opportunities that address conceptual understanding, and the materials include problems and questions that develop conceptual understanding throughout the grade-level.

• In the Teacher’s Edition, both Modules and Lessons begin with The Three Pillars of Rigor where conceptual understanding for the topic is briefly outlined. For example, Lesson 2-2 states, “In this lesson, students continue to develop understanding of percents. They begin to understand that percents greater than 100% represent numbers greater than 1 and percents less than 1% represent numbers that are significantly less than the whole.”
• In Explore & Develop, Explore is “intended to build conceptual understanding through Interactive Presentations that introduce the concept and can be completed by pairs on devices or as a whole class through digital classroom projection.” For example, in Lesson 5-4, “Students will examine what happens to the value of an algebraic expression as the values of each of the variables change. Throughout this activity, students will use Web Sketchpad to explore the changing values of variables by using a slider. Students will use their observations to make conjectures about how the values of the variables impact the value of the algebraic expression.” (6.EE.6)
• Some Interactive Presentations (slide format) introduce vocabulary and methods to complete concepts. These Presentations include Teaching Notes with suggestions for student activities. For example, Lesson 1-2, Learn Slide 2, “Present students with the ratio table showing the relationship between the number of cups of Greek yogurt and the number of cups of flour in the pizza dough recipe. You may wish to have a student reveal how ratio tables show both an additive structure and multiplicative structure. Encourage students to attend to the differences in structures.” (6.RP.1) Related student pages contain examples involving bar models, ratio tables, and double number lines for students to build understanding. Teachers can use presentations during instruction, and students access presentations independently as needed.

Examples of the materials providing opportunities for students to independently demonstrate conceptual understanding include:  

• In Lesson 1-1 Learn, “The students at Madison Middle School are going on a field trip. The principal requires one teacher chaperone for every eight students attending the field trip. Drag a possible combination of students (S) and teachers (T) to see if the bus will move. Then hit GO! to see if the bus will move. The bus will only move if the relationship is maintained of one teacher for every eight students. One of the buses has 16 students on it. How many teachers need to be on the bus to serve as the chaperone? Drag the appropriate number of teachers (T) onto the bus. Then hit GO! to see if the bus will move.” (6.RP.1)
• In Lesson 4-1 Explore, “Students will work in pairs to progress through the activity. Students will first use a thermometer to explore positive integers. They will then explore a thermometer with both positive and negative values. Finally, students will engage in the drag and drop activity that involves placing the negative numbers on the thermometer and explore how they relate to their everyday lives.” (6.NS.5)
• In Module 5, Test Form A, Question 19B, “Explain how using the Distributive Property can help you use mental math to multiply numbers.” (6.EE.3) 
• In Lesson 8-2 Explore, students use Web Sketchpad to explore how the area of a parallelogram is related to the area of triangles. “What do you notice about the base and height of the triangle and the base and height of the parallelogram? Make a conjecture about how to use what you know about the area of the parallelogram to find the area of triangle DEF? Explain your reasoning.” (6.G.1)

The instructional materials reviewed for Reveal Math Grade 6 meet expectations for attending to those standards that set an expectation of procedural skill and fluency.

The structure of the lessons includes several opportunities to develop these skills. The instructional materials develop procedural skill and fluency throughout the grade-level.

• In the Teacher’s Edition, both Modules and Lessons begin with The Three Pillars of Rigor where procedural skill and fluency for the topic is briefly outlined. For example, Module 5, “In this module, students use their understanding to build fluency with using powers and exponents, order of operations, and mathematical properties, as well as evaluating multi-step algebraic expressions and generating and simplifying equivalent algebraic expressions.”
• In Explore & Develop, Develop gives students multiple examples to practice “different strategies and tools to build procedural fluency.” For example, Lesson 6-2, Example 2, students are shown how to solve one-step addition problems with whole numbers using algebra tiles, a bar diagram, and the Subtraction Property of Equality; then in Example 3, students are encouraged to “adhere to the Subtraction Property of Equality” when the problem involves fractions: “Solve 3 3/4 + m = 7 1/2. Check your solution.” (6.EE.7)
• Some Interactive Presentations (slide format) demonstrate procedures to solve problems. For example, in Lesson 3-4, students are shown the “equation method” to divide fractions, “Find the reciprocal of the divisor and multiply.” (6.NS.1)
• Some Checks address procedural skills and fluency. For example, Lesson 8-3, Example 1 Check: Given bases of 6.1 and 10 units and a height of 9.7 units, “Decompose the trapezoid to find its area.” The intent is to use the area formulas for a rectangle and a triangle. (6.G.1)

The instructional materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade-level.

• Lesson 3-1. Example 2, Divide Multi-Digit numbers. Find 5,272 ÷ 64. Move through the steps to solve by annexing the zero.” The steps are all demonstrated and written out. (Divide from left to right, Multiply 8 x 64, then subtract. Multiply 2 x 64, then subtract. There is a remainder. Annex a zero) (6.NS.2)
• Lesson 3-2, Learn - Multiply Decimals, “When multiplying a decimal by a decimal, multiply as with whole numbers. To place the decimal point in the product, find the sum of the number of decimal places in each factor. The product has the same number of decimal places. If there are not enough decimal places in the product, annex zeros to the left of the first non-zero digit.” (6.NS.3)
• In Lesson 9-1, Practice Questions 1-6, “2) Roy made a jewelry box in the shape of a rectangular prism with the dimensions shown. What is the volume of the jewelry box?; 4) The rectangular prism shown has a volume of 115 cubic yards. What is the length of the prism?” (6.G.2)

The instructional materials for Reveal Math Grade 6 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied. 

The instructional materials include multiple opportunities for students to engage in routine and non-routine application of mathematical skills and knowledge of the grade-level.

• In the Teacher’s Edition, Modules and Lessons begin with The Three Pillars of Rigor where application for the topic is briefly outlined. For example, in Module 7, “In this module, students draw on their knowledge of tables, equations, and the coordinate plane … to apply their understanding of relationships between two variables to solve real-world problems.”
• Each Module includes a Performance Task that addresses application. For example, in Module 6, Performance Task, “The students in the chorus will be performing at an amusement park. After the performance, the students will get to go on the rides. The students sell scones at breakfast during the school year to help raise money for the trip. Part A. Parents volunteer to make the scones. They are given a recipe. The recipe calls for 6 1/2 cups of flour. Duyl has already added 2 3/4 cups of flour. How much more flour will he need to add? Use a bar diagram to model this situation. Then write and solve an addition equation to find the answer. How can you check your answer? Explain.”
• Some Checks address application. For example, in Lesson 9-3, Apply, Check, “The dimensions of two climbing walls that are in the middle of an obstacle course are shown. How much greater is the surface area of Wall B than Wall A?” (6.G.4)
• Some Exit Tickets address application. For example, in Lesson 5-5, Exit Ticket, “Samantha and her sister have 250 roses and 175 peonies. If they want to use all of the flowers, how many identical centerpieces can they make, if they want to have as many as possible? Write a mathematical argument that can be used to defend your solution.” (6.NS.4)

The instructional materials provide opportunities for students to independently demonstrate the use of mathematics flexibly in a variety of contexts.

• In Lesson 1-5, Apply, Inventory, “The manager of an office supply store decides to hold a Buy 2, Get 1 Free sale on all reams of paper. A ream of paper holds 500 sheets of paper. The sale is held for one week and a total of 154 reams were sold (not including the ones given away for free). If each ream of paper cost the store $4.50, how much money did the store lose by giving away the free reams of paper?” (6.RP.3)
• In Lesson 1-8, Apply, “Keshia can ride her bike 15 miles in 90 minutes. She wants to ride in a bike-a-thon that consists of two trail options, a 56-mile trail or a 36-mile trail. At her current rate, how many more minutes will it take her to ride 56 miles than 36 miles? If she wants to ride for about 4 hours, which trail should she choose?” (6.RP.3)
• In Lesson 3-5, Example 1, “Faye is making party favors. She is dividing 3/4 pounds of cashews into 12 packages. How many pounds of cashews are in each package?” (6.NS.1)
• In Lesson 6-2, Practice Question 12, “Create. Write and solve a real-world problem that can be solved with a one-step addition equation.” (6.EE.6,7)
• In Lesson 8-1, Practice Question 10, “Create. Draw and label a parallelogram with a base that is 2 times its height and has an area that is less than 100 square yards.” (6.G.1)

The instructional materials for Reveal Math Grade 6 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. Many of the lessons incorporate two aspects of rigor, with an emphasis on application, and practice problems for students address all three aspects of rigor.

All three aspects of rigor are present independently throughout the materials, and examples include:

• In Lesson 4-1, Explore and Develop, Learn, students place positive and negative numbers on a number line to represent a quantity in a familiar situation, emphasizing conceptual understanding. “Select the button to see where negative and positive integers are on the number line. Watch animations to see how integers are used in real life.” Example, “A football team has a 10 yard loss on a play. Write an integer to represent the situation.” (6.NS.5,6)
• In Lesson 3-4, Practice, students divide fractions by fractions, emphasizing procedural skill. “Divide. Write in simplest form. 1) 5/6 ÷ 5/12; 4) Chelsea has 7/8 pound of butter to make icing. Each batch of icing needs 1/4 pound of butter. Write and solve an equation that models the situation. Then interpret the quotient.” (6.NS.1)
• In Lesson 8-4, Explore and Develop, Apply, Home Improvement, students use area formulas to solve real-world and mathematical problems. “Takeru is planning to paint the walls of his bedroom which is in the shape of a rectangular prism. The bedroom has one window and two doors. The dimensions of the window and doors are shown in the table. If one gallon of paint covers about 150 square feet, how many gallons of paint are needed to cover the walls of a room that is 20 feet long, 15 feet wide, and 8 feet high?” (6.G.1)

Examples of the materials integrating at least two aspects of rigor include:

• In Lesson 2-4, Explore and Develop, Examples 1 and 2, students develop understanding of finding the percent of a number using the rate per 100 and mental math, equivalent ratios, bar diagrams, and double number lines. In Example 3, students develop procedural skill by using ratio tables and equivalent ratios to find the percent of a number when the percent is greater than 100. The goal for the lesson is students “Come up with their own strategy to solve an application problem involving attendance: Book Fair - Students were asked which night they planned on attending the book fair. The results of the survey are shown in the table. Twenty percent of the students who planned to attend on Wednesday attended on Thursday instead. Twenty-five percent of the students who planned to attend on Thursday attended on Wednesday instead. Which day, Wednesday or Thursday, had a greater actual attendance? By how many students?” (6.RP.A)
• In Lesson 10-1, students develop understanding of the difference between a statistical and non-statistical question, and students practice the steps to answer a statistical question by collecting data and organizing the data in a table or graphical display for analysis/summary. Throughout the lesson, students have multiple opportunities to demonstrate understanding and practice the skills. For example, “Practice Question 5: Suppose you want to determine the number of siblings each of your classmates have. You survey them using the question How many siblings do you have? The responses were 1, 4, 2, 3, 0, 1, 0, 5, 1, 2, 2, 3, 0, 1, 2, 0, 1, 1, 6, 2 siblings. Organize the data by completing the table and analyze the results.; Practice Question 8: “Multiselect. Which of the following are statistical questions? Select all that apply.” (6.SP.1)

The instructional materials reviewed for Reveal Math Grade 6 meet expectations that the Standards for Mathematical Practice (MPs) are identified and used to enrich mathematics content within and throughout the grade-level.

All 8 MPs are clearly identified throughout the materials, including:

• The materials contain a Correlation to the Mathematical Practices PDF which includes explanations and descriptions of the MPs and examples of MPs located in specific lessons.
• Within the digital module opener and lesson, the Standards tab contains a list of the MPs found in that specific module/lesson. The same list is part of the Teacher Edition PDF. Throughout each lesson, the program indicates each opportunity for students to engage in the practices, with an MP symbol and a description of how to connect the MP to the content within the lesson. 
• In Reflect and Practice, questions intended to engage students in the MPs are specifically noted with an MP symbol. The Teacher Edition states which of the MPs each practice question is intended to align with.
• Performance Task rubrics list which MPs students are intended to engage in during the task.
• Each component of the digital materials (Learn, Explore, Examples, Apply) contains an About this Resource narrative explaining how related MPs should specifically be addressed within the activity. The same information is found in the Teacher Edition PDF in the margin labeled MP Teaching the Mathematical Practices.
• Each lesson includes Launch - Today’s Standards: How can I use these Practices?. The Teacher’s Notes recommend that teachers, “Tell students that they will be addressing these content and practice standards in this lesson. You may wish to have a student volunteer read aloud How Can I meet this standard? and How can I use these practices? and connect these to the standards.”

Examples of the MPs being used to enrich the mathematical content include:

• MP1: In Lesson 1-2, Explore and Develop, Apply - Packaging, students persevere with this non-routine application problem that requires multiple steps. “A toy store sells assorted marbles, sold in small or large bags. The table shows the number of each color of marble in the small bag. The manager of the store wants to maintain the same ratio of each color of marble in the large bag as in the small bag. Each marble costs 20 cents. If the large bag contains 20 green marbles, how much does the large bag cost?”
• MP7: In Lesson 7-1, Practice Question 9, students use the structure of an expression to find the value of a variable given a rule and a value (in this case, the output) for the other variable (the input). “Identify Structure. Complete the table by finding the input values: Students are given the rule 2x-2.5 and the outputs 7.5, 10.5, 13.5.”
• MP8: In Lesson 4-2, Explore and Develop, Example 3, students use regularity in repeated reasoning to make a conjecture about the number of negative signs in an expression. For example, “-[-(-3)].” Students start by identifying the original integer, then they “Talk About It: Compare the opposite of the opposite of a number to the original number.”

There are instances where the labeling of MPs is inconsistent, and examples of this include:

• In Lesson 2-2, the materials identify MPs 1, 2, 3, and 5, and during the lesson, students utilize MPs 2, 5, and 7.
• In Lesson 1-7, the print materials identify MPs 1, 2, 3, and 6, and the digital materials also identify MP4.

The instructional materials reviewed for Reveal Math Grade 6 meet the expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

Examples of the materials prompting students to both construct viable arguments and analyze the arguments of others include: 

• Talk About It! in lesson examples are often opportunities for students to create viable arguments. For example, in Lesson 1-2, “Why might a ratio be more advantageous to use than a bar diagram when finding the quantity of each ingredient needed to make 5 pizzas?”
• In Lesson 1-4, Exercise 6, students construct an argument to defend their chosen display and why they think it is more advantageous than other displays. In Exercise 8, students diagnose and explain why Avery’s solution is incorrect and correct the solution.  
• In Lesson 1-7, Exit ticket, “Mr. Blackwood is buying boxes of snack-size bags of crackers to pack in his family’s school lunches. The first box costs $9.80 and contains 20 bags. The second box contains 30 bags and costs $13.50. Write about it! Which box is the better buy? Write a mathematical argument that can be used to defend your solution.”
• In Lesson 2-5, Practice Question 18, “Justify Conclusions. A store is having a 40% off sale. If you have $38, will you have enough money to buy an item that regularly sells for $65.99? Write an argument to justify your conclusion.”
• Write About It! within lesson examples are often opportunities for students to engage with MP3. In Apply of many lessons, students are prompted to “Write About It! Write an argument that can be used to defend your solution.”
• In Lesson 4-2, Practice Question 19, “Justify Conclusion. A student states that -x is always equal to a negative integer. Is the student correct? Justify your reasoning.” 
• In Lesson 4-3, Practice Question 12, “Justify Conclusions. A student said -5 is less than -4 and |-5| is less than |-4|. Is the student correct? Justify your reasoning."
• In Lesson 5-4, Practice Question 17, “Find the Error. Evaluating the expression 4b + c for b = 2 and c = 3. Find the student’s mistake and correct it.”

The instructional materials reviewed for Reveal Math Grade 6 meet the expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

There are multiple locations in the materials where teachers are provided with prompts to elicit student thinking.

• In Resources, there is a Correlation to the Mathematical Practices, Grade 6, which defines the Standards for Mathematical Practice. For example, MP3 is defined, there are examples where MP3 “Students are required to justify their reasoning and to find the errors in another’s reasoning or work. Look for the Apply problems and the exercises labeled as Make a Conjecture, Find the Error, Use a Counterexample, Make an Argument, or Justify Conclusions. Many Talk About It! question prompts ask students to justify conclusions and/or critique another student’s reasoning. In the Teacher Edition, look for the Teaching the Mathematical Practices tips labeled as this mathematical practice.”
• There are Questions for Mathematical Discourse in the Develop and Explore of each lesson. For example, in Lesson 6-1, the teacher notes suggest, “Guide students through the example using these questions for mathematical discourse, Why is 3 not a solution?; Once you know that 3 is not a solution, how do you know to check numbers greater than 3, as opposed to less than 3?; Once you know that 4 is a solution, do you need to check whether 5 is a solution? Explain.”
• Talk About It! is designed to elicit student justification. For example, in Lesson 5-5, Learn, “Talk About It! When is making a list of the factors difficult to do?” The teacher’s notes provide guidance, “Encourage students to think about the process they use to find the factors of very large numbers in order to make a plausible argument and justify their reasoning.” 
• The materials also prompt teachers to have students share their responses to Write About It!. Teacher guidance throughout the materials states, “As students respond to the Write About It! prompt, have them make sure their argument uses correct mathematical reasoning. If you choose to have them share their responses with others, encourage the listeners to ask clarifying questions to verify that the reasoning is correct.” The Write About It! prompts typically read, “Write an argument that can be used to defend your solution.” 
• The Teacher Edition includes Teaching the Mathematical Practices tips which involve developing arguments. In Lesson 10-4, Teaching the Mathematical Practices, Exercise 7: “Students determine the validity of the statement. Encourage students to use the structure and characteristics of a box plot to determine the statement is false.; Exercise 9, “Encourage students to explain why the student’s thinking is correct.”
• Teacher’s Notes often give prompts and suggestions for facilitating arguments. For example, in Lesson 2-2, Practice Question 11, Collaborative Practice states, “Find the Error. A student said that to represent 0.2% with a 10x10 grid, you should shade 2 squares in the grid. Find the student’s error and correct it.” The teacher notes include, “Have students work together to prepare a brief explanation that illustrates the flawed reasoning. For example, the student in the Exercise thinks that 0.2% is equivalent to 2%. Have each pair or group of students present their explanation to the class.”

The instructional materials reviewed for Reveal Math Grade 6 meet the expectations for explicitly attending to the specialized language of mathematics.

The materials use precise and accurate mathematical terminology and definitions, and the materials support students in using them. Teacher’s guides, student books, and supplemental materials explicitly attend to the specialized language of mathematics.

• In Resources, there is a Correlation to the Mathematical Practices, Grade 6, which defines the Standards for Mathematical Practice. For example, MP6 is defined, there are examples where MP6 can be found, and states, “Students are routinely required to communicate precisely to partners, the teacher, or the entire class by using precise definitions and mathematical vocabulary. Look for the exercises labeled as Be Precise. Many Talk About It! prompts ask students to clearly and precisely explain their reasoning. In the Teacher Edition, look for the Teaching the Mathematical Practices tips are labeled as this mathematical practice.”
• In each Module introduction, What Vocabulary Will You Learn? prompts teachers to lead students through a specific routine to learn the vocabulary of the unit.
• Many Lessons have a “Language Objective.” For example, in Lesson 9-2, “Students will describe how nets can help find the surface area of rectangular prisms, correctly using a noun (rectangle) and the adjective form of the noun (rectangular).”
• In each lesson, Math Background briefly describes key concepts/vocabulary or directs teachers to an online component to learn background. Definitions are not included, but are accessible in the glossary. Glossary definitions are precise and accurate, and there are definitions for math content and math models. In addition, the glossary references the lesson where the vocabulary is introduced.
• The lesson Launch includes a vocabulary section that introduces new vocabulary for the lesson. During Develop and Explore, the new vocabulary is always bolded and defined. For example, in Lesson 5-5, Learn: “A common factor is a number that is a factor of two or more numbers.”
• In Lesson 4-5, What Vocabulary Will You Learn?, students use the prefix quad- to begin to understand “quadrant”. Teachers ask students to consider other words they know with quad- as a prefix.
• When students see vocabulary in successive lessons, What Vocabulary Will You Use? assists teachers in facilitating discussions that help students apply the vocabulary they have previously learned.
• In Lesson 5-1, Teaching Notes for Interactive Slideshows, “Students will learn the definitions of exponent, power, and base. Play the animation for the class. Students will learn how to write a power using a base and an exponent and how to label each part of the expression, including expressions with multiple bases and exponents."
• In Lesson 2-6, Practice Question 11, “Be Precise. Of the number of sixth grade students at a middle school, 120 prefer online magazines over print magazines. Of the number of seventh grade students, 140 prefer online magazines. A student said that this means a greater percent of seventh grade students prefer online magazines than sixth grade students? Is the student correct? Use precise mathematical language to explain your reasoning.”
• Each Module includes a Vocabulary Test. “This summative assessment asset is designed for students to demonstrate their knowledge, understanding, and proficiency of the vocabulary covered in this module.”