7th Grade - Gateway 2

The instructional materials reviewed for Reveal Math Grade 7 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, Module 2 states, “In this module, students draw on their understanding of proportional relationships related to ratio reasoning and properties of operations to solve algebraic equations involving percents.”
• 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 4-1, Explore, “Students will use an algebra tiles tool. The tool includes an equation mat and algebra tiles that represent x, -x, 1, and -1. Throughout this activity, students will use the algebra tiles to model and solve one-step equations with integers.” (7.EE.4)
• 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 7-2, Explore and Develop - Learn, “Ask students to compare and contrast the steps for writing an equation from a real-world problem with the steps for writing an inequality from a real-world situation. Point out the steps are the same, but the symbols used are different. Students should pay close attention to the phrases used in an inequality situation to determine the correct symbol to use.” (7.EE.4b) Teachers can use presentations during instruction. Students may access presentations independently as needed.
• Some Checks address conceptual understanding. For example, Lesson 10-1 Check, “Classify the likelihood of each event as impossible, unlikely, equally likely, likely, or certain: spinning a number less than 5 on a spinner divided into 4 equal sections labeled 1 through 4; choosing a weekday when randomly selecting dates from a given year; it rains, given the chance of rain is 25%; drawing a red marble from a bag containing only 10 marbles; flipping a coin and it landing on heads.” (7.SP.5)
• Some Exit Tickets address conceptual understanding. Lesson 5-5 Exit Ticket, “The expression 6x + 17y - (3x + 5y) represents the number of decorations and streamers that would be left over. Find the simplified expression that represents the number of decorations and streamers that were given to each of three neighboring classrooms. Explain how you found this expression.” (7.EE.1) 

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

• Lesson 1-2, Explore and Develop - Example 1, “The recipe for a homemade glass cleaner indicates to use a ratio of 1 part vinegar and 12 tablespoons of water to make the cleaner. Is the relationship between the vinegar and water in the recipe and the vinegar and water in Elyse’s cleaning solution a proportional relationship? Explain. Talk about it! Would this ratio be maintained if she used 1 cup of vinegar and 4 cups of water? Explain.” (7.RP.A)
• Lesson 5-1, Explore and Develop - Explore, students learn to combine like terms and use the distributive property to simplify algebraic expressions. “How can algebra tiles be used to simplify an expression? Situation: Patrick, Santiago, and Kaya work at a restaurant. Each week, Patrick works three more than twice the number of hours that Santiago works. Kaya works 2 hours less than Santiago. An expression for the amount of time each student worked is shown in the table. Talk about it!: How could you write an expression to represent the total number of hours the three students worked?” (7.EE.4)
• Lesson 11-1 Exit Ticket, “Suppose you wanted to determine the number of students in your entire school who prefer having a certain type of pet (cat, dog, or other). Design an unbiased sampling method you can use and explain why your sampling method is unbiased.” (7.SP.1)

The instructional materials reviewed for Reveal Math Grade 7 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 7, “In this module, students will use their understanding of inequalities and equations to build fluency in solving and graphing one- and two-step inequalities.”
• In Explore & Develop, Develop gives students multiple examples to practice “different strategies and tools to build procedural fluency.” For example, Lesson 3-5, Explore and Develop - Exercise 1, “ Move through the steps to evaluate the expression -4(3) + 7. Multiply -4(3). Add.” Similar practice throughout: “Find -4(-5)(-2) - (-8).” (7.NS.3)
• Some Interactive Presentations (slide format) demonstrate procedures to solve problems. For example, in Lesson 2-1, Explore and Develop - Example 1: Procedures are shown to find the percent of increase using equivalent ratios. “Step 1 Identify the part and the whole…. Step 2 Find the percent of increase.” (7.RP.3)
• Some Checks address procedural skills and fluency. For example, Lesson 6-2 Check, “Solve -2 + 2/3w = 10.” (7.EE.4)

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

• Lesson 4-1, Practice Questions 1-12, “Write each fraction as a decimal. Determine if the decimal is a terminating decimal. 4) -5/6.” (7.NS.2d)
• Lesson 5-5, Explore and Develop - Example 1, “Simplify -2(x+3) + 8x. Write your answer in factored form. Step 1) Write the expression: -2(x+3) + 8x; Step 2) Distributive Property: -2(x+3) + 8x = -2x - 6 + 8x; Step 3) Combine like terms: -2x + 8x - 6 = 6x - 6; Step 4) Write in factored form: = 6(x-1).” (7.EE.1)
• Lesson 6-4, Practice Questions 1-4, “Solve each equation. Check your solution. 1) 4(x+8) = 44.” (7.EE.4a)
• Lesson 7-6, Practice Questions 1-3, “Solve each inequality. Graph the solution set on a number line. 1) -3x - 3 > 12.” (7.EE.4b)

The instructional materials for Reveal Math Grade 7 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 Lesson 2-7, “In this lesson, students use their knowledge of proportional relationships and percents to … apply their understanding to percent error to solve real-world problems.”
• Each Module includes a Performance Task that addresses application. For example, in Module 4, Performance Task, Operations with Rational Numbers, Part E, “When Noah arrives at work on Thursday, he finds out that one of the other employees is unable to work for the next few days. The company asks him to cover the other employee’s properties in addition to his own. This means that Noah will have to mow 1 1/3 times as many acres on Thursday, 2.5 times as many acres on Friday, and 1 5/16 times as many acres on Saturday. Explain how many total acres Noah will now have to mow over the next three days. Noah knows he will not have time to do all the work himself, so he asks his brother to mow half of the total number of acres. How many acres will Noah’s brother mow?” (7.NS.3)
• In Lesson 7-2, Reflect and Practice, Apply, Practice Question 8, “Open Response. Teddy has two piggy banks. The difference in the amount of money between the two banks is no more than $10. One piggy bank has $7.31 in it. Determine the possible amount of money in the other piggy bank. Then interpret the solution.” (7.EE.4)
• Some Checks address application. For example, in Lesson 1-2, Apply, Check: ”One type of yarn costs $4 for 100 yards. Another type of yarn costs $5 for 150 yards. Is the relationship between the number of yards and the cost a proportional relationship between the two types of yarn? Explain.” (7.RP.2)
• Some Exit Tickets address application. For example, in Lesson 2-2, Exit Ticket, “Suppose you purchase school supplies for $20 and lunch meat for $5 at a store. Sales tax of 6% is added to all non-food items in your state. What is the total cost of all the items? Explain how you found the total cost?” (7.RP.3, 7.EE.2)

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

• In Lesson 2-1, Apply, “The first known motion picture was filmed in 1888 and lasted for only 2.11 seconds. Today, we watch movies that last an average of about two hours. What is the percent of change in the times from 1888 to today? Round your answer to the nearest whole percent if necessary.” (7.RP.A, 7.EE.3)
• In Lesson 4-6, Practice Question 14, “Jake is enclosing his vegetable garden with fencing. The table shows the dimensions of his rectangular garden. Fencing is sold in 2.5-foot sections and costs $25.99 per section. How much will it cost to fence in the entire garden?” (7.NS.3)
• In Lesson 8-4, Example 2, “The scale of the floor plan is 1 inch = 3 feet. What is the actual area of Bedroom 3?” (drawing shows floor plan with dimensions in fractional parts) (7.G.1)
• In Lesson 9-3, Practice Question 13, ”Suppose a swimming pool is in the shape of a composite figure that has a curved side that is not a semicircle. Explain how you could estimate the area of the swimming pool.” (7.G.6)
• In Lesson 10-6, Practice Question 2, “Open Response. Leigh designs and conducts a computer simulation with 30 trials and uses the data from the simulation to create the relative frequency bar graph shown. The graph shows the relative frequency of the number of spins needed for a spinner divided into 6 equal sections labeled A through F to land on each letter at least once. Using the graph, what is the experimental probability that more than 10 spins are need to land on each letter at least once? Write this probability as a percent.” (7.SP.8)

The instructional materials for Reveal Math Grade 7 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 1-3, Explore and Develop, Example 1, students develop an understanding of recognizing proportional relationships between quantities. Students use a table to answer the following, “Is the amount of money she earns proportional to the number of hours she spends babysitting?” Think About It! asks, “Think of a way that Carrie could be paid so the amount she made was not proportional to the number of hours she worked. Explain your reasoning.” (7.RP.2)
• In Lesson 9-1, Explore and Develop, Example 1, students develop procedural skill in finding the circumference of a circle given a diameter. “Big Ben is a famous clock tower in London, England. The diameter of the clock face is 26 feet. Find the circumference of the clock face. Use 3.14 for $$\pi$$. Round to the nearest hundredth if necessary. Think About it! What formula can you use to find the circumference if you know the diameter?” The next slide includes step-by-step guidance in using $$C=\pi d$$.” Practice Question 2: “A circular fence is being used to surround a doghouse. How much fencing is needed to build the fence? Use 3.14 for $$\pi$$. Round to the nearest hundredth if necessary.” (7.G.4)
• In Lesson 3-4, Explore and Develop, Apply, students apply properties of operations with rational numbers in a real-world situation. “Natalie had $165 in her bank account at the beginning of the summer. Over the next 10 weeks, she worked at a summer camp and added $160 to her savings each week, while spending only $40 each week. Once she gets back to school, she plans to spend $105 per week. For how many weeks can she make withdrawals until her balance is $0?” (7.NS.3)

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

• In Lesson 3-3, students develop conceptual understanding of multiplication of integers, and engage in application in Question 14: “Mrs. Rockwell lost money on an investment at a rate of $4 per day. What is the change in her investment, due to the lost money, after 4 weeks?” Question 18 involves conceptual understanding and procedural skill: “The product of two integers is −24. The difference between the two integers is 14. The sum of the two integers is 10. What are the two integers?” (7.NS.2a)
• In Lesson 7-6, Explore and Develop, Example 5, students develop an understanding of two-step inequalities and procedural skill in writing and solving two-step inequalities and graphing the solution set on a number line. Students write two-step inequalities and interpret the solution based on real-world contexts, for example: “Meredith is given a $50 monthly allowance to buy lunch at school. Any remaining money can be spent on entertainment. Meredith would like to have at least $12 left at the end of the month to go to the movies with her friends. It costs Meredith $2.50 per lunch that she buys at school. Write and solve an inequality to determine the number of lunches Meredith can buy and have at least $12 left. Then interpret the solution.” (7.EE.4)

The instructional materials reviewed for Reveal Math Grade 7 meet expectations that the Standards for Mathematical Practice 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 7-5, Explore and Develop, Apply, students use variables to represent quantities when making sense of a real-world situation. “The students at Westlake Middle School are raising money for buses to go on a science field trip. Each bus holds 44 students and costs $150 for the day. If at least 275 students go on the trip, how much money will they need to raise for buses?”
• MP2: In Lesson 9-1, Explore and Develop, Apply, students reason about finding the circumference of the neighbor’s garden using only radius measurements, rather than finding the diameter. In Practice Question 13, students consider a similar problem: “How would the circumference of a circle change if its radius was doubled? Provide an example to support your reasoning.”
• MP7: In Lesson 11-4, Launch, Today’s Standards: “I can make use of the structure of a double box plot or double line plot to compare centers and variability.”

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

• In Lesson 5-2, Apply, the digital materials identify MP4 with the Theater problem, but the print materials do not identify the same problem with MP4.

The instructional materials reviewed for Reveal Math Grade 7 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 2-4, Example 2, “Talk About It! The final price had a discount of 20% followed by a discount of 30%. Is this the same thing as finding 20% + 30% or 50% of the original prices? Use the values in the Example to justify your reasoning.” 
• In Lesson 1-4, Practice  Question 7, “Make an Argument. Determine if a line can have a constant rate and not be proportional. Write an argument to defend your response.”
• In Lesson 9-1, Practice Question 14, “Justify Conclusions. Use mental math to determine if the circumference of a circle with a radius of 5 inches will be greater than or less than 30 inches. Write an argument that can be used to justify your solution.”
• 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 10-3, Practice Question 11, “Find the Error. The spinner shown has 8 equal-size sections. A student said that the theoretical probability of spinning a multiple of 3 on the spinner is 5/8. Find the student’s error and correct it.”

The instructional materials reviewed for Reveal Math Grade 7 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 7, which defines the Standards for Mathematical Practice. For example, MP3 is defined, there are examples connected to MP3, and states, “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 Develop and Explore of each lesson. For example, in Lesson 7-6, Example 1, the teacher notes suggest, “Guide students through the example using these questions for mathematical discourse: Explain why you need to reverse the inequality symbol when solving this inequality.; If -5x - 12 < 8, what must be true about -5x - 12? Explain without calculating the value of x.”
• Talk About It! is designed to elicit student justification. For example, in Lesson 7-6, “As the students discuss the Talk About It! question on Slide 4 (Suppose when Jesse solved the inequality, he claimed the solution is x < -4. Find his error and explain how to correct it.), encourage them to use correct mathematical vocabulary when explaining the flaw in Jesse’s work.”
• 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 2-3, “As students discuss the Talk About It! question, encourage them to make a case for when Method 1 might be the more helpful method to use.”
• Teacher’s Notes often give prompts and suggestions for facilitating arguments. For example, in Lesson 4-5, Learn - Teaching Notes Slide 1 states, “Be sure that students understand and can be able to explain why it is often easier to write all of the numbers as decimals before computing if the fractions or mixed numbers would have decimal forms that terminate (repeat, zeros).”

The instructional materials reviewed for Reveal Math Grade 7 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 7, 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 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 8-1: “Students will describe angles with precise mathematical vocabulary: acute, obtuse, right, vertical, adjacent, congruent.”
• 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 2-5, Learn: “If you borrow money or deposit money, the principal is the amount of money borrowed or deposited.”
• In Lesson 1-1, What Vocabulary Will You Learn?, “What are some synonyms of the word relationship?” Teacher notes: “Recommended Use: Tell students that they will be using this key vocabulary term in this lesson. Facilitate a class discussion using the question shown on the screen.”
• 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 7-4, Teaching the Mathematical Practices, the teaching notes state, “Attend to Precision. Students should notice how an inequality changes when they multiply or divide each side of the inequality by a certain value. Students should calculate and compare accurately and efficiently, paying careful attention to the symbols of inequalities.”
• 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.”