7th Grade - Gateway 2

The instructional materials for enVision Florida Mathematics Grade 7 meet the expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

The structure of the lessons includes several opportunities that address conceptual understanding.

• In the Teacher Edition, every Topic begins with Math Background: Rigor, where conceptual understanding for the Topic is outlined.
• Lessons are introduced with a video, “Visual Learning Animation Plus,” at PearsonRealize.com to build conceptual understanding.
• Links within the digital program to outside resources, such as Virtual Nerd, include videos for students that introduce concepts.
• In the Student Practice problems, Do You Understand? reviews conceptual understanding.

Materials include problems and questions that develop conceptual understanding throughout the grade level and provide opportunities for students to demonstrate conceptual understanding independently throughout the grade. For example:

• In Lesson 1-4, students use number lines to build their understanding by writing matching equations. In Example 2, “Ian’s football team lost two yards on a running play. Then they received a 5-yard penalty. What is the team’s total change in yards? Write a subtraction expression to represent the change in yards. Write an equivalent addition expression.” This example also shows the problem using a number line. (7.NS.1.1c and 7.NS.1.1d)
• Lesson 1-6, Example 2, “Why is it easier to show three groups of -500 on the number line than -500 groups of 3?” (7.NS.1.2a and 7.NS.1.2c)
• In Lesson 4-3, students simplify expressions by combining like terms with both integer and rational coefficients, as well as with two variables. (7.EE.1.1) Do You Understand? Question 1, “How are properties of operations used to simplify expressions?” Practice and Problem Solving Question 17, “Explain whether 11t - 4t is equivalent to 4t - 11t. Support your answer by evaluating the expression for t = 2.”
• In Lesson 1-1, students are shown how positive and negative numbers relate to zero on a number line by combining opposite quantities. Problems include determining the change in a temperature drop and the end height of a car on a roller coaster after dropping and rising. (7.NS.1.1a)

Physical manipulatives are not a part of the materials. When manipulatives are to be used by teacher and students, they are referenced in digital format.

The instructional materials for enVision Florida Mathematics Grade 7 meet the expectations that they attend to those standards that set an expectation of procedural skill and fluency.

The structure of the lessons includes several opportunities to develop these skills.

• In the Teacher Edition, every Topic begins with Math Background: Rigor, where procedural skills for the content is outlined.
• In the Student Practice problems, Do You Know How? is the second section, which provides students with a variety of problem types to practice procedural skills.
• There is additional practice of procedural skills online.

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

• In Lesson 1-5, students use the same procedure for adding and subtracting signed rational numbers as they do when adding and subtracting integers. (7.NS.1.1b and c) For example, Practice and Problem Solving, Question 11: “Simplify each expression. a) $$50\frac{1}{2}+(-12.3)$$ b) $$-50\frac{1}{2}+(-12.3)$$ c) $$-50\frac{1}{2}+12.3$$.”
• In Lesson 5-3, students use the distributive property to organize information in word problems in order to write and solve equations. (7.EE.2.3 and 7.EE.2.4a) For example, Do You Know How? Question 4: “A family of 7 bought tickets to the circus. Each family member also bought a souvenir that cost $6. The total amount they spent was $147. How much did one ticket cost?”
• In Lesson 1-9, students divide rational numbers. (7.NS.1.2b) For example, Practice and Problem Solving, question 15: “Find the quotient. Express your answer as a simplified fraction. $$\frac{3}{8}\div3.8$$”
• In Lesson 7-1, students use probability to describe chance, likelihood, and fairness. (7.SP.3.5) For example, Practice and Problem Solving, question 8: “A spinner has 8 equal-sized sections. Six of the sections are green. A) What is the probability that the spinner will land on green? ____ out of 8, or ____/4, or ____%. B) Use words to describe the probability. It is ____ that the spinner will land on green.”

In addition, each cumulative assessment spirals through all previous topics, reviewing key information with a variety of problems to reinforce skills.

The instructional materials for enVision Florida Mathematics Grade 7 meet the expectations that the materials are 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 structure of the lessons includes several opportunities for students to engage in application.

• In the Teacher Edition, every Topic begins with Math Background: Rigor, where applications of the content are outlined.
• In the Student Practice problems, Practice & Problem Solving provides students with a variety of problem types to apply what they have learned.
• Each Topic includes a Performance Task, where students apply math of the Topic in multi-step, real-world situations.
• Every Topic also includes a 3-Act Mathematical Modeling application problem.
• Each Topic includes a STEM project which is application; this incorporates more science or engineering.

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 as well as provide opportunities for students to demonstrate the use of mathematics flexibly and independently in a variety of contexts. Non-routine problems are typically found in Performance Tasks and STEM activities.

• In Lesson 2-1, Practice and Problem Solving, students apply knowledge of solving multi-step problems with rational numbers to solving problems with ratios, rates, and unit rates. (7.RP.1.1 and 7.RP.1.3) Question 9: Given 3 bags of rice, “Which package has the lowest cost per ounce of rice?”
• In Lesson 1-10, Do You Know How?, students solve problems using rational numbers operations. (7.NS.1.3) Question 6, page 66: “The temperature of a cup of coffee changed by -54$$\degree$$F over $$22\frac{1}{2}$$ minutes. What was the change in temperature each minute?” In Practice and Problem Solving, Question 11, page 67: “Brianna works as a customer service representative. She knows that the amount of her yearly bonus is $155, but $2.50 is taken away for each customer complaint about her during the year. What is her bonus if there are 12 complaints about her in a year?”
• In Topic 5 STEM Project (7.EE.2.3 and 7.EE.2.4), students research filtration systems, decide which one they would purchase, and plan a fundraiser. Part of planning is writing an equation to represent the amount of money they will earn from a fundraiser to purchase the filtration system.
• In Topic 2 3-Act Mathematical Modeling: Mixin' It Up (7.RP.1.1 and 7.RP.1.2a), students attempt to make liquid in a water glass have the same flavor as that of a large water cooler. Question 15, "A classmate usually adds six drops to 16 ounces of water. Use your updated model to predict the number she would use for the large container."

The instructional materials for enVision Florida Mathematics Grade 7 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately.

All three aspects of rigor are present in program materials. With few exceptions, lessons are connected to two aspects of rigor with an emphasis on application. Student practice includes all three aspects of rigor, though there are fewer questions for conceptual understanding.

There are instances where all three aspects of rigor are present independently throughout the program materials.

• Lesson 8-1: Students build conceptual knowledge of scale using scale drawings to find actual lengths and widths of things like a kitchen island, flooring needed in a living room, and the dimensions of a tennis court. Students use the scale to write proportions in order to solve for actual measurements.
• Lesson 5-3: Students learn and practice procedural skills using the distributive property to solve for problems that include a negative number.
• Lesson 7-3: Students apply the concepts of experimental and theoretical probability by comparing them: “Hakeem randomly draws equal-sized cards labeled with letters A, B, C, D, and F from a hat and records the results in the table. Compare the theoretical and experimental probabilities of randomly drawing a card that is labeled with the letter C.”

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials.

• In Lesson 6-2, students build conceptual understanding of using random samples to make and compare inferences about populations and determine if the inferences are valid. In Lesson 6-3, students practice procedures to compare two populations using measures of center and variability. In Lesson 6-4, students apply their knowledge of measures of center and variability and make inference about two populations.
• Lesson 1-3: Students build conceptual understanding of adding integers and apply this in real-world problems. Students start by analyzing the level of water in a pool. All of the examples in the lesson utilize a number line and students are prompted in the exercises to use a number line. Finally they apply it in situations such as Practice & Problem Solving question 13, “A submarine traveling 200 meters below the surface of the ocean increases its depth by 45 meters. Adam says that the new location of the submarine is -155 meters. Describe an error Adam could have made that would result in the answer he gave.”

For some standards that emphasize conceptual understanding, the materials do not provide students a consistent opportunity to develop understanding of the mathematical content within the standard and quickly transition to developing procedural skills around the mathematical content. An example of this includes:

• Lesson 4-2 Simplifying Expressions identifies a connection to 7.EE.1.1 with an emphasis on conceptual understanding. “Students develop a deeper understanding of the Associative and Commutative Properties and become fluent in simplifying expressions by combining like terms.” The lesson starts with students looking at how Marco and Andrea sorted a pile of number tiles that included variables: Marco did it by variable, Andrea used coefficients. Next, example 1 uses pictures of algebra tiles to simplify “-2c + 3c - 5 - 4c + 7.” Example 2 is “-3 + $$\frac{1}{3}$$x + (-4.5) - $$\frac{1}{5}$$x” and uses properties of operations to solve. The rest of the lesson and practice uses procedural steps. Students do not have an opportunity to develop understanding of like terms.

The instructional materials reviewed for enVision Florida Mathematics 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 eight MPs are clearly identified throughout the materials in numerous places, including:

• The Program Overview book begins by listing the eight Topics and their connections to standards and practices.
• The Table of Contents in the Program Overview book connects every lesson to standards and practices.
• The Math Practice and Problem Solving Handbook includes a list of the Mathematical Practice Standards and real-world scenarios modeled through questions and answers.  
• The online tools offer a video, “Math Practices Animation,” for each MP, with explanations of the Math Practices as well as problems that demonstrate the practice.  
• Topic Overviews contain bulleted descriptions of how MPs are addressed and what mathematically proficient students should do.
• Topic Planner Tables at the beginning of each Topic in the Teacher Edition connect standards and practices to descriptions of each lesson.
• Lesson Overviews include indications of Math Practices within a lesson. For example, in Lesson 1-5, page 31A states, “MP.8.1 Express Regularity in Repeated Reasoning: Students will generalize about the solutions of equations of the form $$x^2=p$$, where p is a positive rational number and of the form $$x^3=p$$.”
• In Student Practice problems, MPs are labeled with descriptions within problems. For example, Lesson 3-1, Practice and Problem Solving Question 17, “Make Sense and Persevere: 153 is 0.9% of what number? Tell which equivalent ratios you used to find the solution.”

The MPs are consistently used to enrich the mathematical content.

• MP.2.1 enriches the mathematical content when students interpret and compare statistical measures and reason about data sets in both qualitative and quantitative forms. Lesson 6-3 Do You Understand? Question 5 says, “The box plots describe the heights of flowers selected randomly from two gardens. Use the box plots to answer 4 and 5. Make a comparative inference about the flowers in the two gardens.” Practice and Problem Solving, Question 9 says, “A family is comparing home prices in towns where they would like to live. The family learns that the median home price in Hometown is equal to the median home price in Plainfield and concludes that the homes in Hometown and Plainfield are similarly priced. what is another statistical measure that the family might consider when deciding where to purchase a home?”
• MP.1.1 enriches the mathematical content when students examine the relationships between the quantities and solve for the whole. Lesson 3-2, Question 11: “A restaurant customer left $3.50 as a tip. The tax on the meal was 7% and the tip was 20% of the cost including tax. What was the total bill?”
• MP.4.1 enriches the mathematical content when students use a table as a mathematical model to represent a real-world situation. They use the quantities in the table to write expressions that represent relationships in the context of the situation. Lesson 1-10, Model with Math, Solve & Discuss It!: “Stefan estimates the income and expenses for renting a phone accessory store in the mall. He enters the amounts in the table below. Should Stefan rent a phone accessory store? Explain.”

Because the Mathematical Practices are labeled in so many places, they are not always consistent and are often overidentified. The identification is broad, rather than targeted, with labels being most relevant at the lesson level. For example:

• In Lesson 2-4, the Table of Contents lists MPs 2.1, 3.1, 4.1, & 7.1, but only MPs 2.1 and 7.1 are listed in the Lesson Overview. MPs 2.1 and 7.1 are integrated into the lesson; however, the other MPs are not a major part of the lesson.
• All 3-Act Math lessons identify all eight MPs, and the questions within 3-Act Math lessons are identical in each topic.
• Multiple MPs are identified for every lesson.
  

The instructional materials reviewed for enVision Florida Mathematics Grade 7 meet the expectations that the instructional materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

Student materials consistently prompt students to both construct viable arguments and analyze the arguments of others.

• Lesson 7-7, Practice and Problem Solving: “Construct Arguments: How is the difference between the simulated probability and the theoretical probability of an actual event related to the number of simulated trials conducted?”
• Lesson 5-3, Explain It!: “Six friends go jet skiing. The total cost for the adventure is $683.88, including a $12 fee per person to rent flotation vests. Marcella says they can use the equation 6r + 12 = 683.88 to find the jet ski rental cost, r, per person. Julia says they need to use the equation 6(r + 12) = 638.88. A) Whose equation accurately represents the situation? Construct an argument to support your response. B) What error in thinking might explain the inaccurate equation?”
• Lesson 5-5, Practice and Problem Solving, Question 2: “Why is -x <3 equivalent to x> -3? Provide a convincing argument.”
• Lesson 4-5, Practice and Problem Solving, Question 17: “Ryan says the expression 3 + 5y cannot be factored using GCF. Is he correct? Explain why or why not.”
• Lesson 2-1, Explain It?: “In a basketball contest, Elizabeth made 9 out of 25 free throw attempts. Alex made 8 out of 20 free throws attempts. Janie said that Elizabeth had a better free-throw record because she made more free throws than Alex. A) Do you agree with Janie’s reasoning? B) Decide who had the better free-throw record. Justify your reasoning using mathematical arguments.”
• Lesson 8-5, Do You Understand?: “Are there any circles for which the relationship between the diameter and the circumference cannot be represented by pi? Explain.”
  

The instructional materials reviewed for enVision Florida Mathematics Grade 7 meet the expectations that the instructional materials assist 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.

• Solve & Discuss It! or Explain It! at the beginning of each lesson include guidance for teachers to Facilitate Meaningful Mathematical Discourse. In Lesson 6-2, the materials prompt teachers to “Ask students to share their solutions. If needed, project Anna’s and Armando’s work and ask: 'How are Anna’s and Armando’s approach similar? How are they different?'” In Lesson 2-1, Explain It!, “How does Janie’s statement compare with Kyle’s argument to oppose it? What additional reasoning does Kyle use to oppose Janie’s statement?”
• In the Visual Learning portion of the lesson, there are sections labeled, Elicit and Use Evidence of Student Thinking and Convince Me. In Lesson 7-3, the materials prompt teachers with, “Is it possible for the theoretical probability to be $$\frac{1}{2}$$ while the experimental probability is 1? Give an example.”
• The 3-Act Mathematical Modeling activities prompt teachers to ask students about their predictions. “Ask about predictions. Why do you think your prediction is the answer to the Main Question? Who had a similar prediction? How many of you agree with that prediction? Who has a different prediction?”
• When MP.3.1 is identified as the emphasis of the lesson, teachers are provided with question prompts in the Lesson Overview and “look fors” such as: “How can you justify your answer? What mathematical language, models, or examples will help you support your answer? How could you improve this argument? How could you use counterexamples to disprove this argument? What do you think about this explanation? What question would you ask about the reasoning used?” In Lesson 6-4, the materials prompt teachers with, “As students work through the Explain It, listen and look for students who apply what they know about measures of center to understand the patterns in the geyser’s eruptions.”
  

The instructional materials reviewed for enVision Florida Mathematics Grade 7 meet the expectations that materials use accurate mathematical terminology.

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

• Each Topic Overview lists the vocabulary being introduced for each lesson. In Topic 7 Probability, the vocabulary listed for the lessons includes: outcomes, probability, event, theoretical probability, experimental probability, relative frequency, sample space, probability model, compound event, and simulation.  
• New vocabulary terms are highlighted in the text and definitions are provided within the sentence where each term is found. In Lesson 7-3, the terms relative frequency and experimental probability are highlighted and definitions provided within the sentence each term is found. “The relative frequency is the ratio of the number of times an event occurs to the total number of trials.”
• A Glossary in the back of Volume 1 lists all the vocabulary terms.
• A Vocabulary Review is included in the Topic Review. Students are provided with explicit vocabulary practice. In Topic 7, Use Vocabulary in Writing: “A restaurant serves either skim milk or whole milk in glasses that are small, medium, or large. Use vocabulary words to explain how you cold determine all the possible outcomes of milk choices at the restaurant. Use vocabulary words in your explanation.”
• Online there is an Animated Glossary and a Vocabulary Game. The video is another way to expose students to the vocabulary terms as it provides a visual and audio definition of each term.
• Teacher question prompts attend to precision using appropriate terminology. In Lesson 3-2, Pose Purposeful Questions, “Multiply the numerator and denominator of $$\frac{20}{100}$$ by an integer to get an equivalent fraction with a numerator of 260. What is the multiplier? What is the equivalent fraction?”
• Each mid-Topic checkpoint includes a vocabulary section where students demonstrate understanding of the terms.