8th Grade - Gateway 2

The instructional materials for enVision Florida Mathematics Grade 8 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.

The instructional materials do provide students opportunities to demonstrate conceptual understanding independently throughout the grade. For example:

• Lesson 6-2, Do You Understand?, “What do you notice about the corresponding coordinates of the pre-image and image after a reflection across the x-axis?” (8.G.1.1a, b, c and 8.G.1.3)
• Lesson 3-2, Do You Understand? Question 2, “How can you use a graph to determine that a relationship is NOT a function?” (8.F.1.1)
• Lesson 3-3, Practice and Problem Solving Question 11, “Justin opens a savings account with $4. He saves $2 each week. Does a linear function or a nonlinear function represent this situation? Explain.” (8.F.1.2 and 8.F.1.3)
• Lesson 6-1, Do You Understand? Question 2, “Triangle L’M’N’ is the image of triangle LMN after a translation. How are the side lengths and angle measures of the triangles related? Explain.” (8.G.1.1a,b,c, and 8.G.1.3)

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 8 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 2-3, students solve multi-step equations using the distributive property. (8.EE.3.7b) For example, Do You Know How? Question 5: “Solve the equation -3(x-1) + 7x = 27.” Practice and Problem Solving Question 12, “What is the solution of the equation 3(x + 2) = 2(x + 5)?”
• In Lesson 5-4, students solve systems of equations using elimination. (8.EE.3.8b) For example, Practice and Problem Solving, Question 8: “Solve the system of equations using elimination: 2y - 5x = -2; 3y + 2x = 35.”
• Lesson 6-4, students describe a sequence of transformations involving floor plans using a coordinate plane. (8.G.1.3) For example, Try It!: “Ava decided to move the cabinet to the opposite wall. What sequence of transformations moves the cabinet to its new position?”

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

The instructional materials for enVision Florida Mathematics Grade 8 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 independently demonstrate the use of mathematics flexibly in a variety of contexts. Non-routine problems are typically found in Performance Tasks and STEM activities.

• In Topic 2 STEM Project (8.EE.2.6), students use linear equations to explore demography, connecting linear equations to predictions of population growth. "Develop a linear equation to represent population growth after x years.”
• Topic 3 Performance Task, Form A (8.F.1 and 8.F.2): “Sofia and Hector are captains of their track and field teams. As captains, they help their teammates train for the different events.” The following questions provide different information in graph and table form and five different questions about this scenario.

The instructional materials for enVision Florida Mathematics Grade 8 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 4-1: Students develop conceptual understanding about constructing scatter plots to interpret the relationship of paired data.
• Lesson 2-3: Students develop procedural skills using the distributive property to solve multi-step equations.
• Lesson 1-10: Students apply scientific notation using operations: “The total consumption of fruit juice in a particular country in 2006 was about 2.28 x $$10^9$$ gallons. The population of that country that year was 3 x $$10^8$$. What was the average number of gallons consumed per person in that country in 2006?”

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

• In Lesson 7-2, students build conceptual understanding about using of the Converse of the Pythagorean Theorem to identify right angles. In Lesson 7-3, students use the Pythagorean Theorem and its converse in real-world problems like solving for the longest poster to fit in a given box and whether the angle of a ramp meets recommendations of horizontal distance for every foot of vertical rise.
• Throughout Topic 3, conceptual understanding about functions is developed and applied using real-world problems. Lesson 3-1 Question 11: Students identify if a relation is a function. “Taylor has tracked the number of students in his grade since third grade. He records his data in the table below. Is the relation a function? Explain.” Lesson 3-3, Question 1: Students compare linear and nonlinear functions. “Justin opens a savings account with $4. He saves $2 each week. Does a linear function or a nonlinear function represent this situation? Explain.” Lesson 3-5, Question 2: Students investigate intervals of increase and decrease. “How would knowing the slope of a linear function help determine whether a function is increasing or decreasing?” Lesson 3-6, Question 9: Students sketch functions from verbal descriptions: “Melody starts at her house and rides her bike for 10 minutes to a friend’s house. She stays at her friend’s house for 60 minutes. Sketch a graph that represents this description.”

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 7-1 Understand the Pythagorean Theorem has an emphasis on conceptual understanding and procedural fluency: “Students learn and understand the Pythagorean Theorem.” The lesson starts with: “Kelly drew a right triangle on graph paper. Kelly says that the sum of the areas of squares with side lengths a and b is the same as the area of the square with side length c. Do you agree with Kelly? Explain.” Example 1 shows pictures that develop the proof of the theorem and includes $$a^2+b^2=c^2$$. Example 2 immediately moves into Use the Pythagorean Theorem. The rest of the lesson and practice uses procedural steps.

The instructional materials reviewed for enVision Florida Mathematics Grade 8 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 TE connect standards and practices to descriptions of each lesson.
• Lesson Overviews include indications of Math Practices within a lesson. For example, in Lesson 6-2, states, “MP.2.1 Reason Abstractly and Quantitatively: Students will analyze the relationship between random samples and populations to make inferences about populations. They will compare different samples from the same populations.”
• In Student Practice problems, MPs are labeled with descriptions within problems. For example, Lesson 7-3 Do you Understand, question 2 says, “Look for Structure: How is using the Pythagorean Theorem in a rectangular prism similar to using it in a rectangle?”

The MPs are consistently used to enrich the mathematical content. For example:

• MP.7.1 enriches the mathematical content when students justify that a relationship is proportional when represented as a graph and then flexibly use graphs to describe proportional relationships. Lesson 2-6 identifies MP7.1 as an emphasis for students to analyze and solve Linear Equations. Do You Understand? For example, Question 3 “Why is the slope between any two points on a straight line always the same?” Practice and Problem Solving Question 9 says, “The points (2.1, -4.2) and (2.5, -5) form a proportional relationship. What is the slope of the line that passes through these two points?”
• MP.2.1 enriches the mathematical content as students expand their knowledge about rational numbers and the relationships with decimals and fractions when explaining their answer. Lesson 1-1, Question 16: ”When writing a repeating decimal as a fraction, why does the fraction always have only 9s or 9s and 0s as digits in the denominator?”
• MP.8.1 enriches the mathematical content when students recognize the mistake as using a + b = c instead of $$a^2+b^2=c^2$$, demonstrating that they’ve practiced using the equation $$a^2+b^2=c^2$$ and see the patterns of the equation in similar situations. Lesson 7-2, Question 13: “Three students draw triangles with the side lengths shown. All three say their triangle is a right triangle. Which students are incorrect? Which mistake might they have made?”

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 1-5, the Table of Contents lists MPs 2.1, 3.1, 4.1, 7.1, & 8.1, but MPs 2.1 and 7.1 are listed in the Lesson Overview. MP.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 8 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 3-4, Practice and Problem Solving, Construct Arguments: “Suppose another store sells a similar package, modeled by a linear function with initial value $7.99. Which store has the better deal? Explain.”
• Lesson 1-2, Explain It!: “Sofia wrote a decimal as a fraction. Her classmate Nora says that her method and answer are not correct. Sofia disagrees and says that this is the method she learned (graphic shown). Is Nora or Sofia correct?”
• Lesson 3-1, Practice and Problem Solving, Question 10: “During a chemistry experiment, Sam records how the temperature changes over time using ordered pairs (time in minutes, temperature in ℃). Is the relation a function? Explain.”
• Lesson 3-6, Explain It!: “The Environmental Club is learning about oil consumption and energy conservation around the world. Jack says oil consumption in the United States has dropped a lot. Ashley says oil consumption in China is the biggest problem facing the world environment. A) Do you agree or disagree with Jack’s statement? Construct an argument based on the graph to support your position. B) Do you agree or disagree with Ashley’s statement? Construct an argument based on the graph to support your position.”
• Lesson 1-2, Practice and Problem Solving, Question 13: “Deena says that 9.565565556… is a rational number because it has a repeating pattern. Do you agree? Explain.”
• Lesson 2-1, Practice and Problem Solving, Question 14 page 109: “Your friend solved the equation 4x + 12x - 6 = 4(4x + 7) and got x = 34. What error did your friend make? What is the correct solution?”
• Lesson 4-2, Practice and Problem Solving, question 3 page 220: “How does the scatter plot of a nonlinear association differ from that of a linear association?”

The instructional materials reviewed for enVision Florida Mathematics Grade 8 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 6-7 Solve and Discuss It!, the materials prompt teachers to “Ask students to share their solutions. If needed, project Melanie’s and Nick’s work and ask: 'How does Nick’s observation differ from Melanie’s? Do you think Melanie’s Answer or Nick’s answer is more complete? Explain.'”
• In the Visual Learning portion of the lesson, there are sections labeled, Elicit and Use Evidence of Student Thinking and Convince Me. In Lesson 8-4, the materials prompt teachers with, “A cone and a sphere have the same radius and height. Which will make the cone have the same volume as the sphere, doubling the radius of the cone or doubling the height of the cone? Explain.”
• 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 1-1, the materials prompt teachers with, “As students work through the Explain It, listen and look for different ways in which students analyze Sofia’s work. Incorporate their critiques into the classroom discussion.”

The instructional materials reviewed for enVision Florida Mathematics Grade 8 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 2 Analyze and Solve Linear Equations, the vocabulary listed for the lessons includes: slope, y-intercept, and slope-intercept form.  
• New vocabulary terms are highlighted in the text and definitions are provided within the sentence where each term is found. In Lesson 2-8,  the term y-intercept is highlighted, and the definition is provided within the sentence. “The y-coordinate of the point where the line crosses the y-axis is the y-intercept.”
• 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 4 Review, page 247, Use Vocabulary in Writing: “Describe the scatter plot at the right. Use vocabulary terms in your description.”