6th Grade - Gateway 2

The instructional materials for enVision Florida Mathematics Grade 6 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, Do You Understand? Question 6, students use number lines and diagrams to develop the concept of division of fractions by whole numbers. “What division equation is represented by the diagram?” (6.NS.1.1)
• In Lesson 1-5, Example 2, students use bar diagrams and area models to understand the concept of dividing fractions. “How much of a $$\frac{3}{4}$$-cup serving is in the $$\frac{2}{3}$$ cup of yogurt?” (6.NS.1.1)
• Lesson 4-8 poses the essential question, “What does it mean for one variable to be dependent on another variable?” In Practice Problem 19, students solve: “The number of oranges in a bag and the cost of the bag of oranges are related. What is the independent variable in this relationship? Explain.” (6.EE.3.9)
• Lesson 4-10 relates variables to the coordinate plane. Students use tables to discover relationships between dependent and independent variables and graph them appropriately. Practice Problem 7: “Complete the table and graph to show the relationship between the variables in the equation d = 5 + 5t.” (6.EE.3.9)
• In Lesson 5-1, students analyze and use bar diagrams and double number lines to build an understanding of ratios. Example 2: “The ratio of footballs to soccer balls at a sporting goods store is 5 to 3. If the store has 100 footballs in stock, how many soccer balls does it have?” Students use a bar diagram to show the initial ratio of 5:3 and then use the same diagram to show 100 footballs to determine the correct number of soccer balls. In Example 3, students use a double number line to find, “Chen can ride his bike 3 miles in 15 minutes. At this rate, how long will it take Chen to ride his bike 18 miles?” (6.RP.1.1 and 6.RP.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 6 meet the expectations that they attend to those standards that set an expectation of procedural skill and fluency. The instructional materials develop procedural skill and fluency throughout the grade.

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

• In the Teacher’s Edition, every Topic begins with Math Background: Rigor, where procedural skill and fluency for the content is outlined.
• In the Student Practice problems, Do You Know How? provides students with a variety of problem types to practice procedural skill and fluency.
• Each topic ends with a fluency review puzzle.
• There is additional practice of procedural skills and fluency 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 level.

• Lesson 3-3: Students use Order of Operations to evaluate numerical expressions. (6.EE.1.1) For example, Do You Know How? Question 6: “Evaluate. $$(8.2 + 5.3)\div5$$."
• Lesson 1-2: Students use the division algorithm to develop and maintain fluency in dividing whole numbers and decimals. (6.NS.2.2 and 6.NS.2.3) For example, Practice and Problem Solving question 25: “Divide. 187.2 ÷ 8."
• Lesson 1-1: Students practice fluently adding, subtracting and multiplying decimals (6.NS.2.3). For example, “Do You Know How?” Question 10. “Find the difference. 15 – 6.108.”
• Lesson 3-1: Students develop procedural skills with whole number exponents. (6.EE.1.1) For example, Do You Know How? Question 13: “Evaluate each power. $$7^3$$.”
• Lesson 3-6: Students generate equivalent expressions. (6.EE.1.4) For example, Practice and Problem Solving Question 18: “Write equivalent expressions, 2x + 4y.”
• Lesson 7-2: Students find the area of a triangle. (6.G.1.1) For example, Practice and Problem Solving Question 13: “The vertices of a triangle are A(0,0), B(3,6), and C(9,0). What is the area of the triangle?”

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 6 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.

• Topic 7 Performance Task Question 3: "Suppose you make dog blankets for Delia's company. Create a blanket with the following features: The finished blanket has an area of 28 inches by 18 inches; It has at least three sections of color, but no more than five; It has one section with an area of 81 square inches; It uses at least three different polygons, including a parallelogram that is not a rhombus."
• In Lesson 1-5, Practice and Problem Solving, students use models to divide fractions by fractions. (6.NS.1.1) For example, Question 26: “A large bag contains $$\frac{12}{15}$$ pound of granola. How many $$\frac{1}{3}$$-pound bags can be filled with this amount of granola? How much granola is left over?”
• In Lesson 4-3, Practice and Problem Solving, students write and solve addition and subtraction equations. (6.EE.2.7) For example, Question 17: “You have some baseball cards. You give 21 baseball cards to a friend and have nine left for yourself. How many baseball cards were in your original deck? Write and solve an equation to find t, the number of baseball cards in your original deck.”
• Topic 7, 3-Act Mathematical Modeling: That's a Wrap (6.G.1.4 and 6.EE.1.2c): Students determine how many stickers are needed to cover the surface area of a box. Question 15, "A classmate says that if all dimensions of the gift were doubled, you would need twice as many squares. Do you agree? Justify his reasoning or explain his error."

The instructional materials for enVision Florida Mathematics Grade 6 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 8-7, Practice and Problem Solving emphasizes application. Students use what they’ve learned about mean and median and apply it to describe the center, spread, and overall shape of data. (6.SP.1.2, 6.SP.2.5b, 6.SP.2.4, and 6.SP.2.5c) Question 10: “Describe the pattern in the dot plot. Then write about a situation that this data could represent. Explain why your situation has this pattern.”
• Lesson 1-6 emphasizes procedural skill. Students divide mixed numbers by mixed numbers and whole numbers. (6.NS.1.1) Practice and Problem Solving Question 21: “$$16\div2\frac{2}{3}$$”
• Lesson 4-6 emphasizes conceptual understanding. Students use and identify properties of equality to write equivalent equations. (6.EE.1.4) Practice and Problem Solving question 15: “This scale was balanced. Find the number to add that makes the scale become balanced again. Then complete the equation to make it true. 12 + ____ = 2 + 7 + 3 + 16.”

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

• Lessons 7-1, 7-2, 7-3, and 7-4: Students apply conceptual understandings of formulas when solving for area in real-world problems. (6.G.1) Practice and Problem Solving Question 21: Students determine if a custom truck with rectangular dimensions of 13.5 ft by 8.5 ft can fit in a parallelogram-shaped parking space with an area of 209 $$ft^2$$.
• Lesson 7-1: Find Areas of Parallelograms and Rhombuses emphasizes conceptual understanding and procedural skill. (6.G.1.1) Example 1 shows students the conceptual connection between parallelograms and rectangles. Example 2 introduces the formula for finding area. In Example 3, students practice the procedure for finding the area.
• Lesson 4-1: “Students are introduced to the concept of an equation and understanding that a solution of an equation is a value for the variable that makes the equation true.” Students also apply their understanding when they “solve one-step equations and use equations with one variable in mathematical and real-world problems.” The lesson includes two examples using a pan balance, then moves into bar diagrams. Students test solutions using substitution and begin to translate situations to equations. Practice and Problem Solving Question 21: “Gerard spent $5.12 for a drink and a sandwich. His drink cost $1.30. Did he have a ham sandwich for $3.54, a tuna sandwich for $3.82, or a turkey sandwich for $3.92? Use the equation s + 1.30 = 5.12 to justify your answer.”

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 3-7 Simplifying Algebraic Expressions  identifies a connection to 6.EE.1.3 with an emphasis on conceptual understanding: “Students use conceptual understanding when they identify like terms and equivalent expressions.” The lesson starts with an expression, x + 5 + 2x + 2, and shows steps to simplify, indicating that Alma used the commutative property and the property of multiplication. There are further examples and practice problems that are similar. Students do not have an opportunity to develop understanding of why 2x and 2 cannot be combined.

The instructional materials reviewed for enVision Florida Mathematics Grade 6 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 5-2, “MP.5.1 Use Appropriate Tools Strategically: Students will use a multiplication table to generate equivalent ratios.”
• In Student Practice problems, MPs are labeled with descriptions within problems. For example, Lesson 2-5, Practice and Problem Solving Question 22, “Use Structure: Suppose a, b, and c are all negative numbers. How do you find the distance between points (a,b) and (a,c)?”

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

• MP.4.1 enriches the mathematical content when students use unit rates to model the relationships between quantities presented in real-world problems, as well as identifying important quantities and using them to complete a double number line diagram to model ratio relationships. Lesson 5-6 identifies MP4 as an emphasis of instruction for Ratios and Rates. Practice and Problem Solving Question 19: “Katrina and Becca exchanged 270 text messages in 45 minutes. An equal number of texts was sent each minute. The girls can send 90 more text messages before they are charged additional fees. Complete the double number line diagram. At this rate, for how many more minutes can the girls exchange texts before they are charged extra?”
• MP.5.1 is used to enrich the mathematical content by using a tool (the multiplication table) that is already familiar to the students and having students connect that to ratios. In Lesson 5-2, students use a multiplication table to generate equivalent ratios. Practice and Problem Solving Question 19, “Equivalent ratios can be found by extending pairs of rows or columns in a multiplication table. Write three ratios equivalent to ⅖ using the multiplication table.”
• MP.1.1 is embedded in the problem as students go beyond a solution and show their understanding in a picture and words. In Lesson 1-5, students work with dividing fractions and mixed numbers. Practice and Problem Solving Question 27: “Higher Order Thinking. Find $$\frac{3}{4}$$ divided by $$\frac{2}{3}$$. Then draw a picture and write an explanation describing how to get the answer.”

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 7-4, the Table of Contents lists MPs 1.1, 4.1, 6.1, & 7.1, with MP1.1 listed in the Lesson Overview. MP.1.1 is 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.

The instructional materials reviewed for enVision Florida Mathematics Grade 6 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 7-2, Do You Understand? Question 3: “In example 1, if the other diagonal were used to divide the parallelogram into two triangles, would the area of each of these triangles be half the area of the parallelogram? Explain.”
• Lesson 2-2, Practice and Problem Solving, Question 28: “A classmate ordered these numbers from greatest to least: 4.4, 4.2, -4.42, -4.24. Is he correct? Construct an argument to justify your answer.”
• Lesson 3-5, Practice and Problem Solving, Question 31: “Katrina says that the expression 5,432 + 4,564 + 23,908 ÷ 61n can be evaluated by adding 5,432 + 4,564 + 13,908 and then dividing by the value of 61n. Do you agree? Explain.”
• Lesson 3-6, Explain It!: “Juwon says all three expressions are equivalent. (Graphics show three expressions Juwon has solved for A., B., and C.) Do you agree with Juwon that all three expressions are equivalent? Explain.”
• Lesson 6-1, Practice and Problem Solving, Question 22: “Kyle solved 18 of 24 puzzles in the puzzle book. He says that he can use an equivalent fraction to find the percent of puzzles in the book that he solved. How can he do that? What is the percent?”
• Lesson 2-3, Practice and Problem Solving, Question 39: “Alberto and Rebecca toss horseshoes at a stake. Whoever’s horseshoe is closer to the stake wins a point. Alberto’s horseshoe is 3 feet in front of the stake. Rebecca’s horseshoe is 2 feet past the stake. Alberto says that -3 is less than 2, so he wins a point. Is Alberto correct? Explain.”

The instructional materials reviewed for enVision Florida Mathematics Grade 6 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 Francisco’s and Abby’s work and ask: 'How does Francisco’s model show that Tom’s friends ate the same amount of vegetable pizza as pepperoni pizza? How does Abby’s model show that $$\frac{2}{5}$$ = $$\frac{4}{10}$$?'”
• In the Visual Learning portion of the lesson, there are sections labeled, Elicit and Use Evidence of Student Thinking and Convince Me. In Lesson 8-5, the materials prompt teachers with, “Can the mean absolute deviation ever have a negative value? Explain.”
• The 3-Act Mathematical Modeling activities prompt the teacher 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?”

The instructional materials reviewed for enVision Florida Mathematics Grade 6 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 5 Understand and Use Ratio and Rate, the vocabulary listed for the lessons includes: ratio, terms, circumference, diameter, equivalent ratios, Pi, rate, and unit rate.   
• New vocabulary terms are highlighted in the text and definitions are provided within the sentence where each term is found. In Lesson 5-5, the terms rate and unit rate are highlighted and defined. “A rate is a special type of ratio that compares quantities with unlike units of measure.”
• 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 1, students use the word bank of four terms to fill in the blanks on three items. Students also provide a mathematical example of the term used in each item. Students are to Use Vocabulary in Writing with this prompt: “Explain how to use multiplication to find the value of $$\frac{1}{3}\div\frac{9}{5}$$. Use the words multiplication, divisor, quotient, and reciprocal 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. Lesson 7-5, page 414 Elicit and Use Evidence of Student Thinking: “How many bases does this solid figure have? What is the shape of the base(s)? What are the shapes of the other faces? Is this solid a prism or a pyramid?”
• Each mid-Topic checkpoint includes a vocabulary section where students demonstrate understanding of the terms.