6th Grade - Gateway 2

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

Materials include problems and questions that develop conceptual understanding throughout the grade level. According to the Teacher Resource Program Overview, “Problem-Based Learning The Solve & Discuss It in Step 1 of the lesson helps students connect what they know to new ideas embedded in the problem. When students make these connections, conceptual understanding takes seed. Visual Learning In Step 2 of the instructional model, teachers use the Visual Learning Bridge, either in print or online, to make important lesson concepts explicit by connecting them to students’ thinking and solutions from Step 1.” Examples from the materials include:

• Topic 1, Lesson 1-3, Solve & Discuss It!, students use a model to connect multiplying a fraction by a fraction. “The art teacher gave each student half of a sheet of paper. Then she asked the students to color one fourth of their pieces of paper. What part of the original sheet did the students color? Solve this problem any way you choose.” (6.NS.1)

• Topic 4, Lesson 4-5, Explore It!, students write and solve equations with rational numbers. “The cost of T-shirts for four different soccer teams are shown below. A. Lorna is on Team A. Ben is on another team. They paid a total of $21.25 for both team T-shirts. Write an equation to represent the cost of Ben’s shirt. B. Dario also plays soccer and he says that, based on the price of Ben’s T-shirt, Ben is on Team B. Is Dario correct? Explain.” (6.EE.7)

• Topic 7, Lesson 7-1, Solve & Discuss It!, students develop conceptual understanding of how to find the area of a parallelogram by decomposing the parallelogram and then composing shapes into a rectangle. “Sofia drew the grid below and plotted the points A, B, C, and D. Connect points A to B, B to C, C to D, and D to A. Then find the area of the shape and explain how you found it. Using the same grid, move points B and C four units to the right. Connect the points to make a new parallelogram ABCD. What is the area of this shape?” (6.EE.2c and 6.G.1)

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Practice & Problem Solving exercises found in the student materials provide opportunities for students to demonstrate conceptual understanding. Try It! provides problems that can be used as a formative assessment of conceptual understanding following Example problems. Do You Understand?/Do You Know How? Problems have students answer the Essential Question and determine students’ understanding of the concept. Examples from the materials include:

• Topic 1, Lesson 1-6, Practice & Problem Solving, Problem 31, students independently apply conceptual understanding of multiplying and dividing mixed numbers to explain the solution to an equation. “Higher Order Thinking If 9 ×$$\frac{n}{5}$$ = 9÷$$\frac{n}{5}$$, then what does n equal? Explain.” (6.NS.1)

• Topic 5, Lesson 5-5, Do You Understand?, Problem 3, students independently demonstrate conceptual understanding of problems involving rate and unit rate. “Reasoning A bathroom shower streams 5 gallons of water in 2 minutes. a. Find the unit rate for gallons per minute and describe it in words. b. Find the unit rate for minutes per gallon and describe it in words.” (6.RP.2 and 6.RP.3)

• Topic 6, Lesson 6-1, Explain It!, students create a visual representation of two pizzas in order to compare them and support their argument. “Tom made a vegetable pizza and a pepperoni pizza. He cut the vegetable pizza into 5 equal slices and the pepperoni pizza into 10 equal slices. Tom’s friends ate 2 slices of vegetable pizza and 4 slices of pepperoni pizza. A. Draw lines on each rectangle to represent the equal slices. B. Construct Arguments Tom Says his friends ate the same amount of vegetable pizza as pepperoni pizza. How could that be true?”

The materials reviewed for enVision Mathematics Grade 6 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. 

The materials develop procedural skill and fluency throughout the grade level. According to the Teacher Resource Program Overview, “Students develop skill fluency when the procedures make sense to them. Students develop these skills in conjunction with understanding through careful learning progressions.” Try It! And Do You Know How? Provide opportunities for students to build procedural fluency from conceptual understanding. Examples include:

• Topic 1, Lesson 1-2, Try It!, students divide multi-digit decimals using the standard algorithm. “Divide. a. 65 ÷ 8 b. 14.4 ÷ 8 c. 128.8 ÷ 1.4” (6.NS.3)

• Topic 4, Lesson 4-7, Try It!, students graph solutions of inequalities. “Graph all of the solutions of x < 8…” (6.EE.8)

• Topic 6, Lesson 6-1, Do You Know How?, Problem 11, students use ratio and rate reasoning to find the percentage of line a point represents. “Find the percent of the line segment that point D represents in Example 2.” (6.RP.3)

The materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade level. Practice & Problem Solving exercises found in the student materials provide opportunities for students to independently demonstrate procedural skill and fluency. Additionally, at the end of each Topic is a Concepts and Skills Review which engages students in fluency activities. Examples include:

• Topic 2, Lesson 2-2, Practice & Problem Solving, Problem 15, students write the number based on the position of the corresponding point on the number line. “Write the number positioned at each point. 15. A [-3.25]” (6.NS.6 and 6.NS.7)

• Topic 3, Lesson 3-1, Practice & Problem Solving, Problem 17, students identify the exponent for each expression. “Write the exponent for each expression. 17. 9×9×9×9” (6.EE.1)

• Topic 6, Lesson 6-2, Practice & Problem Solving, Problem 12, students write a given number in two other forms of notation, fraction, decimal, or percent based other than the given notation. “Write each number in equivalent forms using the two other forms of notation: fraction, decimal, or percent. 12.7%” (6.RP.3c)

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

The materials include multiple opportunities for students to independently engage in routine and non-routine application of mathematical skills and knowledge of the grade level. According to the Teacher Resource Program Overview, “3-Act Mathematical Modeling Lessons In each topic, students encounter a 3-Act Mathematical Modeling lesson, a rich, real-world situation for which students look to apply not just math content, but math practices to solve the problem presented.” Additionally, each Topic provides a STEM project that presents a situation that addresses real social, economic, and environmental issues, along with applied practice problems for each lesson. For example:

• Topic 4, Lesson 4-1, Practice & Problem Solving, Problem 25, students apply their understanding of solving equations to answer whether an equation can be used with the given answers.  “Alisa’s family planted 7 palm trees in their yard. The park down the street has 147 palm trees. Alisa guessed that the park has either 11 or 31 times as many palm trees as her yard has. Is either of Alisa’s guesses correct? Use the equation 7n = 147 to justify your answer. (6.EE.5)

• Topic 7, STEM Project, Pack It, students determine how the volume of the packaging relates to the volume of the food items being packed. "Food packaging engineers consider many elements related to both form and function when designing packaging. How do engineers make decisions about package designs as they consider constraints, such as limited dimensions or materials? You and your classmates will use the engineering design process to explore and propose food packaging that satisfies certain criteria.” (6.G.2 and 6.G.4)

• Topic 8, 3-Act Mathematical Modeling: Vocal Ranges, Question 12, students use informal arguments and statistical reasoning to decide who should win a singing competition. "Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?" (6.SP.2, 6.SP.3, and 6.SP.5)

The materials provide opportunities for students to independently demonstrate the use of mathematics flexibly in a variety of contexts. Pick a Project is found in each Topic and students select from a group of projects that provide open-ended rich tasks that enhance mathematical thinking and provide choice. Additionally, Practice & Problem Solving exercises found in the student materials provide opportunities for students to independently demonstrate mathematical flexibility in a variety of contexts. For example:

• Topic 1, Lesson 1-5, Practice & Problem Solving, Problem 26, students use models to divide fractions by fractions, “Be Precise 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?” (6.NS.1)

• Topic 4, Pick a Project 4C, students make a model of a staircase using tables and equations, “Think about what you need in order to make a model of a staircase. Design a staircase following these rules: The staircase must follow a linear pattern. Use identical blocks to model the staircase. Keep track of the number of blocks you need for each step. Make a table of data for the number of blocks used for any number of steps. Write an equation to represent your staircase pattern. Present your model, table, and equation to your teacher.” (6.EE.9)

• Topic 5, Lesson 5-6, Practice & Problem Solving, Problem 20, students apply unit rates to a real-world situation. “Reasoning Which container of milk would you buy? Explain.” Students are given the price of a half-gallon and a gallon of milk. (6.RP.3b)

The materials reviewed for enVision Mathematics Grade 6 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade.

All three aspects of rigor are present independently throughout the program materials. Examples, where materials attend to conceptual understanding, procedural skill and fluency, and application, include:

• Topic 2, Lesson 2-2, Concepts and Skills Review, Problem 4, students use procedural skill and fluency to compare two numbers using greater than, less than or equal to. “Use <, >, or = to compare. 4. 0.25 ___ \frac{1}{4}” (6.NS.7a)

• Topic 4, Lesson 4-1, Practice & Problem Solving, Problem 21, students apply their understanding of solving an equation as a process of answering a question. “Construct Arguments 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. (6.EE.5)

• Topic 8, Lesson 8-6, Convince Me!, students develop conceptual understanding when they determine whether the mean, median, or mode best describes the data in a set. “Gary says that he usually scores 98 on his weekly quiz. What measure of center did Gary use? Explain.” A number line is given with dots of data with the mean, median, and mode labeled. (6.SP.5)

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

• Topic 3, Mid-Topic Performance Task, Part B, students develop conceptual understanding, procedural skill and fluency, and application as they find what the least number of cups and napkins Monique would have to purchase to have an equal number. “Monique and Raoul are helping teachers make gift bags and gather supplies for a student celebration day at Pineville Middle School… Monique wants to have an equal number of cups and napkins. What is the least number of packages of cups and the least number of packages of napkins Monique should buy to have an equal number of cups and napkins? Justify our answer.” (6.NS.4) 

• Topic 5, Assessment Form B, Problem 4, student develop procedural skill and fluency and apply their knowledge about ratios in the real-world context of basketball. “A varsity boys’ basketball team has a ratio of seniors to juniors that is 7:4. Part A if the team is made up only of juniors and seniors, what is the ratio of seniors to total team members? Part B The varsity girls’ basketball team has a ratio of seniors to juniors that is 3 to 2. If each team has 12 juniors, which team has more seniors?” (6.RP.1 and 6.RP.3)

• Topic 7, Lesson 7-1, Convince Me!, students develop conceptual understanding and procedural skill and fluency as they calculate and compare the areas of two shapes. “Compare the area of this parallelogram to the area of a rectangle with a length of 7 cm and a width of 4.5 cm. Explain.” (6.EE.2c and 6.G.1)

The materials reviewed for enVision Mathematics Grade 6 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them, and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Some examples where the materials support the intentional development of MP1 are:

• Topic 3, Lesson 3-7, Solve & Discuss It!, students make sense of problems by using prior knowledge and applying properties of operations to rewrite algebraic expressions through simplifying. “Write an expression equivalent to x + 5 + 2x + 2 by combining as many terms as you can. Solve this problem any way you choose.” 

• Topic 6, Lesson 6-6, Practice & Problem Solving, Problem 20, students make sense of the relationship between the given quantity and percent to solve the problem. “Make Sense and Persevere Sydney completed 60% of the math problems assigned for homework. She has 4 more problems to finish. How many math problems were assigned for homework?” 

• Topic 7, Lesson 7-6, Solve & Discuss It!, students analyze a multistep problem involving the surface area of prisms and consider different ways to find solutions. “Marianne orders a pack of shipping boxes shaped like cubes. When they arrive, she finds flat pieces of cardboard as shown. What is the least amount of cardboard needed to make each box? Explain how you know. Solve this problem any way you choose.” 

Some examples where the materials support the intentional development of MP2 are:

• Topic 1, Lesson 1-2, Practice & Problem Solving, Problem 37, students use proportional reasoning to find a fair division of the money earned. “You and a friend are paid $38.25 for doing yard work. You worked 2.5 hours and your friend worked 2 hours. You split the money according to the amount of time each of you worked. How much is your share of the money? Explain.”

• Topic 5, Lesson 5-3, Solve & Discuss It!, students use quantitative reasoning to determine the meaning of the quantities in problems and determine what needs to be done to find a solution. “Scott is making a snack mix using almonds and raisins. For every 2 cups of almonds in the snack mix, there are 3 cups of raisins. Ariel is making a snack mix that has 3 cups of almonds for every 5 cups of sunflower seeds. If Scott and Ariel each use 6 cups of almonds to make a bag of snack mix, who will make a larger batch?” 

• Topic 8, Lesson 8-3, Practice & Problem Solving, Problem 17, students use reasoning to determine the importance of ordering to find the median. “Reasoning The price per share of Electric Company’s stock during 9 days, rounded to the nearest dollar, was as follows: $16, $17, $16, $16, $18, $18, $21, $22, $19. Use a box plot to determine how much greater the third quartile’s price per share was than the first quartile’s price per share.”

The materials reviewed for enVision Mathematics Grade 6 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Student materials consistently prompt students to construct viable arguments. These opportunities are found in the following activities: Solve & Discuss It!, Explain It!, Explore It!, Practice & Problem Solving, Do You Understand?, and Performance Tasks. Examples include:

• Topic 1, Lesson 1-3, Do You Understand?, Problem 3, students use their understanding of fraction multiplication to construct arguments to support their response. “Construct Arguments Why is adding \frac{3}{9} and \frac{6}{9} different from multiplying the two fractions?”

• Topic 4, Lesson 4-2, Practice & Problem Solving, Problem 20, students use their understanding of properties of equality to write equivalent equations to construct arguments as they form the basis for the procedure used to solve algebraic equations. “Construct Arguments John wrote that 5 + 5 = 10. Then he wrote that 5 + 5 + n = 10 + n. Are the equations John wrote equivalent? Explain.”

• Topic 8, Lesson 8-7, Explain It!, students use their understanding of data distributions to construct arguments. “George tosses two six-sided number cubes 20 times. He records his results in a dot plot. A. Describe the shape of the data distribution. B. Critique Reasoning George says that he expects to roll a sum of 11 on his next roll. Do you agree? Justify your reasoning. Focus on math practices Construct Arguments Suppose George tossed the number cubes 20 more times and added the data to his dot plot. Would you expect the shape of the distribution to be different? Construct an argument that supports your reasoning.”

Student materials consistently prompt students to analyze the arguments of others. These opportunities are found in the following activities: Solve & Discuss It!, Explain It!, Explore It!, Practice & Problem Solving, Do You Understand?, and Performance Tasks. Examples include:

• Topic 1, Lesson 1-2, Practice & Problem Solving, Problem 33, students analyze the arguments of others as they divide whole numbers and decimals and apply these skills to solve mathematical problems. “Critique Reasoning Henrieta divided 0.80 by 20 as shown. Is her work correct? If not, explain why and give a correct response.”

• Topic 2, Lesson 2-1, Explain It!, students analyze the arguments of others as they explain differences between integers. “Sal recorded the outdoor temperature as -4℉ at 7:30 A.M. At noon, it was 22℉. Sal said the temperature changed by 18℉ because 22 - 4 = 18. A. Critique Reasoning Is Sal right or wrong? Explain. B. Construct Arguments What was the total temperature change from 7:30 A.M. until noon? Use the thermometer to help justify your solution.”

• Topic 7, Lesson 7-6, Practice & Problem Solving, Problem 14, students analyze the arguments of others as they explain how to find the surface area of a cube. “Critique Reasoning Jacob says that the surface area of the cube is less than 1,000cm$$^2$$. Do you agree with Jacob? Explain.” An image of a cube is shown with a side length of 10 cm.

The materials reviewed for enVision Mathematics Grade 6 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Use appropriate tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials allow for the intentional development of MP4 to meet its full intent in connection to grade-level content. Examples of this include:

• Topic 2, Lesson 2-3, Practice & Problem Solving, Problem 40, students write and solve equations to represent real-life situations. “Model with Math Find the distance from Alberto’s horseshoe to Rebecca’s horseshoe. Explain.” A diagram is given showing the distances of two horseshoes from the stake.

• Topic 3, 3-Act Mathematical Modeling: The Field Trip, students write an algebraic expression to model costs on a field trip and decide how much money to take. “How much cash should [the teacher] bring [on the field trip]? Model with Math Represent the situation using mathematics. Use your representation to answer the Main Question…What is your answer to the Main Question? Does it differ from your prediction? Explain.” 

• Topic 5, Lesson 5-6, Practice & Problem Solving, Problem 19, 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. “Model with Math 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?” 

The materials allow for the intentional development of MP5 to meet its full intent in connection to grade-level content. Examples of this include:

• Topic 4, Lesson 4-7, Practice & Problem Solving, Problem 29, students use a number line to write and represent solutions to inequalities. “Reasoning Graph the inequalities x > 2 and x < 2 on the same number line. What value, if any, is not a solution of either inequality? Explain.”

• Topic 5, Lesson 5-2, Try It!, students use mathematical tools to help find equivalent ratios. “Which of the following ratios are equivalent to 16:20? 2:3, 4:5, 18:22, 20:25” Students have access to a ratio table and other mathematical tools to help identify equivalent ratios.

• Topic 6, Lesson 6-5, Do You Understand?, Problem 5, students explain how to use a calculator to produce a desired result. “Use Appropriate Tools How can you use a calculator to find the percent of 180 is 108?”

The materials reviewed for enVision Mathematics Grade 6 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students are encouraged to attend to the specialized language of mathematics throughout the materials. A chart in the Topic Planner lists the vocabulary being introduced for each lesson in the Topic. As new words are introduced in a Lesson they are highlighted in yellow and students are encouraged to utilize the Vocabulary Glossary in the back of the text (with an animated version online in both English and Spanish) to find both definitions and examples where relevant. Lesson Practice includes questions that reinforce vocabulary comprehension and the teacher's side notes provide specific information about what math language and vocabulary are pertinent for each section. 

Examples where students are attending to the full intent of MP6 and/or attend to the specialized language of mathematics include:

• Topic 1, Topic Review, Use Vocabulary in Writing, students attend to the specialized language of mathematics as they use mathematical terms in their explanations. “Explain how to use multiplication to find the value of \frac{1}{3}÷ \frac{9}{5}. Use the words multiplication, divisor, quotient, and reciprocal in your explanation.”

• Topic 4, Lesson 4-8, Try It!, students attend to precision as they use information from the problem to determine which variable is dependent and illustrate how one is dependent on the other. “A baker used a certain number of cups of batter, b, to make p pancakes. Which variable, p, pancakes or b, batter is the dependent variable? Explain. Convince Me! If the baker doubles the number of cups of batter used, b, what would you expect to happen to the number of pancakes made, p? Explain.”

• Topic 7, Lesson 7-3, Try It!, students attend to precision as they answer questions about the shapes that they previously found the areas of. “When you decompose the trapezoid in Part A of the Try It! into two triangles and a rectangle, are the triangles identical? Explain. What is the height of the two large, identical triangles that compose the kite in Part B of the Try It!?”

• Topic 8, Mid-Topic Checkpoint, Problem 1, students attend to the specialized language of mathematics as they select the answer that describes the definition of mean. “Vocabulary Which of the following describes the mean of a data set? (A) The data value that occurs most often (B) The middle data value (C) The sum of the data value divided by the total number of data values (D) The difference of the greatest and least data values”

The materials reviewed for enVision Mathematics Grade 6 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards.

Students are encouraged to look for and make use of structure as they work throughout the materials, both with the instructor's guidance and independently. Examples of where there is intentional development of MP7 include:

• Topic 3, Lesson 3-6, Practice & Problem Solving, Problem 28, students use structure and properties of operations to determine the equivalence of expressions. “Use Structure Write an algebraic expression to represent the area of the rectangular rug. Then use properties of operations to write an equivalent expression.” Dimensions provided in illustration are l = 2(x - 1) and w = 5. 

• Topic 7, Lesson 7-6, Try It!, students analyze the structure of prisms to find the surface area. “Find the surface area of each prism. [Drawing of a cube and a triangular prism with edge lengths with appropriate lengths given.]” 

• Topic 8, Lesson 8-2, Practice & Problem Solving, Problem 14, students use the structure of a data set to analyze statistical measures. “Look for Relationships Does increasing the 3 to 6 change the mode? If so, how?” Students are given a data set of states traveled to or lived in.

Students look for and express regularity in repeated reasoning as they are engaged in the course materials. Examples of intentional development of MP8 include: 

• Topic 3, Lesson 3-2, Practice & Problem Solving, Problem 3, students use repeated reasoning from examples to answer what is the greatest common factor of two prime numbers. “Generalize Why is the GCF of two prime numbers always 1?” 

• Topic 6, Lesson 6-2, Practice & Problem Solving, Problem 24, students use repeated reasoning to recognize a pattern and generalize what is the same between fractions that are equivalent to 100%. “Generalize What are the attributes of fractions that are equivalent to 100%?”

• Topic 7, Lesson 7-5, Practice & Problem Solving, Problems 20 and 21, students analyze the patterns in a table to create an equation. “Look for Relationships The Swiss mathematician Leonhard Euler and the French mathematician Rene Descartes both discovered a pattern in the number of edges, vertices, and faces of polyhedrons. Complete the table. Describe a pattern in the table. Higher Order Thinking Write an equation that relates the number of edges, E, to the number of faces, F, and vertices, V.”