6th Grade - Gateway 2

The instructional materials for enVisionMATH California Common Core Grade 6 partially meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

The instructional materials present a Problem-Based Interactive Learning activity (PBIL) and a Visual Learning Bridge (VLB) within each lesson to develop conceptual understanding. However, the PBIL and VLB are teacher-directed and do not offer students the opportunity to practice conceptual understanding independently through the use of pictures, manipulatives, and models.

Overall, the instructional materials do not consistently provide students opportunities to independently demonstrate conceptual understanding throughout the grade level.

• In Topic 6 Lesson 6-8, the Overview of PBIL states, “Students divide mixed numbers by using models and improper fractions.” In the teacher-directed PBIL activity, students use a ruler to model the division of 5 1/2 inches by 1 3/8 inches. In the Develop the Concept: Visual section, the process for changing a mixed number to an improper fraction is described, as well as using the reciprocal of a fraction to create an equivalent multiplication problem to solve. The directions for the Independent Practice state, “In 9 through 20, find each quotient. Simplify, if possible.” Students do not demonstrate the conceptual understanding of dividing mixed numbers independently as fractions are shown as sample answers in the 12 problems in the Independent Practice.
• In Topic 9 Lesson 9-2, the Overview of PBIL states, “Students interpret and use ratios and equivalent ratios.” In the teacher-directed PBIL activity, students draw pictures to represent a ratio of rock songs to hip-hop songs. The Develop the Concept: Visual section of the lesson describes the procedural steps of finding equivalent ratios. The directions for the Independent Practice state, “In 9 through 16, write three ratios that are equivalent to the given ratio.” Students do not demonstrate the conceptual understanding of equivalent ratios independently as ratios in reduced form are shown as sample answers in the eight problems in the Independent Practice.
• In Topic 10 Lesson 10-3, the Overview of PBIL states, “Students use unit rates to solve proportions.” In the teacher-directed PBIL activity, students create a table to find the unit rate to solve a word problem. The directions for the Independent Practice state, “In 5 through 12, find the unit rate.” Students do not demonstrate the conceptual understanding of unit rates independently as written unit rates are shown as sample answers in the eight problems in the Independent Practice.
• In Topic 12 Lesson 12-2, the Overview of PBIL states, “Students find the area of a parallelogram.” In the teacher-directed PBIL activity, students create a parallelogram on paper and cut a triangle from one side and move it to the other side to form a rectangle. The directions for the Independent Practice state, “Find the area of each parallelogram or rhombus.” Students do not demonstrate the conceptual understanding of finding the area of a parallelogram independently as solutions are shown as sample answers in the eight problems in the Independent Practice.

The instructional materials for enVisionMATH California Common Core Grade 6 partially 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.

Each topic includes at least one Problem Solving lesson that can be found at the end of the topic. These lessons offer students opportunities to integrate and apply concepts and skills learned from earlier lessons. Within each individual lesson, there is a section titled, Problem Solving, where students practice the application of the mathematical concept of the lesson. However, the applications of mathematics in Problem Solving are routine problems.

The instructional materials have few opportunities for students to engage in non-routine application throughout the grade level. Examples of routine applications, where a solution path is readily available, are:

• In Topic 2 Lesson 2-3, students use addition and subtraction equations to solve word problems. Problem Solving problem 32 states, “The world record for a hot-air balloon flight is 65,000 feet high. Most hot-air balloons fly 62,150 feet below this height. At what height do most hot-air balloons fly? Use the equation h + 62,150 = 65,000.”
• In Topic 2 Stop and Practice, students determine if a statement involving equations is true or false and explain their reasoning. Number Sense problem 23 states, “In the equation 18h = 108, h will be less than 5.”
• In Topic 3 Lesson 3-2, students analyze different tables of values to determine a pattern and write an equation. Problem Solving problem 11 states, “To celebrate their 125th anniversary, a company in Germany produced 125 very expensive teddy bears. The bears, known as the “125 Karat Teddy Bears,” are made of mohair, silk, and gold thread and have diamonds and sapphires for eyes. The chart at the right shows the approximate cost of different numbers of these bears. Based on the pattern, how much does one bear cost?”
• In Topic 6 Lesson 6-8, students use division of mixed numbers to solve real-world problems. Problem Solving problem 30 states, “How many 3/4 ft. pieces can you cut from a 6 1/2 ft. ribbon?”
• In Topic 9 Lesson 9-4, students use ratios to solve word problems. Problem Solving problem 14 states, “Alberta found that 6 cars passed her house in 5 minutes. How many cars would you expect to pass her house in 2 hours?”
• In Topic 10 Lesson 10-8, students apply ratios and unit rates to solve word problems. Independent Practice problem 10 states, “A truck driver started the year making $0.35 per mile. Halfway through the year, she received a raise and began earning $0.43 per mile. She drove 48,000 miles the first 6 months and 45,000 miles the second 6 months. How much did she earn for the year?”

The instructional materials reviewed for enVision Grade 6 partially meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade level.

Mathematical Practice standards are identified in three places within the Teacher Edition: Problem-Based Interactive Learning activity, Guided Practice exercises, and Problem-Solving exercises. Throughout the teacher and student editions, there is a symbol that indicates that one or more MP is being used. Key phrases such as “Look for Patterns," "Use Tools," and "Reason" identify which practice is being highlighted. At the beginning of each lesson, all eight mathematical practices are listed. A check mark is placed beside each practice that is to be addressed within the lesson.

Examples of MPs that are identified but do not enrich the mathematical content include:

• In Topic 7 Lesson 7-1, MP8 is identified with the icon and the key word “Generalize.” Question 8 states, “Which integers do you use for counting?”
• In Topic 7 Lesson 7-4, MP5 is identified with the icon and the key word “Use Tools.” Question 9 states, “How can you use a vertical number line to compare two numbers?”

An example of MPs that are identified and enrich the mathematical content include:

• In Topic 9 Lesson 9-2, MP3 is identified with the icon and the key word “Writing to Explain.” Question 32 states, “The ratio of the maximum speed of Car A to the maximum speed of Car B is 2:3. Explain whether Car A or Car B is faster.” Students construct an argument, although, they do not have to critique the reasoning of others in this problem.

Examples where the MPs are incorrectly labeled:

• In Topic 7 Lesson 7-2, MP5 is identified in the PBIL activity. However, students are given two positive and two negative numbers and directed to order them from least to greatest. Students are instructed to use a number line or thermometer to help them, and both tools are provided within the student edition. Therefore, students are not selecting tools strategically.
• In Topic 11 Lesson 11-2, MP3 is identified with the icon and the key word “Communicate.” Question 8 states, “How is the decimal point moved when changing from a decimal to a percent?” This question does not require a viable argument or a critique of the reasoning of others. According to the teacher edition, students need to state that one moves the decimal point two places over to the right.

Overall, all eight math practices are included within the curriculum and are not treated as separate standards. However, the standards are not used to enrich the content. They are aligned to some of the problems as an explanation to what math practice students might need to use to solve the problem.

The instructional materials reviewed for enVisionMATH California Common Core Grade 6 partially meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

The Teacher Edition contains a Mathematical Practice Handbook which defines each math practice and includes question stems for each MP to help the teacher engage students. MP3 offers the following questions stems: “How can I use math to explain why my work is right?”, “How can I use math to explain why other people’s work is right or wrong?”, and “What questions can I ask to understand other people’s thinking?”

The materials label multiple questions as MP3 or parts of MP3; however, those labeled have little information assisting teachers to engage students in constructing viable arguments or to critique the reasoning of others. The information that the materials provide is not specific and are often hints or reminders to give students while they are solving a problem.

There are some missed opportunities where the materials could assist teachers in engaging students in both constructing viable arguments and analyzing the arguments of others. For example:

• In Topic 3, Lesson 3-2, Problem Solving Question 12 states “Writing to Explain. Explain how you can find a pattern in the chart showing the cost of “125 Karat Teddy Bears.” Use the pattern to write a rule and an equation.” No teacher guidance is given for this question.
• In Topic 6, Lesson 6-4, Problem Solving Question 32 states, “Writing to Explain. A bowl of soup holds 7 ounces. If a spoonful holds 1/6 ounce, how many spoonfuls are in 3 bowls of soup? Explain.” No teacher guidance is given for this question.

Examples where teachers are supported, although generally, to assist students in constructing viable arguments and analyzing the arguments of others include:

• In Topic 10, Lesson 10-8, PBIL identifies MP3 as the mathematical practice being used in the activity. The teacher note states, “Remind students that as they learn to write good mathematical explanations, they will become better able to communicate their own mathematical thinking.”
• In Topic 14, Lesson 14-1, Problem Solving Question 15 states, “Critique Reasoning. Jo says How do you get to school each morning? Is not a statistical question because each student who answers it can give only one answer. Do you agree with Jo? Explain.” Teacher guidance for this MP is “Remind students to check for reasonableness when solving each problem.”