6th Grade - Gateway 2

The instructional materials for Big Ideas Math: Modeling Real Life 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 do not always provide students opportunities to independently demonstrate conceptual understanding throughout the grade-level. 

Each lesson begins with an Exploration section where students develop conceptual understanding of key mathematical concepts through teacher-led activities. For example:

• In Chapter 3, Section 1, Exploration 2, “Using Ratios in a Recipe,” students are directed to “Work with a partner. The ratio of iced tea to lemonade in a recipe is 3:1. You begin by combining 3 cups of iced tea with 1 cup of lemonade. A. You add 1 cup of iced tea and 1 cup of lemonade to the mixture. Does this change the taste of the mixture? B. Describe how you can make larger amounts without changing the taste.” The teaching notes direct the teacher not to teach but to listen to the conversations and take notes on the answers students are getting in order to reference the answers when discussing equivalent ratios in the lesson. (6.RP.1)  
• In Chapter 5, Section 5, Exploration 1, “Finding Dimensions,” students are directed to work with a partner in solving the following? “A. The models show the area (in square units) of each part of a rectangle. Use the models to find missing values that complete the expressions. Explain your reasoning. B. In part (a), check that the original expressions are equivalent to the expressions you wrote.  Explain your reasoning. C. Explain how you can use the Distributive Property to rewrite a sum of two whole numbers with a common factor.” Throughout this exploration, the teacher is encouraged to listen to conversations and have volunteers share their strategies to the class. (6.EE.1)
• In Chapter 5, Section 4, Exploration 1, “Using Models to Simplify Expressions,” students are given three models to “simplify expressions”. Through these models, students are able to see how the distributive property is applied. (6.EE.1)
• In Chapter 6, Section 2, Exploration 2, students are asked to work with a partner as they look at an equation as a balanced scale. They are asked, “A. How are the two sides of an equation similar to a balanced scale? B. When you add weight to one side of a balanced scale, what can you do to balance the scale?  What if you subtract weight from one side of a balanced scale? How does this relate to solving an equation? C. Use a model to solve x + 2 = 7. Describe how you can solve the equation algebraically.” (6.EE.5)
• In Chapter 3, Section 2, Example 1, “Interpreting a Tape Diagram” students are shown how to use a tape diagram to solve problems. For example, “The tape diagram represents the ratio of blue monsters to green monsters you caught in a game. You caught 10 green monsters. How many blue monsters did you catch?” While the problem is modeled using a tape diagram for students, the solution is explained to the students. (6.RP.1)

The instructional materials do not always provide students opportunities to independently demonstrate conceptual understanding throughout the grade-level. The shift from conceptual understanding, most prevalent in the Exploration Section, to procedural understanding is completed within the lesson. The Examples and Concepts, Skills, and Problem Solving sections have a focus that is primarily procedural with limited opportunities to demonstrate conceptual understanding. For example: 

• In Chapter 5, Section 5, Example 1, “Factoring Numerical Expressions”, students are shown how to find the GCF by listing the factors of two numbers and circling the common factors. (6.NS.2)
• In Chapter 6, Section 3, “Concepts, Skills and Problem Solving”, there are missed opportunities to demonstrate conceptual understanding. Problems 16-35 have students “Solve the equation. Check your solution.” Problem 16: “s/10 = 7”; Problem 26: “13 = d ÷ 6” Problem 33: “7b ÷12 = 4.2.” (6.EE.5)

The instructional materials for Big Ideas Math: Modeling Real Life Grade 6 partially meet expectations that the materials are designed so that teachers and students spend sufficient time working with engaging applications of mathematics. 

The instructional materials present opportunities for students to engage in application of grade-level mathematics; however, the problems are scaffolded through teacher-led questions and procedural explanation. The last example of each lesson is titled, “Modeling Real Life,” which provides a real-life problem involving the key standards addressed for each lesson. This section provides a step-by-step solution for the problem; therefore, students do not fully engage in application. In addition, there are few non-routine problems presented. For example:

• In Chapter 2, Lesson 2, there are several, multi-step, routine problems that students solve independently. For example, Problem 64: “You use 1/8 of your battery for every 2/5 of an hour that you video chat. You use 3/4 of your battery video chatting. How long did you video chat?” Problem 67: “You have 6 pints of glaze. It takes 7/8 of a pint to glaze a bowl and 9/16 of a pint to glaze a plate. a. How many bowls can you completely glaze? How many plates can you completely glaze? b. You want to glaze 5 bowls, and then use the rest for plates. How many plates can you completely glaze? How much glaze will be left over?” Part C is an example of a non-routine application: “c. How many of each object can you completely glaze so that there is no glaze left over? Explain how you found your answer.” (6.NS.1)
• Chapter 3, Lesson 2, Example 4, Modeling Real Life, “In a seven-game basketball series, a team’s power forward scores 8 points for every 5 points the center scores. The forward scores 60 more points than the center in the series. How many points does each player score in the series? The ratio of the forward’s points to the center’s points is 8:5. Represent the ratio using a tape diagram.” The tape diagram is given to the students and a step-by-step breakdown explanation is given about how to solve the ratio using a tape diagram. (6.RP.3) 
• Chapter 6, Lesson 1, students explore multiple examples of routine application problems. Problem 11, “After four rounds, 74 teams are eliminated from a robotics competition. There are 18 teams remaining. Write and solve an equation to find the number of teams that started the competition.” (6.EE.7) 

Overall, there are limited opportunities for students to engage in non-routine problems throughout the grade level.

The instructional materials for Big Ideas Math: Modeling Real Life Grade 6 partially meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. 

The instructional materials present opportunities in most lessons for students to engage in each aspect of rigor, however, these are often treated together. There is also an over-emphasis on procedural skill and fluency. For example:

• In Chapter 4, Lesson 1, Percents and Fractions, Introduction, students build conceptual understanding of percent as they consider the meaning of the word “percent.” In the Exploration section, students interpret models with a partner by determining the percent, fraction, and ratio shown by each model. In these sections, students are building conceptual understanding. In Example 1, students use both models and numbers to write percents as fractions. In Example 2, students write fractions as percents without models, using a modeled procedure. In the Concept, Skills, & Problem Solving, the focus is on procedural skill and fluency, and application. 
• In Chapter 9, Lesson 2, Mean, the first two examples provide students an opportunity to build an understanding of mean as a “fair share” or “balanced share”. In the Self-Assessment section, students write answers to questions that focus on conceptual understanding. For example, “Is the mean always equal to a value in the data set? Explain. Explain why the mean describes a typical value in a data set. What can you determine when the mean of one set is greater than the mean of another data set? Explain your reasoning.” In Example 1, students learn a procedure to find the mean. In Example 2, students learn to compare means using a double bar graph. In Example 3, students learn how an outlier affects the mean. In Concepts, Skills, & Problem Solving, students find the mean in both isolated number sets and story problems. 
• In Chapter 6, Lesson 3, Solving Equations Using Multiplication or Division, students build conceptual understanding in Exploration 1, where they solve an equation using a tape diagram, and Exploration 2, where they solve an equation using a balance. The lesson shifts to a focus on procedural understanding and fluency, providing annotated step-by-step solutions in Examples 1: “Solve w/4 =12, Solve 2/7x=6”; Example 2: “Solve 65=5b”; and Example 3: “The area of a rectangular LED ‘sky screen’ in Beijing, China is 7500 square meters. The width of the sky screen is 30 meters. What is the length of the sky screen?”

The instructional materials reviewed for Big Ideas Math: Modeling Real Life Grade 6 partially meet expectations that the instructional materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

“You be the Teacher” found in many lessons, presents opportunities for students to critique the reasoning of others, and construct arguments. For example:

• Chapter 2, Lesson 1, Problems 52 & 53, You Be the Teacher, “Your friend finds the product. Is your friend correct? Explain your reasoning.” The student is provided work to examine.
• Chapter 5, Lesson 5, Problem 56, You Be the Teacher, “Your friend factors the expression 24x + 56. Is your friend correct? Explain your reasoning.” The student is provided work to examine.
• Chapter 9, Lesson 5, Problem 22, You Be the Teacher, “Your friend finds and interprets the mean absolute deviation of the data set 35, 40, 38, 32, 42, and 41. Is your friend correct? Explain your reasoning.” The student is provided work to examine.

The Student Edition labels MP3 as “MP Construct Arguments,” however, these activities do not always require students to construct arguments. “Construct Arguments” was labeled only twice for students and “Build Arguments” was labeled once for students. For example:

• Chapter 2, Section 7, Exploration 1, Construct Arguments is identified in the Math Practice blue box with the following question, “Why do the quotients in part (b) have the relationship you observed?”
• Chapter 9, Lesson 1, Exploration 2, Build Arguments is identified in the Math Practice blue box with the following question, “How can comparing your answers help you support your conjecture?”
• Chapter 10, Lesson 4, Exploration 1, Construct Arguments is identified in the Math Practice blue box with the following question, “Explain why the shapes of the distributions in Exploration 1 affect which measures best describe the data.”

The instructional materials reviewed for Big Ideas Math: Modeling Real Life Grade 6 partially meet 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 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 Chapter 3, Lesson 2, during the self-assessment teachers are prompted to have students share their thinking. “Discuss the language and clarity of explanations. Then ask, “If you were unclear before, did any of the explanations help you make sense of this question?”
• In Chapter 8, Lesson 8, Example 4, students are shown how to set up and solve an inequality. Laurie’s Notes identify MP3, “Mathematically proficient students take into account the context of the problem. Students should consider the number of times a person needs to ride a bus in a 30-day period when deciding whether a 30-day pass saves money.” There is no support provided to assist teachers to engage students in MP3.