6th Grade - Gateway 3

The instructional materials reviewed for Utah Middle School Math Project Grade 6 meet expectations that the underlying design of the materials distinguishes between problems and exercises.

The chapters begin with a non-routine problem that introduces new concepts and is labeled as an Anchor Problem for the chapter. The chapters are sectioned into Class Activities, Homework, Spiral Reviews, and Self Assessments. 

Typically, each Class Activity has problems that teachers guide students through as a class. Occasionally, problems are intended to review previous grades’ concepts in order to connect them to Grade 6 concepts. Most often, Class Activities are for the students to apply new learning. The mathematical concepts in each Class Activity are reinforced by accompanying Homework components. 

The instructional materials reviewed for Utah Middle School Math Project Grade 6 meet expectations that the design of the assignments is not haphazard; exercises are given in intentional sequences. 

Students are presented with an Anchor Problem at the beginning of each chapter to introduce new concepts. Anchor Problems are sometimes referenced throughout the chapter. 

Within each chapter, concept development is sequential. During Class Activities, the teacher introduces new concepts or builds upon prior knowledge. Students work individually or as a whole class when engaged in the Class Activities. The Homework component reinforces the mathematical concepts taught during the previous Class Activity. Spiral Reviews are used to provide continued practice of concepts learned throughout the year. 

The progression of lessons is intentional and assists students in building their mathematical understandings and skills. Students begin with activities to build conceptual understanding and procedural skill, and progress to applying the mathematics with more complex problems and procedures. 

The instructional materials reviewed for Utah Middle School Math Project Grade 6 meet expectations for having a variety in what students are asked to produce.

Throughout the Class Activities, students produce answers and solutions, discuss ideas, make conjectures, explain solutions and justify reasoning, make sketches and diagrams, and use appropriate models. These aspects are found individually within problems as well as in combination with others, such as providing an explanation of a solution and including a diagram. Examples include:

• In Chapter 2, 2.1d Class Activity, students produce a variety of work in relation to fractions, decimals and percents. Question 1 states, “$$\frac{3}{5}$$ of the 6th grade class at a certain school own a cell phone. a. Make a tape diagram to represent this situation. b. What percent of the students own a cell phone? c. What percent of the students do not own a cell phone? d. What is the ratio of students who own a cell phone to the ratio of students who do not own a cell phone? e. If there are seventy-five 6th graders at this school, how many own a cell phone?”
• In Chapter 3, 3.1a Homework, Problems 1-3, students make a model (number line), explain their method, and provide answers. The materials state, “In the space below, construct a number line showing the integers from −5 to 5. Explain the process you used to create your number line. What steps did you take? What tools did you use? How many points are 3 units away from 0 on your number line? Explain.”

The instructional materials reviewed for Utah Middle School Math Project Grade 6 meet expectations that manipulatives are faithful representations of the mathematical objects they represent and, when appropriate, are connected to written models. 

Although the use of manipulatives is limited, teachers are given general guidance, other options, and explanations on how to connect the manipulatives to written methods. Examples include:

• In Chapter 1, 1.0 Anchor Problem, Connected Gears, Question c. states, “If the smaller gear makes 24 revolutions, how many revolutions will the larger gear make? If the smaller gear makes 24 revolutions, the larger gear will make 16 revolutions.” The teacher notes state, “Again, students can use a variety of strategies including manipulatives, tape diagrams, partial tables, and numeric methods.”
• In Chapter 6, 6.2e Class Activity, Activity 3 states, “Prior to doing this lesson, you may want to introduce students to the algebra tiles if you plan to use them in the lesson. Depending on your students, you may choose not to use the physical tiles and move right into having students draw pictures and models to represent the problems.”
• In Chapter 5, students “can explore nets with online interactive manipulatives as you investigate the problem above.” Additional guidance for teachers includes, “online tools for viewing 3D objects and their nets can really help students that have difficulty visualizing how an object and its net are related.” Using physical 3D shapes and nets, in addition to viewing them on a 2D platform, could assist students in visualizing nets and 3D objects.

The instructional materials reviewed for Utah Middle School Math Project Grade 6 meet expectations for supporting teachers in planning and providing effective learning experiences by providing quality questions to help guide students’ mathematical development. Examples include:

• In Chapter 1, 1.0 Anchor Problem, students investigate ratios and determine that the quantities in a ratio are connected. The teacher notes guide the students’ mathematical development by prompting the teacher in a variety of ways. For example, “You may want to show students animations of gears to make sure they know how gears work. Ask questions such as, ‘If the smaller gear is rotating clockwise, which direction will the larger gear rotate?’ ‘If you turn the larger gear one full rotation, does the smaller gear make one revolution, more than one revolution, or less than one revolution?’”
• Class Activities are the guided parts of lessons where a teacher facilitates students’ work. In Chapter 6, 6.3j Class Activity, teachers guide students in their understanding of inequalities. For example: “In this lesson, students are making sense of problems, asking the following questions: What are the constraints in the given situation? When I solve for the inequality, what answers make sense in the context? For example, in problem 2 below, it does not make sense to order part of a pizza based on the way the problem is set up. In #10, students need to think realistically about the degree of precision someone will use when cutting fabric or paper.” 

The instructional materials reviewed for Utah Middle School Math Project Grade 6 meet expectations for containing a Teacher Edition Workbook that has useful annotations and includes suggestions on how to present the content in the student edition. Where applicable, materials include teacher guidance to support and enhance student learning. The Teacher Edition Workbook offers suggestions and annotations, labeled in red, on how to present the content. Examples include:

• In Chapter 2, 2.1f Class Activity states, “This lesson is intended to introduce students to three different types of percent problems: 1) Find a percent given a part and the whole; 2) Find a part of a quantity given a percent and the whole; 3) Find the whole given a part and a percent. Students have already practiced changing a fraction to a percent in the previous lessons (e.g., Express $$\frac{15}{20}$$ as a percent). These types of problems are included again here so that students can compare the different types of percent problems.”
• In Chapter 5, 5.1a Class Activity, guidance is included to help students graph regular and irregular shapes on graph paper and use graphs to determine the area. The materials state, “Encourage students to share their reasoning and different methods for finding the area with each other. As students try to find the area of each figure, they must recognize that they can decompose different parts of the figure in order to recompose them into whole square units that they can count. As they do this they must step back and shift their perspective of how to view a square unit of area.”

The instructional materials reviewed for Utah Middle School Math Project Grade 6 meet expectations for containing a Teacher Edition Workbook that contains full, adult-level explanations and examples of the more advanced mathematical concepts in the lessons so that teachers can improve their own knowledge of the subject, as necessary. Examples include:

• Each chapter resource list contains a Mathematical Foundations book, a resource to help teachers understand the mathematics of the chapter and to expand their understanding of the mathematical concepts. 
• Each Mathematical Foundation book includes problems, explanations of problems, examples, and connections to prior and future work, and CCSSM alignment. 
• The Teacher Edition Workbook provides clear, step-by-step solutions to problems. 

The instructional materials reviewed for Utah Middle School Math Project Grade 6 meet expectations for containing a digital Teacher Edition Workbook that explains the mathematics in the context of the overall mathematics curriculum throughout the grades. Each chapter contains a Connections to Content section where teachers gain an understanding of the mathematical content in the lessons as well as where the content fits in the scope of mathematics from Kindergarten to Grade 12. Knowledge required from prior chapters and/or grade is explicitly identified in this section. Examples include:

• In Chapter 5, Prior Knowledge states, “In 3rd grade they recognize area as an attribute of plane figures and investigate concepts of area measurement. In 4th grade they apply area formulas to real-world and mathematical problems. Volume is studied in 5th grade where students learn to recognize it as an attribute of solid figures and investigate concepts of volume measurement. This prior work with area and volume help them to develop competencies in shape composition and decomposition and form a foundation for understanding the formulas for area and volume and the coordinate plane.” 
• The Teacher Edition Workbook connects the learning in Grade 6 to future grade levels and explains how standards build on one another throughout the program. For example, in Chapter 2, Connections to Content states, “In Grade 7, students move from concentrating on analysis of data to production of data. They understand that good statistical answers depend on a well-developed plan for collecting data. They investigate random sampling, and in turn, concepts related to probability. In 8th grade, students extend their knowledge of shape, center, and spread to the analysis of bivariate data, (collection of counts with 2 variables or characteristics) as related to their work with linear functions.”

The instructional materials reviewed for Utah Middle School Math Project Grade 6 meet expectations for providing opportunities for ongoing review and practice for students in learning both concepts and skills.

Over the course of each chapter, responsibility for the learning process transfers from the teacher to the student. Students move from scaffolded support within the Class Activities to independent problem solving within the Homework. Examples include:

• Anchor Problems engage students in standards that are to be taught in the chapter. The Anchor Problems often guide the teacher to return to the problem while working through the concepts in the chapter. In Chapter 4, Anchor Problem 4.0 states, “This anchor problem should be revisited throughout the chapter as students gain more knowledge of how to analyze and interpret the shape, center, and spread of a data distribution.” 
• Mathematical concepts are reinforced by an accompanying Homework component for each Class Activity that is designed for individual practice.
• The materials provide frequent opportunities for ongoing review and practice in the Spiral Review located with the Homework. The Spiral Review consists of problems from standards covered both from within the chapter and from previous chapters. For example, in Chapter 4, Spiral Review includes review questions on the following concepts: multiplying, adding, and subtracting decimals, measurement conversions and finding volume. Examples include: “1. $$5 × 3.21$$, 6. $$5 + 0.682 - .03$$, 11. 54 inches = ____ ft, 13. Find the volume of a rectangular prism that has a length of 7 inches, a width of 5 inches, and a height of 9 inches.”

The instructional materials reviewed for Utah Middle School Math Project Grade 6 meet expectations for providing strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners. The sequencing and scaffolding are built into lesson development so that teachers pose problems as they progress through more rigorous processes or skills. Examples include:

• In Chapter 1, Section 1 Overview provides assistance for teachers, “As students progress, their pictures often become more abstract. For example, a student may draw 5 circles (of the same size) to represent the three cups of flour and two cups of sugar. Another way to illustrate a ratio is to draw a tape diagram. A tape diagram is simply a rectangle composed of smaller, equal-sized pieces to represent each of the smaller parts of the ratio. For the ratio of 3 cups of flour to 2 cups of sugar, a tape diagram would look like Figure 1.”
• In Chapter 2, 2.1b Class Activity, teacher guidance includes, “Refer to the Anchor Problem Part 1 to help students to make sense of what they are being asked to do. If they are trying to express a part as a decimal, the whole is equal to 1. If they are trying to express the part as a percent, the whole is 100. Once they have identified the value of each box, they can use repeated reasoning to determine the value of the shaded portion of the grid.”
• In Chapter 4, 4.4b Class Activity, Question 4, students determine “what possible arguments could Roman give to his principal that on average at least 40 candy bars are sold from the vending machine each day?” Guidance for teachers includes, “assess how much guidance your students might need to answer the question above. You can provide them with a series of discussion questions to scaffold their thinking. Possible discussion questions are below.” Questions include specific direction for students to determine answers from the histogram.

The instructional materials reviewed for Utah Middle School Math Project Grade 6 meet expectations for frequently embedding tasks with multiple entry-points that can be solved using a variety of solution strategies or representations. Class Activities and Homework provide tasks that include multiple entry points that can be solved using a variety of strategies or representations. Examples include:

• In Chapter 5, 5.1a Class Activity, “As students try to find the area of each figure, they must recognize that they can decompose different parts of the figure in order to recompose them into whole square units that they can count. As they do this they must step back and shift their perspective of how to view a square unit of area.”
• In Chapter 6, 6.0 Anchor Problem, “A local charity has a benefit to raise money. You are on the planning committee and have been tasked to determine the number of tickets that must be sold for the charity to raise at least $5,000 after all expenses have been covered.” The materials also list different information students will need while working on the problem. Students work with decimals, fractions, expressions, equations and inequalities. Because this problem involves so many components, there are multiple entry-points for a student to make sense and solve the problem.

The instructional materials reviewed for Utah Middle School Math Project Grade 6 meet expectations for providing a balanced portrayal of various demographic and personal characteristics, and no examples of bias were found. Examples include:

• Pictures, names and situations present a variety of ethnicities and interests. In Chapter 2, 2.2a Class Activity, Question 4 states, “Eduardo is painting a mural depicting the beauty of the four seasons in his neighborhood. He wants to divide the spring portion ($$\frac{1}{4}$$ of the mural) into 4 equal parts showing graduation activities, planting of gardens, melting of snow, and children playing outside. What portion of the mural will depict graduation activities?”
• In Chapter 5, 5.1a Class Activity, Problem 9, Antonia is wearing a head wrap.