6th Grade - Gateway 1

The instructional materials reviewed for The Utah Middle School Math Project Grade 6 meet expectations for assessing grade-level content. The assessments are aligned to grade-level standards.

There are multiple summative Self-Assessments within each unit that include a scoring rubric to help students identify their understanding of the concepts being assessed. All Self-Assessments have answer keys provided in the Teacher Workbook. Examples include:

• In Chapter 2, 2.2f, Problem 1 states, “How much chocolate will each person get if 3 people share $$\frac{1}{2}$$ a pound of chocolate equally? Complete the following to answer this question. $$\frac{1}{2}÷3=$$ ____ because ____ $$× 3 =\frac{1}{2}$$.” (6.NS.1)
• In Chapter 3, 3.1j, Problem 2 states, “Construct a number line to show the location of the integers from -8 to 8. Explain how you used ideas about symmetry and opposites to construct your number line.” (6.NS.6)
• In Chapter 4, 4.2f, Problem 2 states, “Find and interpret the mean, median, and mode for each set of data below. Then determine which measure of center best represents the data. Be sure to justify your answer. Number of sit ups: 78, 86, 86, 96, 90, 71, 110, 102, 92, 80, 106, 100.” (6.SP.2)
• In Chapter 6, 6.2i, Problem 5 states, “Write the expression 6x + 42 as the product of two factors.” (6.EE.3)

There are no Self-Assessments for Chapter 0: Fluency and Chapter 5: Geometry.

The instructional materials reviewed for Utah Middle School Math Grade 6 meet expectations for spending a majority of class time on the major clusters of the grade. The instructional materials contain chapters which identify the total number of weeks for instruction. Within each chapter are sections consisting of multiple Class Activities (lessons). While the materials do not specify the total number of days, some Class Activities include a statement that they may take multiple days. There are 105 Class Activities in Grade 6. 

• The approximate number of chapters devoted to major work of the grade (including assessments and supporting work connected to the major work) is 4 out of 7 chapters, which is approximately 57%.
• The number of Class Activities devoted to major work of the grade (including assessments and supporting work connected to the major work) is 75 out of 105 Class Activities, which is approximately 71%.
• The number of weeks devoted to major work of the grade (including assessments and supporting work connected to the major work) is 18 out of 28, which is approximately 64%. 

Class Activities are the best representation of the amount of class time spent on major work of the grade, and supporting work connected to major work of the grade as it includes all lessons. Thus, approximately 71% of instructional time is spent on major work of the grade.

The instructional materials reviewed for Utah Middle School Math Grade 6 meet expectations for supporting content enhancing focus and coherence simultaneously by engaging students in the major work of the grade. Overall, the lessons that focus on supporting content also engage students in major work where natural and appropriate.

Examples of supporting content connected to the major work of the grade include:

• In Chapter 0.2e, the Class Activity connects supporting cluster 6.NS.B to major cluster 6.EE.A, as students use the GCF and the Distributive Property to find equivalent expressions. In Question 5 students, “Use the distributive property to find all the equivalent expressions for each sum given. Circle the expressions that contain a factor that is the GCF of the two addends in the original sum. Check and see if this expression follows the same principle as the expressions with the GCF from the numbers 1 and 2 above, a. 45 + 60, b. 42 + 70, c. 20 + 60.”
• In Chapter 1.1c, Homework connects supporting cluster 6.NS.B to major cluster 6.RP.A, as students use ratio reasoning to solve problems. In Question 1, students determine “The ratio of sugar to flour used in a sugar cookie recipe is 1 cup sugar to 2 cups flour. Determine the amount of each ingredient needed to double, triple, quadruple, and half the recipe.”
• In Chapter 2.2d, Homework connects supporting cluster 6.NS.B to major cluster 6.NS.A, as  students divide fractions by fractions using the ability to fluently find common multiples. In Question 1, students determine “How many halves fit into three-fourths?”
• In Chapter 5.1b, Class Activity connects supporting cluster 6.G.A to major cluster 6.EE.A, as students use a formula to find the area of parallelograms. In Question 4, students “Describe in words and write a formula to find the area of any parallelogram.”

The instructional materials reviewed for Utah Middle School Math Grade 6 meet expectations that the amount of content designated for one grade-level is viable for one school year.

According to the publisher, each chapter has a designated amount of time spent in weeks. There is no guidance from the publisher on the suggested length of daily work including Class Activities, Homework, Self-Assessment, and Anchor Problems. 

The guidance for length of daily work is generalized in the overall time for the length of the unit. According to the Mathematical Foundations for Chapter 0, it is intended to “be inserted into the natural flow of the course where appropriate.” Since the publisher did not indicate a number of weeks for Chapter 0, the review team allocated two weeks for Chapter 0. Therefore, the following units are assigned the following number of weeks:

• Chapter 0: 2 weeks 
• Chapter 1: 4 weeks
• Chapter 2: 5 weeks
• Chapter 3: 4 weeks
• Chapter 4: 6 weeks
• Chapter 5: 3 weeks
• Chapter 6: 4 weeks 

The total number of 28 weeks in the materials would be equivalent to an average of 140 days (28 weeks x 5 days/week) of instruction including assessments.

The instructional materials reviewed for Utah Middle School Math Grade 6 meet expectations for being consistent with the progressions in the Standards.

The instructional materials clearly identify content from prior and future grade-levels and use it to support the progressions of the grade-level standards. Content from prior and future grade levels is identified in the Connections to Content section found at the beginning of each chapter. Examples include:

• In Chapter 2, Chapter Overview, Connections to Content, Future Knowledge states, “In Chapter 6 of this text, students will learn how to write and solve equations to represent the different types of percent problems studied in this chapter. In 7th grade, students will continue to focus on proportional relationships, learning how to set up and solve a proportion to solve percent problems, including problems involving discounts, interest, taxes, tips, and percent increase and decrease.”
• In Chapter 3, Chapter Overview, Connections to Content, Future Knowledge states, “In Grade 7, students will learn to operate with positive and negative rational numbers. In Grade 8, the number system is expanded to include irrational numbers. Students come to understand that irrational numbers are points on the real number line even though they cannot be represented with an exact decimal value.”
• In Chapter 6, Chapter Overview, Connections to Content, Future Knowledge states, “In 7th grade, students will encounter expressions with positive and negative rational numbers. As coursework progresses, students will write expressions to model different types of functions such as exponential and quadratic functions. Being able to examine numeric expressions and identify abstract patterns is an important part of being able to write explicit rules to model a function. In later grades, students will see more complex equations.”

The instructional materials attend to the full intent of the grade-level standards by giving all students extensive work with grade-level problems. Examples include:

• In Chapter 2, 2.2d Homework, Questions 1-8, students solve word problems involving division of fractions. Students are instructed to “Create a model of your choice to answer the questions below. Then, write a number sentence to represent the problem.” Question 1 states, “How many halves fit into three-fourths?” (6.NS.1)
• In Chapter 4, 4.1a Homework, Questions 1-6, students determine if a question is a statistical question. The materials state, “Yesterday, Ruth and Carl invited 10 friends to go out to lunch. The questions below came up during the meal. Decide whether or not each question is a statistical question, and justify your decision.” Question 1 states, How much does each person’s meal cost?” Question 3 states, “Would Carl rather have burgers or pizza?” (6.SP.1)
• In Chapter 5, 5.1a Class Activity, students solve 14 questions as they find the area of figures in square units. Question 7 states, “Gloria is painting a feature wall in her bedroom. The dimensions of the wall measure 14 feet by 11 feet. The gallon of paint she purchased will cover 400 square feet. Does she have enough paint to do two coats on the wall? Justify your answer.” Students also complete a Homework activity with an additional 9 questions. For example, Question 6 states, “How many 4-inch square tiles are needed to cover a table that measures 24 inches by 40 inches? Draw and label if needed.” (6.G.1)
• In Chapter 6, 6.1c, Class Activity 3 contains questions for students to write algebraic expressions for each phrase. Activity Three states,  a.“The sum of a number $$n$$ and twenty. b. The sum of twenty and a number $$n$$. c. Four less than a number $$c$$.” (6.EE.2)

The instructional materials relate grade-level concepts explicitly to prior knowledge from earlier grades. Examples of explicit language from the Teacher Edition Workbook for teachers to use include:  

• In Chapter 2, Chapter Overview, Connections to Content, Prior Knowledge states, “In this chapter students draw on their work with ratio from Chapter 1 as they explore the meaning of percent – a part to whole ratio with a whole equal to 100. They express parts of a whole using fraction, decimal, and percent notation. To do this, they construct models learned previously (area models, including hundred grids, tape diagrams, double number lines, tables, etc.). Students know how to express a fraction as a decimal by creating an equivalent fraction with a denominator of 10 or 100 (4.NF). Students will rely on their ability to operate fluently with rational numbers (5.NF and 6.NS). Students use understanding of a rational number, $$\frac{a}{b}$$, as both a groups of $$\frac{1}{b}$$ (3.NF and 4.NF) and $$a÷b$$ (5.NF). Students have also used models to divide whole numbers by unit fractions and unit fractions by whole numbers (5.NF). They will build on this knowledge to divide fractions by fractions.”
• In Chapter 5, Chapter Overview, Connections to Content, Prior Knowledge states, “In previous grades students have investigated writing and solving simple equations. Working with area and volume provides a context for developing and using these equations. Students have also classified triangles and quadrilaterals and have developed an understanding of their properties and relationships. They have also learned how to graph points in the coordinate plane. 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.” 
• In Chapter 6, Chapter Overview, Connections to Content, Prior Knowledge states, “In previous grades, students worked with the properties of operations with whole numbers, fractions, and decimals. In 5th grade, students learned how to use whole number exponents to represent powers of ten. Students have been writing numerical expressions throughout their elementary course work. Additionally, students have been writing and solving equations, representing the unknown with question marks, boxes, and letters.”

The instructional materials reviewed for Utah Middle School Math Project Grade 6 meet expectations for fostering coherence through connections at a single grade, where appropriate and required by the Standards.

The instructional materials include learning objectives that are visibly shaped by CCSSM cluster headings. Examples of chapter and section headings shaped by cluster headings include:

• In Chapter 1, Section 1.1: Representing Ratios and Section 1.2: Rates, Graphs, and Equations are visibly shaped by cluster 6.RP.A, understand ratio concepts and use ratio reasoning to solve problems.
• In Chapter 2, Section 2.2: Division of Fractions is visibly shaped by cluster 6.NS.A, apply and extend previous understandings of multiplication and division to divide fractions by fractions.
• In Chapter 3, Section 3.1: The Symmetry of the Number Line, Section 3.2: Absolute Value and Ordering, and Section 3.3: Negative Numbers in the Real World are visibly shaped by cluster 6.NS.C, apply and extend previous understandings of numbers to the system of rational numbers.
• In Chapter 6, Section 6.2: Writing, Simplifying, and Evaluating Algebraic Expressions and Section 6.3: Equations and Inequalities in One Variable are visibly shaped by cluster 6.EE.B, reason about and solve one-variable equations and inequalities.

Examples of problems and activities that connect two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important include:

• In Chapter 1, Homework 1.2g connects major cluster 6.RP.A to major cluster 6.EE.B as students write equations to represent proportional relationships. Question 2 states, “Jack and Carol both submitted drawings for the cover of the yearbook. The student body voted on whose drawing will be on the cover. For every vote Jack receives, Carol receives three. a. Complete the table to show the relationship between votes for Jack, votes for Carol, and total votes. b. Write an equation that shows the relationship between votes for Jack, j, and votes for Carol, c. c. Write an equation that shows the relationship between votes for Jack, j, and total votes, t.”
• In Chapter 2, Homework 2.2d connects major cluster 6.EE.B to major cluster 6.NS.A as students write and solve equations using rational numbers requiring a fraction divided by a fraction. Question 6 states, “Lisa has $$\frac{3}{4}$$ of a cup of sugar; it’s $$\frac{1}{2}$$ of what she needs. How much sugar does she need?”
• In Chapter 4, Class Activity 4.2d connects supporting cluster 6.SP.B to supporting cluster 6.NS.B as students perform operations with decimals to find measures of central tendency. Question 11a states, “Find the mean, median, and mode for each set of data. Round your answer to the nearest hundredth. 13.22, 11.05, 10.77, 15.04, 12.3, 12.89, 14.7, 16.3, 13.9.”
• In Chapter 5, Class Activity 5.1d connects supporting cluster 6.G.A to supporting cluster 6.NS.B as students use decimal calculations to find the area of trapezoids. Question 5c states, “Find the area of each trapezoid.” A trapezoid is shown with bases of 8.5 units and 11.5 units and a height of 7 units.