6th Grade - Gateway 2

The instructional materials reviewed for Utah Middle School Math Grade 6 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

The materials include Anchor Problems, Class Activities, and Homework that develop conceptual understanding throughout the grade-level. There is one Anchor Problem at the beginning of each chapter. The Class Activities are question sets for teachers to guide student discussion and learning. The Homework questions are assigned as independent work for students. Examples that elicit conceptual understanding include: 

• In Chapter 2, 2.2a Class Activity introduces students to the concept of dividing fractions using models and equations (6.NS.1). Class Activity 4 states, “You have $$\frac{1}{2}$$ of a pizza left over from dinner last night. You invite two friends over and the 3 of you share the leftover pizza. What portion of a whole pizza will you each be getting? Solve using a model and an equation.”
• In Chapter 3, 3.0 Anchor Problem states, “Create a model to represent the elevations of the locations shown in the table. You may need to create another number line to zoom in on the elevations of Badwater Basin, Furnace Creek, Beatty Junction, and Stovepipe Wells. Students can think of it as magnifying that portion of the graph to better distinguish the points.” (6.NS.6)
• In Chapter 5, 5.4a Class Activity introduces drawing nets to help students calculate the surface area of three-dimensional figures (6.G.4). Question 2 states, “Draw a net for the “Family Size” cereal box shown. Try to draw the net differently than you do for the cereal box described in number 1 [previous question]. Be sure to label all the dimensions.”
• In Chapter 6.1a, Class Activity, students generate as many expressions as they can to represent different scenarios, then simplify the expressions to determine whether they “work” or “don’t work,” (6.EE.3). Question 2 states, “Carmen has 420 tickets to spend at the prize counter at an arcade. She buys 3 packs of Nerds that cost 50 tickets each. How many tickets does Carmen have left?” Students complete a table with the headings Expression, Simplified Form, Does It Work, and Ideas. 

The materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. In the Homework assignments, students complete conceptual problems independently that are similar to the conceptual exercises done with teacher guidance and support in the Anchor Problems and Class Activities. This independent practice reinforces their conceptual understanding. Examples where students independently demonstrate conceptual understanding include: 

• In Chapter 1, 1.1a Homework, Question 2, students “Write three different ratio statements about the picture below. Use words like ‘to’, ‘per’, ‘for every’, ‘each’, and ‘ratio’. Consider the relationship between birds and branches and also the relationship between birds and body parts of a bird. For example, ‘Each bird has two wings.’” (6.RP.1)
• In Chapter 2, 2.2c Homework, students “solve each problem using the model of your choice.” Question 3 states, “Sasha has 18 yards of string for her kite; it’s one and half times what she needs. How much string does she need?” (6.NS.1)
• In Chapter 3, 3.2a Homework, Question 1, students “Use the number line below to answer the questions. a. Is the value of A positive or negative? b. Is the distance from zero to A positive or negative? Explain.” (Students use a number line with 0 in the middle. A is located to the left of 0.) (6.NS.7c)
• In Chapter 6, 6.3h Homework, students “1) Write an inequality to represent the situation. 2) Create a number line diagram to represent the situation. 3) Give two values that make the inequality true. Make sure your values make sense in the problem.” Question 1 states, “A minimum of three people need to show up for a workout class for the instructor to hold class.” Question 2 states, “A maximum of 45 people can be in the school library at one time.” (6.EE.8)

The instructional materials reviewed for Utah Middle School Math Project Grade 6 meet expectations for attending to those standards that set an expectation of procedural skill and fluency.

The instructional materials develop procedural skill and fluency throughout the grade level. Chapter 0 is specifically dedicated to fluency in Grade 6. The Section 0.1 Overview states, “This section specifically addresses a student’s ability to fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.” In the Section 0.2 Overview, the first Class Activity focuses on, “divisibility rules...being able to draw upon them makes working with factors and multiples much easier for students, especially when dealing with larger numbers.” The Section 0.3 Overview states, “students build upon their knowledge of operations with multi-digit whole numbers and extend similar reasoning to multi-digit decimals.” 

Examples of procedural skill and fluency developed throughout the grade-level include: 

• In Chapter 0, 0.1b Class Activity, Part 3, Questions 1-8, students “Estimate each quotient. Then use the standard algorithm to find the exact quotient. 1. $$14\lceil322$$, 2. $$12\lceil5484$$, 3. $$115\lceil8625$$, 4. $$205\lceil21115$$, 5. $$20\lceil361$$, 6. $$41\lceil4970$$, 7. $$352\lceil6890$$, 8. $$6\lceil108.75$$.” (6.NS.2)
• In Chapter 0, 0.3d Class Activity, Question 7 states, “$$13.6 + 901.15$$.” (6.NS.3)
• In Chapter 6, 6.1c Class Activity, Activity 2, Question J states, “Equivalent or Not? $$4n + 2n$$ and $$(4 + 2)n$$.” (6.EE.A)
• In Chapter 6, 6.2a Class Activity, Part I, Activity 4 states, “Simplify the expression. If the expression is already simplified, write ‘already simplified’. There are 26 expressions for students to simplify. For example, Question h states, “$$3(7x)-10x$$.” (6.EE.A)

The instructional materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade-level. Examples include: 

• In Chapter 0, 0.1b Homework, Questions 5-14, students “Estimate each quotient. Then use the standard algorithm to find the exact quotient. Express remainders as decimals. 5. $$\frac{78}{6}$$, 6. $$\frac{352}{16}$$, 7. $$\frac{540}{18}$$, 8. $$\frac{49815}{405}$$, 9. $$\frac{578}{32}$$, 10. $$\frac{4635}{45}$$, 11. $$\frac{6996}{212}$$, 12. $$\frac{1018}{72}$$, 13. $$\frac{326}{8}$$, 14. $$\frac{40613601}{4263}$$.” (6.NS.2)
• In Chapter 0, 0.2e Homework, students “Use the distributive property and the GCF to write an equivalent expression for each given sum. In Question 5, “$$16 + 36$$.” (6.EE.A)
• In Chapter 1, Spiral Review, Question 2, students answer, “What is $$108 ÷ 8$$?” (6.NS.2)
• In Chapter 3, Spiral Review, Question 3, students “Write an equation to show the relationship between number of pies p and cups of cherries c.” (6.EE.A)
• In Chapter 4, Spiral Review, Question 1a, students calculate, “$$25 ÷ 4$$.” (6.NS.2)
• In Chapter 6, 6.1c Homework, students “Write an algebraic expression for each phrase. A number j increased by four.” (6.EE.A)

The instructional materials reviewed for Utah Middle School Math Project Grade 6 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. 

The instructional materials include multiple opportunities for students to engage in routine and non-routine application of mathematical skills and knowledge of the grade-level. Both routine and non-routine applications are found in the Anchor Problems and Class Activities for students. Routine applications are also found in the Homework questions for students. 

The materials provide multiple opportunities for students to engage in routine applications of mathematics. Examples include:

• In Chapter 3, 3.3b Class Activity, Activity 7 states, “In February of 2011, Nowata, Oklahoma experienced a 110-degree rise in temperature over a 7-day period. On February 10, 2011, the low temperature was -31℉, the coldest temperature ever recorded in Oklahoma. On February 17, 2011, the temperature at one point during the day was 110 degrees hotter than the temperature on February 10, 2011. What was the high temperature on February 17, 2011 in Nowata, Oklahoma?” (6.NS.5)
• In Chapter 5, 5.2c Class Activity, Question 2 states, “A coordinate grid represents the map of a city. Each square on the grid represents one city block. a. Heather’s apartment is at the point (5, 7). She walks 4 blocks south, then 8 blocks west, then 4 blocks north, and then finally 8 blocks east back to her apartment. How many blocks did she walk total? Describe the shape of her path. Mark and label her apartment and highlight her walk.” (6.G.3)

The materials provide multiple opportunities for students to engage in non-routine applications of mathematics. Examples include:

• In Chapter 1, the Anchor Problem states, “Two gears are connected as shown in the picture below. The smaller gear has 8 teeth and the larger gear has 12 teeth. a. Find a way to determine the number of revolutions the small gear makes based on the number of revolutions the large gear makes. Organize your results. a. If the smaller gear makes 24 revolutions, how many revolutions will the larger gear make? b. If the larger gear makes 20 revolutions, how many revolutions will the smaller gear make? c. If the smaller gear makes 24 revolutions, how many revolutions will the larger gear make? d. If the larger gear makes 1 full revolution, how many revolutions does the smaller gear make? e. If the smaller gear makes 1 full revolution, how many revolutions does the larger gear make? f. Create four different representations of the relationship between the number of revolutions the large gear makes and the number of revolutions the small gear makes. Make up a question that can be answered using each representation.” (6.RP.3)
• In Chapter 6, Alternative Anchor Problem, Part 1 states, “The side length of a square is unknown. Write an expression for the perimeter of the square. Write an expression for the area of the square. Write an expression to show the perimeter of 3 copies of the square. Assume the squares are not touching. Write an expression to show the area of 3 copies of the square. The side length of the square from above is tripled. Write an expression for the new perimeter of the square. Compare the perimeter of the new square to the perimeter of the original square. Write an expression for the new area of the square. Compare the area of the new square to the area of the original square.”

The instructional materials provide opportunities for students to independently demonstrate the use of mathematics flexibly in a variety of contexts. Examples of students independently demonstrating the use of mathematics flexibly in a variety of contexts include:

• In Chapter 0, 0.3d Homework, Question 20 states, “Seth wants to buy a new skateboard that costs $167. He has $88 in the bank. a. If he earns $7.25 an hour pulling weeds, how many hours will Seth have to work to earn the rest of the money needed to buy the skateboard?” (6.NS.3)
• In Chapter 1, 1.1e Homework, Question 1 states, “Students in Mrs. Benson’s gym class are voting on whether the next sport they learn how to play should be basketball or soccer. The tape model shows the results of the survey. a. If there are 35 students in Mrs. Benson’s gym class, how many voted for soccer? Solve this problem using at least two different methods. Explain the methods you used and how they are related.” (6.RP.3)
• In Chapter 5, 5.4b Homework, Question 7 states, “Carly and Nadia are painting their bike ramp. They have one quart of paint which will cover 100 ft2. Do they have enough paint to do the two coats? Justify your answer.” (6.G.4).
• In Chapter 6, 6.3j Homework, Question 1 states, “Iya sells friendship bracelets for $4. Write and solve an inequality to represent the number of bracelets Iya needs to sell to make at least $150?” (6.EE.8)

The instructional materials reviewed for Utah Middle School Math Project Grade 6 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. All three aspects of rigor are present independently in the materials, and multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. 

Examples of where the materials attend to conceptual understanding, procedural skill and fluency, and application independently include: 

• Conceptual Understanding: In Chapter 4, 4.2c Class Activity, Question 1 states, “Carly’s teacher says that the center of this data is the balance point. Discuss with a neighbor what you think she means by this. Then draw a triangle under the dot plot where you think the balance point is.” (6.SP.3)
• Procedural Skills and Fluency: In Chapter 6, Spiral Review, Question 1, students “Write an algebraic expression for each phrase. a. Twice the sum of a number n and 5. b. The sum of twice a number n and 5. c. The sum of twice a number n and ten.” (6.EE.A)
• Application: In Chapter 2, 2.2d Class Activity, Activity 4, students apply their knowledge of division of fractions to real-world situations. Students “create a model of your choice to answer this question. Then, write a number sentence to represent the problem.” Question c states, “Josephine has run 7.5 miles, which is $$\frac{2}{5}$$ of her training distance for the day. How far was she planning on running today?” (6.RP.3)

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

• In Chapter 0, 0.3b Class Activity connects all three aspects of rigor. In Part 1, Question 4, students engage in conceptual understanding as they “Assume the side length of the large square is 0.1. What is the area of the large square? What is the side length of a small square? What is the area of the small square?” In Part 3, Question 4, students solve real-world problems using procedural skills, “You work part time at a bookstore and get paid $12.05 per hour. In the entire month of March you worked 78.25 hours. How much money did you make in March?”
• In Chapter 4, 4.3d Class Activity, students engage in conceptual understanding of box plots and data, use procedures to find the Interquartile Range, and apply their understanding and skills to determine how the data best fits the discussion. Question 1 states, “The three box plots below represent the test scores for three different classes. Examine each plot and then discuss the questions that follow. a. What is the same about these box plots and what is different? b. Find the IQR for each plot and use it to compare the variability of each set of class scores. c. Make an argument for each class that supports the claim that this class performed the best on the test.” (6.SP.3-5)
• In Chapter 5, 5.3a Class Activity, Question 10 connects conceptual understanding and procedural skills as students “make a prediction about which rectangular prism below has the greatest volume, which one has the smallest volume? Find the volume of each prism to check your predictions.” (6.G.2)

The instructional materials reviewed for Utah Middle School Math Project Grade 6 meet expectations for identifying the Standards for Mathematical Practice and using them to enrich mathematics content within and throughout the grade-level.

Examples of the Standards for Mathematical Practice (MPs) being clearly identified in two ways include:

• At the beginning of each chapter, there is a section labeled “Standards of Mathematical Practice.” This section provides an explanation about how the MPs are applied specifically in the chapter, and includes an example problem the students will see later in the materials. 
• In the Anchor Problems and Class Activities, the MPs are identified by icons that match the MPs at the beginning of the chapter. 

Examples of the MPs being used to enrich the mathematical content:

• MP1: In Chapter 4, Standards for Mathematical Practice states, “The example problem given shows how students must make sense of practical problems and turn them into statistical investigations. They must make sense of what statistical arguments can be made about the data. They must determine what statistical measures might be used to support their arguments and how to go about finding them. Throughout the solving process, students must stop and evaluate their progress.”
• MP2: In Chapter 2, 2.2c Class Activity, Activity 1a states, “Eli has 8 pints of ice cream. It’s $$\frac{2}{3}$$ of what he needs. How much does he need? Draw a model of your choice to answer this question. Then, write a number sentence to represent the problem.” Students decontextualize a fraction model to quantitatively reason with fractions. 
• MP4: In Chapter 0, 0.3a Class Activity states, “Marta has created the model below. She claims that this model can be used to represent the sum of 24 and 38. 1. If Marta’s claim is true, what is the value of the small square? 2. What is the value of a rod (long rectangle)? 3. Find the sum of 24 and 38 using the addition algorithm and discuss how this relates to the model above.” Students create a model that represents the sum of numbers and could be used to solve other problems. 
• MP5: In Chapter 1, Standards for Mathematical Practice states, “Throughout this chapter, students are exposed to a variety of tools that can be used to represent ratio relations and solve real world problems. These tools include concrete manipulatives such as tiles and chips, pictures, tape diagrams, tables, graphs, and equations. The chapter starts with more concrete tools then progresses into tools that are more abstract. For example, students start by modeling ratio relations with chips and tiles (concrete), then pictures (pictorial), and then tape diagrams (abstract). They understand that these all show the same relationship but more abstract representations can be more efficient and flexible for solving problems. They make connections between the different tools. For example, they consider patterns in tables and on graphs. They understand that an equation shows the explicit relationship between two variables given in a table.”
• MP6: In Chapter 4, 4.1b Class Activity states, “When creating data displays it is so important to attend to the precision of labeling your model so that all the information that the model presents is clearly understood.” Question 7 states, “Marta records the high temperatures for each day she goes swimming in the month of August. She has recorded her data on the dot plot below. a. Marta did not label her dot plot with units or a title. Determine the appropriate units for this data and how the data was collected. Then give the dot plot an appropriate title.”
• MP7: In Chapter 3, 3.2a Class Activity, Activity 8 states, “This problem requires students to make sense of the structure of the ordered pairs. Devise a strategy for finding the distance between two points without graphing. Then, find the distance between the two points. a. (3,157) and (3, 84).”
• MP8: In Chapter 6, 6.1d Class Activity, Question c states, “Marin has 50 tickets to spend on rides at a carnival. Each ride takes 6 tickets. Write different expressions to represent the number of tickets Marin has left based on the number of rides she goes on.”

The instructional materials reviewed for Utah Middle School Math Project Grade 6 meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

Examples from the Student Edition Workbook for prompting students to construct viable arguments include: 

• In Chapter 0, 0.3b Class Activity, Question 2 states, “Dallin has begun to do the following multiplication problem. His work is shown below; he does not know where to place the decimal point in the product. Correctly place the decimal point for him and justify your answer.”
• In Chapter 3, 3.1b Homework, Question 5a states, “The number line below shows the location of 2.5. Explain a method for representing −2.5 on the number line.”
• In Chapter 4, 4.3d Class Activity, Question 1c states, “The three box plots below represent the test scores for three different classes. Examine each plot and then discuss the questions that follow. Make an argument for each class that supports the claim that this class performed the best of the test.”

Examples from the materials for prompting students to analyze the arguments of others include:

• In Student Edition Workbook, Chapter 0, Mathematical Practice Standards, MP3, students solve, “Roxy’s cashier has made some calculations for some of the purchases at the candy store and has made some mistakes, his work is shown below. For problems 5, 6, and 7 go through each transaction and determine the mistake, explain how to perform the calculation correctly and fix the mistake.” Additional information for students includes, “Throughout this appendix, you will find several problems that take on the form of ‘Find, Fix, and Justify.’” For these problems students analyze another student’s work and must identify mistakes in the work. They make arguments as to why something is wrong by pointing out explicit errors observed. Once they fix the mistake they must justify why their reasoning is correct.
• In Student Edition Workbook, Chapter 3, 3.1d Class Activity, Activity 3, Problem 20 states, “Will said, ‘The opposite of the opposite of a number is sometimes positive.’ Is Will’s statement true or false? Explain.”
• In Chapter 5, 5.3c Class Activity, Question 8, “Olivia claims that where s is the side length of a cube is the formula you should use to find the volume of a cube. Harrison claims that the correct formula is where l is the length, w is the width, and h is the height of the cube. a. Who is correct? Why or why not.”

There are a few instances where the MP3 icon is noted in the instructional materials, but students do not engage in MP3 in those questions. Examples include:

• In Chapter 3, 3.1i Class Activity, Activity 1d states, “Write three true statements based on the diagram. For example, ‘All team sports are sports.’”
• In Chapter 6, 6.2a Class Activity, Activity 7a states, “Explain what each word means. Use examples and non-examples to support your ideas. Term, Constant, Coefficient, Like Terms”

The instructional materials reviewed for Utah Middle School Math Project Grade 6 meet expectations for explicitly attending to the specialized language of mathematics.

Examples of the materials providing explicit instruction in how to communicate mathematical thinking using words, diagrams, and symbols include:

• At the beginning of each chapter, in the Student and Teacher Edition Workbooks, there is a list of academic vocabulary for the chapter. 
• In the Mathematical Foundations book, accurate mathematical language is explained to assist teachers in the presentation of materials.
• In Chapter 3, Section Overview 3.3, Negative Numbers in the Real World states, “The first lesson is focused on the vocabulary associated with positive and negative quantities.” Concepts and Skills to Master are listed for teachers. The first concept includes “use and interpret academic vocabulary used to describe situations with positive and negative quantities.”
• In Chapter 6, 6.1c Class Activity, Activity 3, students “write an algebraic expression for each phrase.” In the Teacher’s Edition, additional guidance for the teacher states, “Students will likely need a review of the following vocabulary: sum, difference, product, and quotient. Have students underline key words. When a phrase starts out as ‘the sum of’, ‘the difference of’, ‘the product of’, or ‘quotient of’ students should look for the word and – it helps to identify the different parts of the expression - the addends (addition), the minuend and subtrahend (subtraction), the factors (multiplication) and the dividend and divisor (division).” 

Examples of the materials using precise and accurate terminology and definitions when describing mathematics, and support students in using them, include:

• In Chapter 1, 1.2g Class Activity, Activity 2 states, “Marcus is training for an ultra-marathon where he will be running 100 miles. He can run 7 miles per hour. a. Complete the table below to show the relationship between time and distance for Marcus.” The Teacher’s Edition includes, “when appropriate, introduce the vocabulary dependent variable and independent variable. The decision as to which quantity is the dependent variable and which quantity is the independent variable is usually interchangeable and is driven by the question being asked.” 
• In Chapter 3 Academic Vocabulary includes, “number line, whole number, positive number, negative number, integer, rational number, line symmetry, opposites, scale, quadrant, origin, x-axis, y-axis, absolute value, greater than, less than, deposit, withdrawal, debit, credit, ascend, descend, profit, loss.”
• In Chapter 5, 5.4b Class Activity, Question 3, students “explain in your own words what surface area is and how to find the surface area of a three-dimensional object.”
• In Chapter 6, 6.0a Class Activity, Frayer charts/models are provided so students can create a reference for each property (Commutative Property, Associative Property, Additive Identity Property of Zero, Multiplicative Identity Property of One and Distributive Property). The materials state, “At the start of the chapter, fill out the chart with the definition in words and symbols and any examples and notes that would be helpful at the time. Then, refer to the reference sheet and continue to add examples and notes when you come across a problem in the chapter that relies on one of the properties.”