6th Grade - Gateway 2

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

The materials identify Five Core Structures: Helping Community, Building Concepts, Math Talk, Quick Practice, and Student Leaders as the five crucial components that are the organizational structures of the program. “Building Concepts in the classroom experiences in which students use objects, drawings, conceptual language, and real-world situations - all of which help students build mathematical ideas that make sense to them.”

The instructional materials present opportunities for students to develop conceptual understanding. For example:

• In Unit 1, Lesson 8, students discuss relationships between drawings and ratio tables. Questions include, “How is the ratio table related to the multiplication table? How is the ratio table related to two rate tables? How are the constant increases shown in the drawing and in the ratio table?"

• Unit 3, Lesson 1, “Place value and operations with decimals and fractions are complex. There are many aspects of the symbols, words, and meanings of the operations that must be related and understood. A major recurring teaching task is leading the attention of your students to multiple aspects of a situation, problem, or numeric representation. This might be while you are explaining something, or you may need to help students do this when they are explaining.”

• In Unit 5, Lesson 12, students explore ways to represent constant speed. They use equations, tables, and graphs, and use these relationships to identify constant change.

The instructional materials include opportunities in the Student Activity Book for students to independently demonstrate conceptual understanding. For example:

• Unit 2, Lesson 3, “Erica’s Solution, Here is how Erica found the area of a parallelogram with no vertices over the base.” Erica’s solution includes finding the area of a rectangle, subtracting the area of two right triangles, to find the area of the parallelogram. Problem 1, “Can you compute the area of the parallelogram using the formula A=bh? Explain.”

• Unit 6, Lesson 3, Problem 25, “A box in the shape of a rectangular prism has a volume of 90 in^2 and a height of 2 1/2 inches. What are possible whole number dimensions for the length and width of the base?” Students use their understanding of volume to find the missing dimensions for the base.

• Unit 7, Lesson 3, Check Understanding, “Draw one drawing to represent the fraction 4/5 and one drawing to represent the ratio 4/5.”

The instructional materials for Math Expressions Grade 6 meet expectations for attending to those standards that set an expectation of procedural skill and fluency.

The instructional materials provide regular opportunities for students to attend to the standards. For example, 6.NS.2, fluently divide multi-digit numbers using the standard algorithm; and 6.NS.3, fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

The instructional materials develop procedural skill and fluency throughout the grade-level. Each lesson includes a “Quick Practice” described as “routines [that] focus on vitally important skills and concepts that can be practiced in a whole-class activity with immediate feedback”. Quick Practice can be found at the beginning of every unit on the pages beginning with the letters QP. Student materials and instructions are also found in the Teacher Resource Book on pages beginning with Q. Examples include:

• Unit 2, Teacher Resource Book, Find Perimeter and Area, “The Student Leader chooses one of the slips of paper and reads the name of a figure, for example: Triangle. A volunteer sketches the figure on the board, chooses a base, draws a height, and describes how to find the area and perimeter of the figure.” Students practice using the formulas for area and perimeter.

• Unit 7, Teacher Resource Book, Find Unit Rates, “Student Leader 1 asks the class for the ratio given in the table on the board, for example: the ratio of apple juice to cherry juice is 8 to 3. Student Leader 2 writes ‘1’ in the second column of the second row in the table, and says: “Say the unit rate using cups.” The class responds, “There are 8/3 cups of apple juice per 1 cup of cherry juice.”

The instructional materials provide opportunities for student to independently demonstrate procedural skill and fluency throughout the grade-level. These include: Path to Fluency Practice, and Fluency Checks. For example:

• Unit 3, Lesson 5, Path to Fluency problems in exercises 20 – 57 provide computational practice in multiplying and dividing decimals. (6.NS.3)

• Fluency Checks are provided throughout the materials. For example, in Unit 4, Lesson 3, Fluency Check includes 15 four-digit by two-digit division problems.

In addition, Homework and Remembering activity pages found at the end of each lesson provide additional practice to build procedural skill and fluency.

The instructional materials for Math Expressions 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.

Students engage with application problems in many lessons for the standards that address application in solving real-word problems. In Unit 4, Lesson 3, Student Activity Book, students solve contextual problems using visual 3-D models and given dimensions to find surface area. “Daniel needs to know how many square feet of metal it takes to build this warehouse including the roof. The warehouse will have a concrete floor. How many square feet of metal does he need?”

Each lesson includes an Anytime Problem listed in the lesson at a glance, and Anytime Problems include both routine and non-routine application problems. For example, Unit 2, Lesson 4, Anytime Problem, “In Ms. Park’s class, there are 16 students with glasses or brown hair (or both). If 7 students have glasses and 14 students have brown hair, how many students have both glasses and brown hair?”

The instructional materials present opportunities for students to engage routine applications of grade-level mathematics. Examples include:

• Unit 1, Lesson 9, Student Activity Book, students create ratio tables to represent various situations. For example, “Noreen makes 2 drawings on each page of her sketchbook. Tim makes 5 drawings on each page of his sketchbook.” “John can plant 7 tomato vines in the time it takes Joanna to plant 4 tomato vines.”

• Unit 1, Lesson 10, Student Activity Book, students solve routine proportion problems. “Noreen did 72 push-ups while Tim did 32 push-ups. When Tim had done 12 push-ups, how many had Noreen done?”

• Unit 7, Lesson 7, Student Activity Book, Question 5, students use representations to solve rate problems. “At a factory, an assembly line processes 100 cans every 3 minutes. How long will it take the assembly line to produce 250 cans?” This question requires applying mathematical representations to answer. It also requires students to know which representations could be used in order to solve problems.

Remembering pages at the end of each lesson are designed for Spiral Review anytime after the lesson occurs. One feature of the Remembering problems are those titled Stretch Your Thinking, which often present opportunities for students to engage with non-routine problems. For example:

• Unit 3, Lesson 9, Remembering, Stretch Your Thinking, Exercise 19, “Lucy has 3 3/4 feet of ribbon. She needs 1 1/2 feet for one project. She needs 1 5/6 piece for another project. Will Lucy have more or less than 1/2 foot of ribbon left after she completes both projects? Explain.”

• Unit 5, Lesson 14, Remembering, Exercise 4, “Jenna is having a sidewalk sale. She pays $12 for a permit. She collects $1.50 for every item she sells.” Stretch Your Thinking, Exercise 5, “Look at the situation in Exercise 4. Suppose at a second sale Jenna pays $15 for a permit and sells each item for $1.75. If she sells 30 items at each sale, at which sale does she make more money? Explain.”

• Unit 8, Lesson 5, Remembering, Stretch Your Thinking, Exercise 8, “The average number of people at a concert over 3 nights is 125. On the first night, there were 151 people at the concert. The same number of people attended on the second and third nights. How many people attended on each of the second and third nights? Explain.”

The instructional materials for Math Expressions 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 represented in the materials. For example:

• Each lesson has a 5-minute Quick Practice providing practice with skills that should be mastered throughout the year.

• There are Performance Tasks throughout the series, where students use conceptual understanding to perform a mathematical task. For example, Unit 3 Performance Task, “If the students had 3 pounds of frozen chopped spinach, what is the maximum number of servings of oeufs a la florentine that they could make? Explain and show your work.”

• Fluency Checks are included throughout the series, where students practice procedural skills and fluency. For example, Unit 3, Lesson 5, Multiplication and Division, Homework and Spiral Review, Fluency Check 1, students solve multiplication with decimals similar to Problem 8, “0.87 x 29.”

• Application problems are embedded into practice in the Student Activity Book. For example, Unit 5, Lesson 10, Homework and Remembering, Problem 1, "Benny is writing a report. For every 7 paragraphs, he uses 4 pieces of art. How many pieces of art will Benny use is his reports is 56 paragraphs long?"

Examples where student engage in multiple aspects of rigor:

• Unit 7, Lesson 2, Ratio as Quotient, introduces students to using unit rates to describe any ratio. Students use two recipes and division to find the unit rate for each recipe. Problem 1, “Find the amounts of cherry juice in each drink for 1 cup of orange juice. Remember that when you divide both quantities in a ratio table by the same number, you get an equivalent ratio.” Students engage in application and procedural skill and fluency, however, the reminder emphasizes procedural skill.

• Student Activity Book, Unit 2, Lesson 5, Solve Real World Problems, students solve application problems which involve conceptual understanding of area and perimeter. Problem 25, “An attic playroom is in the shape of a rhombus with a base of 12 ft and a height of 6 ft. How much carpeting is needed? How much wood molding is needed to go around the room?” Students are instructed to solve the problems and draw pictures to help them.

• Student Activity Book, Unit 9, Lesson 7, students solve: “Victor’s checking account has a balance of $10 and is charged a $2 service fee at the end of each month. Question 1: “Suppose Victor never uses the account. Complete the table below to show the balance in the account each month for 6 months. Then use the data to plot points on the coordinate plane to show the decreasing balance over time.”

The instructional materials reviewed for Math Expressions Grade 6 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 clearly identified in a variety of places throughout the materials. For example:

• The Mathematical Practices are identified in both volumes of the Teacher’s Edition. Within the introduction, on page I13 in the section titled The Problem Solving Process, the publisher groups the Mathematical Practices into four categories according to how students will use the practices in the problem solving process. Mathematical Practices are also identified within each lesson.

• Each time a Mathematical Practice is referenced, it is listed in red with a brief description of the practice.

• At the beginning of each Unit is a section devoted to the Mathematical Practices titled Using the Common Core Standards for Mathematical Practices which includes guidance for the teacher. Within this section, each Mathematical Practice is defined in detail. In addition, an example from the Unit is provided for each practice. For example, in Unit 7, Lesson 10, Activity 2, Percents of Numbers, identifies “MP1 Make Sense of Problems. Be sure students understand the problem. Discuss what information we know and what we need to find. We know the adult dose. We know what percent the child dose is to the adult dose. We need to find the child dose.”

Examples of Mathematical Practices that are identified, and enrich the mathematical content include:

• Unit 1, Lesson 7, MP6 - Attend to Precision|Verify Solutions. “Once students have identified the unit rate from the graph, they can use that rate to complete the table. Or, they can use points on the graph to complete the table. You can suggest that they use one method first, and then use the other method to verify their solution.”

• Unit 2, Lesson 2, identifies MP7 - Look for Structure. Teachers are guided to “Ask students to think about many different right triangles to which this formula applies; huge right triangles, tiny right triangles, and medium right triangles. Will the formula A= 1/2 bh work for all these right triangles? Explain.”

• Unit 4, Lesson 5, Problem 12, MP2 - Reason Abstractly and Quantitatively | Connect Symbols and Words. “Students should connect that finding 'how many square inches' means finding the surface area as they solve Problems 7-12. Students also need to use the description of the object and the picture to decide how much of the surface area needs to be found. Finally, students need to use the correct operations to calculate the number of square inches needed.”

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

Math Expressions includes a Focus on Mathematical Practices lesson as the last lesson within each unit. Activity 3 of each of these lessons prompts students to determine whether a mathematical statement is true or false or to establish an arguable position surrounding a mathematical statement. These activities provide students opportunities to construct an argument and critique the reasoning of others. Student volunteers ask questions of other students to verify or correct their reasoning. Examples of Focus on Mathematical Practices lessons include, but are not limited to:

• Unit 2, Lesson 10, students develop an argument for the following statement: “When the perimeter of a rectangle increases, the area always increases.” Volunteers share their positions and explanations with the class. The class asks the volunteers questions and verifies or corrects reasoning errors.

• Unit 4, Lesson 6, students develop an argument for the following statement: “When the dimensions of a rectangular prism are doubled, the surface area doubles.” Volunteers share their positions and explanations with the class. The class asks the volunteers questions and verifies or corrects reasoning errors.

• Unit 7, Lesson 7, students develop arguments for the following statements: “All integers are rational numbers. All rational numbers are integers.” Volunteers share their positions and explanations with the class. The class asks the volunteers questions and verifies or corrects reasoning errors.

Puzzled Penguin problems are found throughout the materials and provide students an opportunity to correct errors in the penguin’s work. These tasks focus on error analysis, and many of the errors presented are procedural. Examples of Puzzled Penguin problems include:

• Unit 3, Lesson 10, Puzzled Penguin problem, students solve: “Dear Math Students, I am planting a flower garden. The space I have to plant in is 3 1/2 feet wide and 4 1/2 feet long. I told my friend that the area of my flower garden is 12 1/4 square feet. She says I’ve made a mistake! Can you help me find my mistake and the correct area? Your friend, Puzzled Penguin.”

• Unit 5, Lesson 3, Puzzled Penguin problem, “Dear Math Students, Here’s how I analyzed m * (4 + m). Did I do it right? If not, help me understand what I did wrong. Step 1: I circled the part in parentheses. Step 2: There are no powers so I didn’t need to do anything. Step 3: I circled the multiplication. Step 4: I looked for addition and subtraction and circled the terms. Your friend, Puzzled Penguin.” Each step is illustrated to the side of the narrative. Student analysis needs to identify that the error is in treating the expression (m + 4) as m, and 4.

• Unit 7, Lesson 6, Puzzled Penguin problem, students analyze a ratio table illustrating 2 purple units for every 5 orange units. “I made my own sand mixture. I mixed 2 parts purple and 5 parts orange. Then I wrote this multiplicative comparison. The amount of purple sand is 2/5 times the amount of the total mixture. My friend says I made a mistake. Did I? If I did, can you tell me what mistake I made and help me correct it?” In this example, students need to demonstrate understanding of part to part and part to whole comparisons.

• Unit 8, Lesson 8, Puzzled Penguin problem, students find the error that was made on drawn box plots. “Dear Math Students, I was given this set of data. 21, 21, 22, 23, 24, 25, 26, 26, 26, 29, 30. Here is the box plot I made to represent the data. Can you help me understand what I did wrong?” The box plot does not include a line to identify the median of the data.

In addition, Remembering pages at the end of each lesson often present opportunities for students to construct arguments and/or critique the reasoning of others. For example:

• Unit 4, lesson 3, Remembering, Stretch Your Thinking, Exercise 7, “Jerome has a block of craft foam shaped like a square prism. He cuts the foam block in half. Jerome thinks that because the surface area of the original black was 358 in.^2, the surface area of each piece is 179 in.^2. Is he correct? Explain.”

• Unit 5, Lesson 8, Remembering, Stretch Your Thinking, Exercise 10, students construct arguments as they explain “The diagram shows 4⋅(m⋅3). Change the diagram so this it shows (4⋅m)⋅3. How do the diagrams show that 4⋅(m⋅3) = (4⋅m)⋅3 = 12m?

• Unit 8, Lesson 6, Remembering, Stretch Your Thinking, Exercise 4, students construct arguments to explain “The balance point of a set of 4 numbers is 14. How can you move one point to make the balance point 16?”