6th Grade - Gateway 2

The instructional materials for Eureka 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 problems and questions that develop conceptual understanding throughout the grade level. For example:

• In Module 1, Lesson 1, students develop conceptual understanding when introduced to ratios through the use of various visual models (tape diagrams, bar models and tables). Students develop their understanding of a ratio and how to write it in notation form. (6.RP.A)
• In Module 1, Lesson 13, students develop their understanding of proportional relationships when using ratio tables to write equations that represent the relationship. (6.RP.A)
• In Module 4, Lessons 1-4, students develop conceptual understanding by using tape diagrams to represent and understand the relationships of operations. Students use the tape diagrams to generate equivalent expressions, first using numbers, and then using both numbers and letters. (6.EE.2a-b, 6.EE.3, 6.EE.6)
• In Module 4, Lesson 25, students develop an understanding of inequalities and equations by using bar diagrams, number lines and algebra. (6.EE.B)

The materials provide opportunities for students to demonstrate conceptual understanding independently throughout the grade level. For example:

• In Module 3, Lesson 2, students independently demonstrate an understanding of positive and negative numbers. Students use the number line to represent real-life situations. (6.NS.5)
• In Module 4, Lesson 9, students independently demonstrate an understanding of writing expressions by using tables and visual diagrams. Classwork Exercise 2 states, “Write two expressions to show w increased by 4. Then, draw models to prove that both expressions represent the same thing.” (6.EE.A)
• In Module 4, Lesson 24, Exit Ticket, students independently demonstrate an understanding of solving an equation or inequality. Students state when the given equations and inequalities will be true and when they will be false. (6.EE.6)

The instructional materials for Eureka Grade 6 meet expectations that they attend to those standards that set an expectation of procedural skill and fluency.

The instructional materials develop procedural skill and fluency throughout the grade level. For example:

• In Module 2, Lesson 13, students develop procedural skill and fluency by explaining how the standard algorithm works when dividing multi-digit numbers. The Closing section of the Teacher Materials state, “Explain in your own words how the division algorithm works.” (6.NS.2)
• In Module 2, Lesson 14, students develop procedural skill and fluency by converting decimal division into whole-number division using fractions. The teacher is prompted to ask the following questions in Classwork Example 2, “We determined that when we multiply a divisor by a power of ten, the decimal point is moved to the right the number of times we multiply by a power of ten. How many places does the decimal point move to the right when we multiply the divisor by ten? Explain why the decimal point moves twice to the right when we multiply the divisor by one hundred? We can use decomposition to explain.” (6.NS.3)

The instructional materials provide opportunities to demonstrate procedural skill and fluency independently throughout the grade level. For example:

• In Module 5, Lesson 1, students independently demonstrate procedural skill and fluency with decimal operations. Problem Set Question 9 states, “A parallelogram has an area of 20.3 cm squared and a base of 2.5 cm. Write an equation that relates the area to the base and height, h. Solve the equation to determine the height of the parallelogram.” (6.NS.3)
• In Module 5, Lesson 18, students independently demonstrate procedural skill and fluency of decimal operations when calculating the surface area of a rectangular prism. Problem Set Question 2 states, “Calculate the surface area of each figure below. Figures are not drawn to scale. 2.3 cm, 8.4 cm, 18.7 cm.” (6.NS.3)
• In Module 6, Lesson 8, students independently demonstrate procedural skill and fluency of multi-digit division with the standard division algorithm when calculating mean and mean absolute deviation. Classwork Exercise 1, Problem 1 states, “Use the data in the table provided in Example 1 to answer the following: Calculate the mean of the monthly average temperature for each city.” (6.NS.2)

The instructional materials for Eureka Grade 6 meet expectations that the materials are 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. For example:

• In Module 1, Lesson 19, students engage in grade-level mathematics when solving various problems involving unit rates. Classwork Exercise 1 states, “Bryan and ShaNiece are both training for a bike race and want to compare who rides his or her bike at a faster rate. Both bikers use apps on their phones to record the time and distance of their bike rides. Bryan’s app keeps track of his route on a table, and ShaNiece’s app present the information on a graph. The information is shown below. At what rate does each biker travel? Explain how you arrived at your answer. ShaNiece wants to win the bike race. Make a new graph to show the speed ShaNiece would have to ride her bike in order to beat Bryan.” (6.RP.3b)
• In Module 4, Lesson 32, students engage in grade-level mathematics when writing equations with two-variables representing the total amount of money saved. Classroom Exercise 1 states, “Each week Quentin earns $30. If he saves this money, create a graph that shows the total amount of money Quentin has saved from week 1 through week 8. Write an equation that represents the relationship between the number of weeks that Quentin has saved his money, w, and the total amount of money in dollars he has saved, s. Then, name the independent and dependent variables. Write a sentence that shows this relationship.” (6.EE.9)

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

• In Module 2, Lesson 5, students independently demonstrate the use of mathematics by creating their own story problems by applying their understanding of division. The Exit Ticket states, “Write a story problem using the measurement interpretation of division for the following: 3/4 divided by 1/8 = 6.” (6.NS.1)
• In Module 4, Lesson 29, students independently demonstrate the use of mathematics by by writing equations and creating tables to determine the amount of dog food to purchase. The Exit Ticket states, “A pet store owner, Byron, needs to determine how much food he needs to feed the animals. Byron knows that he needs to order the same amount of bird food as hamster food. He needs four times as much dog food as bird food and needs half the amount of cat food as dog food. If Byron orders 600 packages of animal food, how much dog food does he buy? Let ???????? represent the number of packages of bird food Byron purchased for the pet store.” (6.EE.7)
• In Module 5, Lesson 11, students independently demonstrate the use of mathematics by calculating the volume of a rectangular prism with fractional sides to solve a real-world problem. Classwork Exercise 3 states, “A toy company is packaging its toys to be shipped. Each small toy is placed inside a cube-shaped box with side lengths of ½ in. These smaller boxes are then placed into a larger box with dimensions of 12 in. x 4 1/2 in. x 3 1/2 in. What is the greatest number of small toy boxes that can be packed into the larger box for shipping? Use the number of small toy boxes that can be shipped in the larger box to help determine the volume of the shipping box.” (6.G.2).

The instructional materials for Eureka Grade 6 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately.

Conceptual understanding is addressed in Classwork. During this time, the teacher guides students through a new concept or an extension of the previous day’s learning. Students practice solving procedural problems in problem sets. Fluency is also addressed as an independent component in selected lessons. The materials provide engaging applications of grade-level concepts throughout each lesson. The program balances all three aspects of rigor in every lesson.

All three aspects of rigor are present independently throughout the program materials. For example:

• In Module 4, Lesson 8, students develop conceptual understanding of expressions in which letters stand for numbers. Classwork Example 2 states, “g x 1 = g, Remember a letter in a mathematical expression represents a number. Can we replace g with any number? Choose a value for g, and replace g with that number in the equation. What do you observe? Will all values of g result in a true number sentence? Experiment with different values before making your claim. Write the mathematical language for this property below:” (6.EE.2a).
• In Module 5, Lesson 17, students practice fluency of solving one-step equations using addition or subtraction with some equations requiring decimal calculations. Fluency-Addition and Subtraction Equations-Round 1 Question 28 states, “23.6 = m - 7.1” (6.EE.7)
• In Module 5, Lesson 19, students engage in the application of mathematics when solving real-world problems involving surface area. Problem Set Question 5 states, “A swimming pool is 8 meters long, 6 meters wide, and 2 meters deep. The water-resistant paint needed for the pool costs $6 per square meter. How much will it cost to paint the pool? How many faces of the pool do you have to paint? How much paint (in square meters) do you need to paint the pool? How much will it cost to paint the pool?” (6.G.4)

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. For example:

• In Module 1, Lesson 5, students develop conceptual understanding of equivalent ratios by drawing tape diagrams to solve real-world problems. Classwork Example 1 states, “A County Superintendent of Highways is interested in the numbers of different types of vehicles that regularly travel within his county. In the month of August, a total of 192 registrations were purchased for passenger cars and pickup trucks at the local Department of Motor Vehicles (DMV). The DMV reported that in the month of August, for every 5 passenger cars registered, there were 7 pickup trucks registered. How many of each type of vehicle were registered in the county in the month of August? Using the information in the problem, write four different ratios and describe the meaning of each. Make a tape diagram that represents the quantities in the part-to-part ratios that you wrote. How many equal-sized part does the tape diagram consist of? What total quantity does the tape diagram represent?” (6.RP.3)
• In Module 2, Lesson 4, students practice procedural skill and fluency of dividing a fraction by a fraction as they solve real-world problems. Classwork Example 1 states, “Molly has 1 3/8 cups of strawberries. She needs 3/8 cup of strawberries to make one batch of muffins. How many batches can Molly make? Use a model to support your answer.” (6.NS.1)

The instructional materials reviewed for Eureka Grade 6 meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade level.

All of the eight MPs are identified within the grade-level materials. The Standards for Mathematical Practice are identified at the beginning of each module under the Module Standards. The tab named “Highlighted Standards for Mathematical Practice” lists all of the MPs that are focused on in the module. Each MP is linked to the definition of the practice as in which lessons throughout the series practice can be found.

Each Module Overview contains a section titled, “Focus Standard for Mathematical Practice.” Every practice that is identified in the module has a written explanation with specific examples of how each practice is being used to enrich the content of the module. For example:

• In Module 3, the explanation for MP 7 states, “Look for and make use of structure. Students understand the placement of negative numbers on a number line by observing the patterns that exist between negative and positive numbers with respect to zero. They recognize that two numbers are opposites if they are the same distance from zero and that zero is its own opposite. Students extend their understanding of the number line's structure to the coordinate plane to determine a point’s location. They recognize the relationship between the signs of a point’s coordinates and the quadrant in which the point lies.”

Each lesson specifically identifies where MPs are located, usually within the margins of the teacher edition.

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

Student materials consistently prompt students to construct viable arguments and analyze the arguments of others.

• In Module 5, Lesson 3, students analyze the arguments of others when finding the area of the triangle. Classwork Exercise 4 states, “Joe found the area of a triangle by writing A = 1/2 (11in.)(4 in.), while Kaitlyn found the area by writing A = 1/2 (3 in.)(4 in.) + 1/2 (8 in.)(4 in.). Explain how each student approached the problem.
• In Module 6, Lesson 12, students analyze the arguments of others when determining the median of a data set. Students determine if the strategy of a fictional student has resulted in a correct value for the median, and explain their decision. Exercise 4d states, “Betse argued that the median was halfway between 60 and 85, or 72.5. Do you think she is right? Why or why not?”

The instructional materials reviewed for Eureka Grade 6 meet expectations for explicitly attending to the specialized language of mathematics.

In each module, the instructional materials provide new or recently introduced mathematical terms that will be used throughout the module. The mathematical terms that are the focus of the module are highlighted for students throughout the lessons and are reiterated at the end of most lessons.

Each mathematical term that is introduced has an explanation, and some terms are supported with an example. The terminology that is used in the modules is consistent with the terms in the standards.

The materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams and symbols. For example:

• In Module 1, Lesson 1, teachers are prompted to “Encourage students to be precise about the order in which the quantities are stated (emphasizing that order matters) and about the quantities being compared.” An example is provided for the teacher stating, “That is, instead of saying the ratio of boys to girls, encourage them to say the ratio of the number of boys to the number of girls.”
• In Module 4, Topic D Overview, the instructional material provides explicit instruction in the use of a bar diagram to differentiate between the mathematical terms subtract and subtract from. The Topic D Overview states, “Students also use bar diagrams to differentiate between the mathematical terms subtract and subtract from. For instance, when subtracting b from a, they know they must first represent a in order to take away b, leading to an understanding that the expression must be written a - b.”

The materials use precise and accurate terminology and definitions when describing mathematics and support students in using them. For example:

• In Module 1, Lesson 1, the student materials use precise language when stating ratios between quantities. Problem Set Question 2b states, “What is the ratio of the number of milk cartons remaining to the number of milk cartons taken?”
• In Module 2, Lesson 2, students draw upon previously learned vocabulary in order to represent new vocabulary (multiplicative inverse.) Example 1 Question 1 states, “As you travel to each model, be sure to answer the following questions: How many ½ miles are in 12 miles? Corresponding division expression. Corresponding Multiplication Expression. Write an equation showing the equivalence of the two expressions.”
• In Module 4, Lesson 22, students use the precise language of using formulas in geometry. Example 3 states, “This box has a width, w. The height of the box, h, is twice the width. The length of the box, l, is three times the width. That is, the width, height, and length of a rectangular prism are in the ratio of 1:2:3. For rectangular solids like this, the volume is calculated by multiplying length times width times height. Follow the above example to calculate the volume of these rectangular solids, given the width, w.”