6th Grade - Gateway 3

The instructional materials for Match Fishtank Grade 6 meet the expectations that there is a clear distinction between problems and exercises in the materials.

There are eight units in each grade level. Each unit presents lessons in a consistent structure. During the Anchor Problems, which include guided instruction, step-by step procedures, and problem solving, students work on examples and problems to learn new concepts. Problem Set Guidance contains a variety of exercises that allow students to independently master, and demonstrate their understanding of the material. These can include problems from Open Up Resources Grade 6-8 Mathematics, Open Middle, MARS Formative Assessment Lessons, Robert Kaplinsky, Yummy Math, EngageNY - Great Minds, and others. Each lesson concludes with a Target Task intended for formative assessment. For Example: 

• Unit 1, Understanding and Representing Ratios, Lesson 1, Anchor Problem 1, students describe the association between two groups of shapes using ratios. In the Problem Set Guidance (Open Up Resources Grade 6 Unit 2 Practice Problems, Lesson 1, Problems 1-3), students are given additional practice with this skill. In Problem 3, they are shown a picture of two animals: “Write two different sentences that use ratios to describe the number of eyes and legs in this picture.” 
• Unit 2, Unit Rates and Percent, Lesson 3, Anchor Problem 3, students find unit rates and use them to solve problems.  An example is as follows: “Emeline can type 2 pages in 8 minutes. What are two rates associated with this ratio? Emeline has a deadline in 18 minutes, at which point she needs to be done typing an article. What is the greatest number of pages that Emeline can type before her deadline? Emeline has to type a 7-page article. How much time will it take her?"  The Guiding Questions for teachers help to lead the student through the problem-solving process. 
• Unit 3, Multi-Digit and Fraction Computation, Lesson 3, Anchor Problem 1, students use a visual model to solve a problem that involves dividing a whole number by a fraction. In the Problem Set Guidance, (Illustrative Mathematics Dividing by One-Half ), students are given additional practice with that skill.  An example is as follows: “Solve the four problems below. Which of the following problems can be solved by finding 3÷1/2?” 
• Unit 4, Rational Numbers, Lesson 4, Anchor Problem 1, students define and label opposite numbers on a number line through direct instruction and guiding questions.  
• Unit 7, Geometry, Lesson 2, Anchor Problems, students  find the area of right triangles, and are introduced to the general formula for the area of a triangle. Students continue to practice this skill in the Problem Set Guidance. 
• Unit 8, Statistics, Lesson 3, Anchor Problem 3, students create histograms from tally charts to represent and interpret a data set. In the Problem Set Guidance: (EngageNY Mathematics Grade 6 Mathematics, Module 6, Topic A, Lesson 4, Exit Ticket) additional problems allow for independent practice and mastery,

The instructional materials reviewed for Match Fishtank Grade 6 meet the expectations that the design of assignments is intentional and not haphazard. 

The lessons follow a logical, consistent format that intentionally sequence assignments,  provide for a natural progression, and lead to full understanding for students. For example:

• In the Anchor Problems, students are introduced to concepts and procedures through a problem-based situation. They are guided through the problem solving process via a series of Guiding Questions provided for teachers.
• In the Problem Set Guidance: This portion of instruction connects to the problem learned previously and is the substance of the lesson. It includes a list of suggested resources (including links to resources) or problem types for teachers to create a problem set aligned to the objective of the lesson.  Teachers are encouraged to create a set of problems that best work for the needs of their students or for that particular lesson. 
• Each lesson concludes with an independent Target Task designed to cover key concepts from the lesson and formatively assess student understanding and mastery.

The instructional materials for Match Fishtank Grade 6 meet the expectations that the instructional materials prompt students to produce content in a variety of ways. For example:

• Unit 1, Understanding and Representing Ratios, Lesson 2: Students use double number lines to represent ratios and identify equivalent ratios.
• Unit 3, Multi-Digit and Fraction Computation, Lesson 2: Students use visual models to divide whole numbers by fractions.
• Unit 4, Rational Numbers, Lesson 6: Students use a number line to explain the order of rational numbers.
• Unit 6, Equations and Inequalities, Lesson 1: Students use tape diagrams and balances to represent and solve equations.

The instructional materials reviewed for Match Fishtank Grade 6 meet expectations for having manipulatives that are faithful representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

The series includes a variety of suggestions for physical manipulatives, and links to physical, as virtual manipulatives, for example: 

• Unit 1, Understanding and Representing Ratios, Lesson 1, Anchor Problem 1 uses a ratio shapes handout allowing students to cut out and sort/group shapes as they describe ratios of groups of objects. 
• Unit 3, Multi-Digit Fraction Computation, Lesson 14, Anchor Problem 3  and Problem Set Guidance both use domino cards and a “Factor Game" to teach and reinforce prime factorization. The game is connected to written methods as students transition to writing numbers as a product of prime factors.
• Unit 8, Statistics, Lesson 3, Problem Set Guidance, links to StatKey Descriptive Statistics for One Quantitative Variable to create virtual histograms from data sets.

The instructional materials for Match Fishtank Grade 6 meet the expectations that materials provide teachers with quality questions for students.

In the materials, facilitator notes for each lesson include questions that are provided in Anchor Problem notes for the teacher to guide students' mathematical development and to elicit students' understanding. The materials indicate that questions provided are intended to provoke thinking and provide facilitation through the mathematical practices, as well as, influencing the students to think through their work. For example:

• Unit 2, Unit Rates and Percent, Lesson 4, Anchor Problem and Guiding Questions 2: “Are you able to compare the two options just using the price? Why or why not? Are you able to compare the two options just using the pounds of honey? Why or why not? Why would the unit rate of cost per pound be a useful tool to compare? How can you find the unit rate for each option? What does each unit rate tell you about the options?”
• Unit 4, Rational Numbers, Lesson 3, Anchor Problem and Guiding Questions 2: “What does zero feet mean in this situation? Did the two friends travel in the same or different directions? How do you know? Sketch a number line to represent the elevations of the friends.”
• Unit 7, Geometry, Lesson 12. Anchor Problem 1 and Guiding Questions : “What is the absolute value of each x−coordinate? Of each y−coordinate? What does it mean if two points are symmetrical about an axis? How far is each point from the axis? Would the points (−2,5) and (2,5) line up vertically or horizontally? Which axis would you fold along to match (−2,5) with (2,5)?”
• Unit 8, Statistics, Lesson 4, Anchor Problem 1 and Guiding Questions: “How are the first two graphs similar? How are they different from the third graph? Which graph would you describe as symmetrical? Why? What features make it symmetrical? A skewed distribution has values that are not typical of the rest of the data. These skewed data points can be on the low or the high end. Which graph would you say is skewed left (is skewed toward the smaller values or has a “tail” to the left)? Which graph would you say is skewed right (is skewed toward the larger values or has a “tail” to the right)?”

The instructional materials reviewed for Match Fishtank Grade 6 meet the expectations for containing a Teacher Edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Where applicable, materials also include teacher guidance on the use of embedded technology to support and enhance student learning. 

Tips for Teachers section is present in all lessons, and supports teachers with resources, and an overview of the lesson. These Tips for Teachers sections provide some guidance on how to present the content. For example:

• Unit 3, Multi-Digit Fractions and Computation, Lesson 4, Tips of Teachers: In this lesson, students use visual models and patterns to develop the general rule for dividing by fractions. The focus of this lesson is on the development of the rule rather than on the use of it. Students will have opportunities for a lot of practice in upcoming lessons. 

There is guidance for teachers in the form of Guiding Questions and Notes to use when implementing Anchor Problems, Problem Set Guidance, and Target Tasks. Problem Set Guidance sections are optional problem sets within student-facing materials. There is no student edition, so guidance for ancillary materials is not needed.

The instructional materials for Match Fishtank Grade 6 meet expectations as they explain the role of the grade-level mathematics in the context of the overall mathematics curriculum.

• Each unit opens with a Unit Summary that includes a Lesson Map and a list of grade-level standards addressed within the unit along with future connections.
• Each Lesson provides current standards addressed in the lesson.
• Each Lesson provides foundational standards which are standards that were covered in previous units or grades that are important background for the current lesson.

The instructional materials for Match Fishtank Grade 6 meet the expectations that materials provide strategies for teachers to identify and address common student errors and misconceptions. 

The materials provide notes for teacher guidance for the Anchor Problems that addresses common misconceptions. For example:

• Unit 3, Multi-Digit and Fractional Computation, Lesson 3, Anchor Problem 3 states “There are often common misconceptions around how to treat any remaining fractions of the wholes. In this case, we can count four 2/3 cups in the visual, and we see that there is one third of a cup left over. However, we are counting in servings of 2/3 cup, not 1 cup, so the remaining piece represents 1/2 of a 2/3 cup serving. If students struggle with this concept, there is an additional problem from Illustrative Mathematics’ “Cup of Rice” referenced in the Problem Set Guidance that addresses this further.”
• Unit 6, Equations and Inequalities, Lesson 5, Anchor Problem 2: “A common misconception with problems involving division is for students to divide to find the variable instead of multiply, especially in a problem like the one above where 10 is divisible by 5. Use the diagram to visually represent x as the value that is being divided into 5 pieces, where each piece is equal to 10. Substituting the value of 2 into the equation for x is another way to demonstrate it as an incorrect answer.” 

The instructional materials for Match Fishtank Grade 6 meet the expectations for the materials to provide opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills. 

Each lesson is designed with teacher-led Anchor Problems, Problem Set Guidance and Target Tasks. The lessons contain multiple opportunities for practice with an assortment of problems. The Anchor Problems provided the teacher with guiding questions and notes in order to provide feedback for students’ learning. For example:

• Unit 2, Unit Rates and Percent, Lesson 11, Anchor Problem 2 , Notes include the following: “This is also a great opportunity to engage students’ number sense skills. Discuss the different ways that students can solve 22/100×150. For example, 22/100 can be rewritten as 11/50 or as 0.22. There is further simplification that can happen with the 150 in the numerator and 100 or 50 in the denominator to simplify the problem a great deal.”
• Unit 5, Numerical and Algebraic Expressions, Lesson 3, Anchor Problem 2 is as follows:  Students write expressions to match several situations. The guiding questions for teachers ask students to use their expressions to solve problems. In Notes, the following is included: “Use this Anchor Problem to introduce students to simple expressions that involve variables to represent quantities (MP.2). If students get stuck, have them try out different values for the variables to see how the quantities interact with each other.”

The instructional materials for Match Fishtank Grade 6 meet the expectations for the assessments to clearly denote which standards are being emphasized.

Each unit provides an answer key for the Unit Assessment. The answer key provides each item number and the targeted standard. For example: 

• Unit 8, Statistics, Assessment Item 5 correlates with 6.SP.2. 
• Unit 3, Multi-Digit and Fraction Computation, Assessment Item 6 correlates with 6.NS.3.

The instructional materials reviewed for Match Fishtank, Grade 6 meet the expectation that materials embed tasks with multiple entry­ points that can be solved using a variety of solution strategies or representations.

Students engage in tasks throughout lessons in the Anchor Problems, Target Tasks, Problem Set Activities, the 3-Act Math Modeling, and the Mathematics Assessment Project activities. They all present multiple entry points for students. For example:

• Unit 1, Understanding and Representing Ratios, Lesson 9, Target Task is as follows : “Fix the Egg Mixup: What I did: 2 eggs, 4 tablespoons flour; What I should have done: 2 eggs, 3 tablespoons flour.” Presenting this scenario without posing a question allows multiple entry points and varied strategies for students.

The instructional materials for Match Fishtank Grade 6 meet the expectations that materials provide a balanced portrayal of various demographic and personal characteristics. For example:

• Different cultural names and situations are represented in the materials, ie., Sorah, John, Alberto, Beth, and Pedro.
• The materials avoid pronouns, referencing a role instead, ie., the banker, a biologist, a scientist, the soccer team.