6th Grade - Gateway 2

Materials meet the expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

• Evidence for this indicator is found throughout all of the chapters, including the chapter assessments.
• Generally, lessons develop understanding through the group work that students complete in the lessons.
• There are extensive suggestions in the teacher guide for every lesson describing the purpose of the lesson and how to guide study teams to develop their understanding of a concept.
• Teacher questioning during instruction is designed to lead to conceptual understanding: "How do you see it? How can you tell it’s correct? What is the pattern? Is there a different way?”
• Students are consistently being asked to communicate with their group and explain their understanding.
• Chapters 2, 3, 4, 5, 6 and 7 all include work directly related to the clusters that address conceptual understanding (6.RP.A, 6.NS.A, 6.EE.A, 6.EE.B, 6.EE.C). Conceptual understanding is built through strategies such as:
  • Using rectangles to multiply and work with distribution.
  • Creating a "concept catcher" (page 90).
  • The "Giant 1" for the multiplicative identity.
  • Percent grids to "see" fractions, decimals, percents.
  • Frogs jumping on a number line for integers and absolute value.
  • Algebra tiles for combining like terms, variables, area, perimeter.
• The materials provide evidence of high-quality conceptual problems using concrete representation, algebra tiles, experimenting, verbalization, online activities/tools, multiple representations, and interpretation.
• Students are required to use previous learning to construct new learning.

Materials meet the expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade.

• There is evidence of the opportunity to work with engaging applications of the mathematics in every chapter, including the chapter assessments.
• There are multiple non-routine problems throughout the chapters, such as "Croakie the Talented Frog," "Which is Sweeter?" and "Shopping Shirley."
• Students are frequently presented with problems in real world situations that are relevant to them.
• Students must also apply their understanding through teaching others.
• Chapters 3, 5, 6, 7, 8 and 9 all explicitly provide opportunity for students to engage in application in the standards of Grade 6 (6.RP.A.3, 6.NS.A.1, 6.EE.B.7, 6.EE.C.9) that specify application. Examples include:
  • Birthday Bonanza - 18 million people on Earth share your birthdate; (ratios, proportions).
  • Dora's Dollhouse - 3/4' pieces from an 8' board; (division with fractions).
  • Memory Lane - taking a trip down memory lane; (integers).
  • Training for the Triathlon - swimming, biking, running; (comparing rates).

The materials meet the expectations for the three aspects of rigor not always treated together and not always treated separately. There is a balance of the three aspects of rigor within the grade.

• Conceptual and procedural knowledge as well as application of knowledge and skills are balanced throughout the course, including the chapter assessments.
• There are multiple lessons where two or all three, of the aspects are interwoven.
  • For example: Lesson 5.3.2 - finding area of a parallelogram - starts with an online tech tool called Area Decomposer where students can cut pieces of shapes to try to rearrange them into a rectangle exemplifies conceptual development which moves into fluency by having students work with paper/pencil to draw cuts on original figures and then draw what the final figure would look like, showing how the pieces move. Students then return to concept development by exploring measurements on parallelograms and rectangles to establish what represents the base and height, then to fluency practicing measuring for several parallelograms until they begin to generalize and are prompted to discover that the area of parallelograms is the same as rectangles, therefore just A=b*h.
• There are also multiple lessons where one aspect is the clear focus, which is almost equally split among all three aspects, with perhaps a slight emphasis on conceptual development.
  • For example: Lesson 2.2.3 - exploring relationships between area and perimeter and what happens if one changes using base 10 blocks is a very conceptual lesson.
  • For example: Lesson 7.1.1 on comparing rates is very application based. Students are given a problem-solving situation and data about fundraising for a field trip which they have to analyze to make a recommendation. This is followed by multiple extensions which require further analysis with ratio comparisons, though students are never given that as a solution path - they have to figure out how to best compare options.

The materials meet the expectations for the MPs being identified and used to enrich mathematics content within and throughout each applicable grade.

• There is a clear articulation of connection between MPs and content. Materials regularly and meaningfully connect MPs throughout the lessons.
• There is a chart in the teacher’s guide that aligns the MPs with the course, including an in-depth explanation of how they are “deeply woven into daily lessons.”
• Every unit identifies the MPs used in the teacher chapter overview page.
• In the Teacher's Guide, each unit specifically relates how the listed standards are used in the unit and for each lesson. These are logical connections and integrated with the content.
• Teachers are reminded to encourage the use of MPs in team discussions even if they aren’t identified.
• Most lessons incorporate multiple practice standards as students have the opportunity to deeply engage with authentic mathematics of the grade.
• All eight MPs are represented throughout the course.
• Connections are NOT made in the student materials until the end of the book in the End-of-Course Reflection when students are asked to discuss/reflect on the entire course about them. The questions and problems in this section clearly facilitate students understanding and making connections to the MPs, though there is concern that the reflection could easily be skipped by teachers if instructional time for the regular lessons runs short.

The materials meet the expectations for attending to the full meaning of each practice standard.

• Each practice is addressed multiple times throughout the year; over the course of the year, students have ample opportunity to engage with the full meaning of every MP. For example:
  • MP1 - 4.2.1 Mystery Mascot, 9.2.3 Shopping Shirley.
  • MP2 - 3.1.2 Portions as Percents, 7.1.1 Comparing Rates.
  • MP3 - 9.3.3 How is it Changing?, 6.1.1 Fair Shares.
  • MP4 - 2.3.1 Using Rectangles to Multiply, 5.1.3 Describing Parts of Parts.
  • MP5 - 2.2.1 Exploring Area, 6.2.3 Perimeters of Algebra Tiles.
  • MP6 - 9.3.1 Trail to the Treasure of Tragon, 4.2.1 Enlarging Shapes.
  • MP7 - 9.3.2 How Does it Grow?, 5.3.2 Area of a Parallelogram.
  • MP8 - 4.1.3 Using Variables to Generalize, 7.2.2 Another Division Strategy.
• MPs are embedded in lessons, assessments, mid-year and end of year reflection, and Puzzle Investigator Problems.

There are clear definitions for all the practices as well as where they are addressed in the curriculum.

• There is a section in the Teacher's Guide under Core Structure and Components that:
  
  • Defines each MP and provides a rationale about how this program addresses that practice overall.
  • Shows a chart on page 54 that identifies the problem tasks that integrate multiple MPs.

• The prep section for every lesson in the Teacher's Guide identifies the MP for the lesson and how it's related, for example:
  
  • 2.1.1: "This lesson is an opportunity for the students to engage with mathematics through a variety of graphical representations. This is their first opportunity to focus on models. Students, in discussing these models, identify important quantities and map relationships. They will learn about whether each model makes sense and has served its purpose."

The materials meet the expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards.

• Students are consistently asked to verify their work, find mistakes, and look for patterns or similarities.
• The materials have questions built throughout every lesson to encourage students to construct viable arguments and critique each other’s reasoning with heavy emphasis on group work.
  • Examples include: What can we compare? How else can we represent it? How are they related? Is there another way to see it? How is it the same (or different)? How can you show it?
• Students construct viable arguments through activities such as explaining their thinking or justifying steps.
  • For example in a single lesson (7.1.2), students are asked to: Justify your answer (7-15c). Explain your reasoning (7-16a and 7-19c). How do you know? (7-17b).

The materials meet the expectations for assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards.

• Teachers are encouraged throughout the teacher’s guide to ask students questions, such as “Who agrees? Who disagrees? Why is there a disagreement? Explain.” This would create an environment in the mathematics classroom that is rich with constructing viable arguments and critiquing the reasoning of others.
• Teachers are encouraged to assign tasks that require students to consistently engage in debate. The course is designed for students to work in teams and have them collaborate and explain their thinking to each other.
• The teacher guide links 1-2 MPs to each lesson –
  • For example: 8.1.5 – The goals of today’s lesson is for students to construct three different representations of a single set of data and decide with representation is most useful.
  • Continuing with 8.1.5, the Teacher's Guide prompts teachers to structure this lesson as a Participation Quiz to help focus students on explaining their reasoning and justifying their choice in data.
  • This is true for every lesson – any page you flip to includes question prompts like, “What information do you need? How can you check? What does this mean? Help me understand how? Why did you? Did anyone else?”
• Teachers also are encouraged to assign tasks that require students to consistently engage in debate.
• The course is designed for students to work in teams and have them collaborate and explain their thinking to each other.

The materials meet the expectations for explicitly attending to the specialized language of mathematics.

• Each chapter ends with a vocabulary list of words used in the unit that includes words from previous learning as well as new terms. Students are referred to the glossary and it is suggested that they record unfamiliar words in the Learning Log.
• Each chapter includes a resource page of Concept Map cards with the vocabulary of the chapter.
• Throughout the unit, these terms are used in context during instruction, practice, and assessment.
• Vocabulary terms are bolded in the context of the lesson, then pulled out specifically in “Math Notes” sections in each chapter.
• There are suggestions like “Encourage students to use appropriate vocabulary, referencing the word wall when necessary.” Sometimes they even list specific words that should be included.
• The text makes connections between mathematics terms such as “mean” and “arithmetic average.” The students are required to learn the correct mathematics terminology with support for how they might hear it outside the mathematics classroom.
• There is vocabulary that seems unique to CPM such as the “Giant 1” or “fraction busting” or “the 5-D process” as strategies.
• The terminology that is used in the course is consistent with the terms in the standards.