8th Grade - Gateway 2

Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings meeting the expectations for this indicator.

• 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 for understanding.
• Chapters 1, 3, 5, 6 and 7 all include work directly related to the clusters that address conceptual understanding (8.F.A, 8.EE.B, 8.G.A). Conceptual understanding is built through strategies such as:
  • Patterns (ex: exponent rules, Pythagorean);
  • Graphing/comparing transformations;
  • Multiple representations of Linear Equations; and
  • Algebra Tiles for solving and comparing equations.
• 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 give attention throughout the year to individual standards that set an expectation of procedural skill and fluency meeting the expectations for this indicator.

• There is evidence of the opportunity to develop fluency and procedural skills in every chapter, including the chapter assessments.
• Fluency is especially evident in the constantly spiraling homework. According to the publisher, about one-fifth of the homework is on new material and the rest of the homework review previous concepts.
• The skills are frequently embedded in an engaging activity such as the silent board game or the human graph.
• Standards that emphasize procedural skill and fluency are 8.EE.C.7, 8.EE.C.8b and 8.G.C.9, and they are evident in Chapters 2, 3, 5, 9 and 10. Procedural skill and fluency is developed through strategies such as:
  • Examples and repetition in practice;
  • Chapter closures have problems with solutions - if students miss them, they are directed back to the relevant lesson ("Need Help?") and to additional practice problems that align with what they missed ("More Practice");
  • Math Note boxes reinforce vocabulary and concrete examples (Angle Vocabulary, Line of Best Fit);
  • Learning logs (Pythagorean Theorem, Slope & Steepness);
  • Spiral homework; and
  • Checkpoint problems - with extra practice if not mastered (Solving Equations, Scatterplots & Associations).
• Students would benefit from having more opportunity to develop fluency and procedural skills in solving equations, including simultaneous linear equations. There were limited situations to practice the process before the knowledge was expected to be routinely applied.

The materials meet the expectation 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 Newton’s Revenge, the Line Factory and the TV Antenna.
• 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, 4, 5 and 9 all explicitly provide opportunity for students to engage in application with standards of Grade 8 (8.EE.C.8.C, 8.F.B) that specify application. Application is developed through non-routine problem solving such as:
  • Newton’s Revenge – could someone be too tall to ride the roller coaster (scatterplot, data collection/analysis and prediction);
  • Iditarod Trail Sled Dog Race – when did they meet, who traveled faster, how long was the race (systems);
  • Biking the Triathlon – interpreting lines on a graph – speed, distance, rate (slope, data analysis); and
  • Personal Trainer – collect/organize biking data (scatterplot, line of best fit).

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

• There are multiple lessons where two, or all three, of the aspects are interwoven.
  • Lesson 10.1.2 presents students with a task to provide input to a sports company designing a new bag given the constraint of using only a piece of fabric that is 40" x 52" and that will hold the most. Students are encouraged to build conceptual understanding through modeling with paper and practice fluency by calculating volume of different shapes/sizes, then applying their discoveries to the size given and make a recommendation with rationale. They're further encouraged to generalize their learning about the volume of cylinders versus rectangular prisms.
• 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.
  • Lesson 3.1.1 "What is the Rule?" is clearly focused on fluency and procedural skill because the lesson is entirely 12 in-and-out tables for students to complete and generalize a rule.
  • Lesson 7.1.2 "Is there a relationship?" is all application - organizing and analyzing data from car advertisements about odometer reading and cost.
• In addition, there is a balance of the three aspects of rigor included in assessments, all pre-made individual assessments contain questions on conceptual understanding, procedural skill and fluency, and application.

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 standards. 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 MPs 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. Examples include:
  • MP1 in 8.1.1 "Profit, Profit, Profit," 10.1.2 "Comparing Gym Bags," 10.2.2 "Fifty Nifty Necklaces."
  • MP2 in 5.1.1 "Changing Forms," 6.2.2 "Undoing Dilation," 7.2.2 "Biking the Triathlon," 9.2.2 "Part of Pythagorean Relationship."
  • MP3 in 4.1.1 "Tile Pattern Team Challenge/Presentation," 6.2.1 in group work on dilations and in 7.1.2 in determining line of best fit as a team.
  • MP4 in 5.2.2, which models saving money to purchase bicycles and how saving at different rates creates different equations and graphs; 7.1.3 experimenting with different factors that will affect plant growth and modeling this mathematically; and 9.2.5 with applications of Pythagorean theorem.
  • MP5 in 3.1.3, that says, “Be ready to defend your math position with all the math tools you have.” (i.e., students choose), and in 3.2.4 “Did you use algebra tiles to solve...Why or why not?” where again, students were given the choice.
  • MP6 in 3.1.4 "Precise Labeling," 8.2.4 which asks students to make an exact computation and in 9.2.1 when it says to "Be as specific as you can."
  • MP7 in 3.1.7 "Goofy Graphing," 4.1.1 "Tile Pattern Team Challenge," 6.1.3 "Describing Transformations."
  • MP8 in 8.2.3, that looks at structure of positive exponents to extend pattern to negative exponents, and 9.2.4 where students determine whether decimals repeat, terminate, or neither.
• 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.
• The core structure and components section of the Teacher's Guide defines each MP and provides a rationale of how the program addresses each math practice. On page 52 a chart identifies problem tasks that integrate multiple MPs.
• In the Teacher Guide the prep section for every lesson identifies the MP(s) and connects the MP(s) to the content of the lesson. For example, 4.1.2 states "Students continue reasoning and quantitatively while working with patterns. Today they makes the connection more explicit, looking for and making use of the structure of a linear equation."

The materials meet the expectation 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 being 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.
• Students construct viable arguments through activities such as explaining their thinking or justifying steps.
  • For example in 9.2.1, students are asked to the following: "Justify your conclusion. Explain your reasoning. How do you know? Do you agree? Is Cisco correct? Why or Why not? What was his mistake? Explain your choice."
  • In 10.1.3 students are asked to explain why Dan’s and Jan’s work is different but they have the same answer.
  • In 5.1.1, students are asked to justify how many feet a tree grew in a year.
  • In 2.1.5, students are asked "Why does it work? as they are developing a method to simplify both sides of the Expression Comparison Math.

The materials meet the expectation 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.
  • Teachers ask questions such as “Who agrees? Who disagrees? Why is there a disagreement?”
  • For example in 2.1.1, the Teacher's Guide states that "The goals of today’s lesson is for students to begin using algebra tiles as an appropriate tool. While using this tool, they will begin to look for and make use of the structure of algebraic notation as they combine like terms."
  • Also in 2.1.1, the Teacher's Guide prompts teachers throughout the lesson description to ask questions such as “What is different? How do we know? Why can I?” There are also prompts about having pairs check each other on different problems so that they have to explain their thinking and prompts asking students to justify solutions.
  • 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?"
• The course is designed for students to work in teams and have them collaborate and explain their thinking to each other.
• Teachers also are encouraged to assign tasks that require students to consistently engage in debate.

The materials meet the expectation 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 is bold 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.
• In some instances, the text is slow to introduce vocabulary such as “slope” – it is developed as “change” or “rate” in through many lessons before it’s called slope. Or starting with “flip, slide, turn” before transformations are labeled. It seems intentional that students have the concept before linking vocabulary to it.
• There is vocabulary that seems unique to CPM such as the “Equal Values Method” for solving systems, which is just a specific case of substitution and “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.