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.

• Generally, lessons develop understanding through explicit discussion outlined in the teacher lessons. Conceptual understanding is evident throughout the majority of lessons and lesson plans of teacher instruction. Students are consistently being asked to verify their work and explain for understanding. Teacher questioning during instruction is designed to lead to conceptual understanding.
• Sentence starters often include terms like introduce, discuss, review, demonstrate, compare, explain, challenge, etc.
• 22 of the 48 lessons include significant conceptual development of ideas.
• Units 2 - 4, 7 - 9 and 12-14 all include work related to the major work clusters that address conceptual understanding (8.F.A, 8.EE.B, 8.G.A).
• The materials provide evidence of high-quality conceptual problems, such as discovering patterns, that lead to rules, using concrete representation, verbalization, multiple representations, and interpretation of models. Some examples include:
  • Balance scales for comparing expressions (unit 2)
• • Cups and Counters (similar to algebra tiles) for equations (units 5, 6)
  • Connections between picture, table, graph, rule, slope-intercept, scatterplots (units 3, 4, 5, 7, 8, 10)
  • Looking at an unlabeled graph and describing a situation that would create it as well as generating a table that would work (unit 7)
  • Finding patterns to generalize exponent rules (unit 11)
  • Exploring Pythagorean Theorem with grid paper (unit 12)
• However, it needs to be noted that beyond the lessons, most of the units did not call for students to demonstrate conceptual understanding on the summative assessments.

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 abundant evidence of the opportunity to develop fluency and procedural skills. Examples include:
  • Every packet starts with a warm-up that is a review of basic background skills.
  • In every lesson, students have a practice section that includes ample "naked" practice such as the integer problems in 1.1 where there are 35 computation problems involving all 4 operations or 6.2 with 10 solving equations problems.
  • Besides an abundance of practice throughout the lessons, there are also Skill Builder activities in each unit designed to develop procedural skill and lead to fluency.
  • There is also a Knowledge Check at the end of each unit that reviews the skills learned.
  • The last thing in each lesson is a Home-School Connection where students have a page of problems to take home and do with their family that also reviews the skills in the unit.
• Procedural skill and fluency that develop the major clusters that emphasize it (8.EE.C.7, 8.EE.C.8.B , 8.G.C.9) is predominantly evident in 5 of the 16 units.
• In the teacher’s guide, there are often multiple "Introduce, Explore/Summarize, and Practice” sections depending on lesson content that develop procedural skill and fluency.

The MPs are 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 to the CCSSM within and throughout the grade.
• Every unit identifies the MP used in the unit both on the student and teacher overview page.
• In the teacher guide, each unit specifically relates how the listed standards are used in the unit. These are logical connections and integrated with the content.
• The MPs have also been identified for the quizzes, proficiency challenges, tests, and tasks.
• Compared to the others, MPs 4 and 8 are relatively underrepresented.

Materials attend to the full meaning of each practice standard.

• Materials attend to the full meaning of each practice standard, though opportunities are limited.
• Each practice is addressed multiple times throughout the year, though modeling and repeated reasoning are relatively under-represented compared to the others.
• There are opportunities to engage in every mathematics practice fully at least a couple of times during the year. For example:
  • “Reason abstractly and quantitatively” as well as “Look for and express regularity in repeated reasoning” has students developing integer rules through discovering patterns.
  • Or “Make sense and persevere” when students must make sense of a money saving problem and need to identify relevant information and extend a simpler problem from previous work.

Materials assist teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards.

• Materials assist teachers in engaging students in constructing viable arguments and analyzing the arguments of others.
• The teacher guide encourages the teacher to put students in pairs/groups and have them explain their thinking to each other (Lesson 1.1).
• The teacher guide encourages the teacher to ask the students to explain their thinking orally and in writing (Lesson 2.2).
• Suggested questions are provided for students to explain their thinking in the lesson summary (Lesson 4.1).
• Suggested questions for introducing the lesson to relate previous learning (Lesson 5.1).
• There were some instances where this practice was connected/described in the teacher guide, but not carried through in the lesson (Units 10 and 15).

Materials explicitly attend to the specialized language of mathematics.

• Materials very explicitly attend to the specialized language of mathematics.
• Correct mathematical terminology is consistently used, enforced, and reinforced.
• Explicit detail is always used in student-teacher discussion and explanation of process
• Each unit starts with a vocabulary list of words used in the unit and students have a “resource guide” to refer to. Throughout the unit, these terms are used in context during instruction, practice, and assessment.
• Teacher notes include hints such as “avoid sloppy language” such as the negative number is bigger rather than the negative has a greater absolute value. Or attending to differences such as expression/equation, solve/simplify. This is evident and strong throughout each unit.
• The terminology that is used in the modules is consistent with the terms in the standards.