8th Grade - Gateway 2

The instructional materials for Grade 8 partially meet the expectations to develop conceptual understanding of key mathematical concepts, especially when called for in specific content standards or cluster headings. Overall, the instructional materials present inquiry labs and some visual examples as a way to develop conceptual understanding. However, the materials lack a fully developed conceptual understanding in some areas that are called for in the common core standards.

• Conceptual understanding is called for in 8.EE.B. This standard is covered in chapter 3.
  • Lesson 1 covers linear proportional relationships and rate of change. The examples offer many visuals in graph and tables and the guided practice encourages students to explain if a relationship is linear of not.
  • Lesson 2 covers slope. The examples offer only one problem for students to get a visual example of what slope means. (Example 1 shows a treadmill and explains rise over run.) There is one example where students count rise over run on a graph, and one example where students find the change in yand change in x in a table. After those three examples, students are expected to use the slope formula. There is a lack of problems intended for students to form their own meaning of slope.
  • There is an inquiry lab that calls for students to use similar triangles to explain why the slope is the same between any two points, but there are only two problems for students to practice this. Also, the inquiry lab comes after students have already practiced slope using the formula.
• Conceptual understanding is called for in 8.F.A and this topic is covered in chapter 4.
  • Lesson 4, 5 and 6 offer some concrete examples where students will develop an understanding of functions by looking at graphs, tables, and equations.
  • The inquiry lab and lesson 3 gives a formal definition of functions, but the lessons fail to give students concrete examples, where one can find the value of one thing when another is changing. Rather than giving concrete examples, the materials quickly jump to abstract explanations of domain and range in lesson 2 and function notation in lesson 3.

• Conceptual understanding is called for in 8.G.1 and this topic is covered in chapters 6 and 7.
  • The inquiry labs in these two chapters help to give students a conceptual understanding of transformations and similarity by using hands on activities.
    • The first inquiry lab in chapter 6 begins by having students physically move objects to understand transformations.
    • The second inquiry lab in chapter 6 has students use tracing paper to understand rotational symmetry.
    • Another inquiry lab in chapter 6 uses a ruler to connect scale factor and dilatation.
    • The first inquiry lab in chapter 7 has students use patty paper to see congruence of triangles.
    • Another inquiry lab in chapter 7 shows students how to use geometer's sketchpad in a way that helps them understand congruence and similarity.

The instructional materials for Grade 8 partially meet the expectations that teachers and students spend sufficient time working with engaging applications of the mathematics without losing focus on the major work of each grade. Overall, the materials have multiple opportunities for application but in many of those application problems students are directed on the procedure to follow to solve the problem.

• Application problems are called for in 8.F.B and this topic is covered primarily in chapter 4.
  • The lessons on 8.F.B give representations of relationships and functions with graphs, equations, and tables. Lesson 5 has students compare functions when multiple representations are given
  • Even though there are many application problems, the materials rarely provide students with opportunities to pick their own strategy for solving the problem. They are usually guided using one given strategy for each step of the problem.
  • The Power Up performance tasks at the end of each lesson offer students multi-step abstract questions where they solve problems by using a variety of solution paths.
  • There is a 21st Century career lesson that explores how physical therapist use functions to do their jobs.
  • The unit project has students work collaboratively to find the cost of growing a vegetable garden and project the profits. In doing this, students use equations and functions to complete the project.
• Application problems are called for in 8.EE.C.8.C. This topic is covered in chapter 3, lessons 6, 7 and 8.
  • Lesson 6 explains how to write an equation in point slope form and slope intercept form. Lesson 7 explains how to solve systems of equations by graphing. Lesson 8 explains how to solve systems of equations algebraically. All three lessons have some application problems, but the application problems are often one-step, routine problems where students are told how to proceed in a step-by-step manner.

The instructional materials reviewed for Grade 8 partially meet the expectation for identifying and using MPs. Overall, the materials clearly identify the MPs and incorporate them into the lessons, however the MPs are sometimes over-identified.

• The MPs are incorporated into each lesson so they are used to enrich the content and they are not taught as a separate lesson.
• There is a Mathematical Practice Handbook at the start of the textbook. This handbook explains each practice standard and gives example problems for each standard.
• There is a table of contents that specifically addresses the MPs and it lists the pages where you could find each of the practices. All of the MPs are represented.
• Each lesson identifies several MPs. For example, chapter 1, lesson 1 states that it incorporates MP1, 3, 4, 6, 7 and 8. The materials point to these MPs in the student practice section of lesson 1.
• Items are sometimes over identified. In the sidebar of the teacher edition, teaching strategies are suggested. Often those strategies are identified as attending to multiple strategies. For example, in the "Pairs Discussion" in chapter 5, lesson 1, students work in pairs to complete a graphic organizer. Then they share and revise their responses. This activity claims to incorporate MP1, 2, 3, 4, 5 and 6. However, there is no explanation or description as to how these practices are incorporated.

The materials reviewed for Grade 8 partially meet the expectations for appropriately prompting students to construct viable arguments and analyze the arguments of others. Overall, there are problem structures that lead a student to explain and justify their reasoning. However, there are few opportunities for students to analyze the arguments of others.

• In the practice problems nearly every lesson includes questions that are specifically labeled with the heading "Justify Conclusions." These questions ask students to explain how they got their answers.
• In some lessons, the questions are labeled in bold with the heading "Construct a Viable Argument." These questions often ask students to explain if something is true or not.
• Even though there are several prompts that ask the students to justify their answers or construct their own viable arguments, there are some missed opportunities for higher frequencies of this type of problem. Some of these missed opportunities include:
  • Chapter 4, lesson 5, question 8. The question poses a real-world application problem with exchange rates for pounds and euros. Next, the students are asked to analyze four statements and decide which one is true, but the students are not asked to explain their decisions.
  • Chapter 5, lesson 3, question 27. The text asks students to apply what they know about angles and lines to find the values of two missing angles in a given picture of an obtuse triangle. The students were never prompted to explain their reasoning or construct a viable argument on how they found the missing angles.
• In some lessons the questions are labeled in bold with the heading "Find the Error." In these classic error analysis problems students are presented with someone's solution and asked to simply identify the error. This does not attend to the full meaning of the standard, where students would need to refute claims made by others by offering counter examples and counterarguments. There were very few instances where students were asked to find a counter-example.

The materials reviewed for Grade 8 partially meet the expectation of assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others. Overall, the materials direct teachers with many scaffolding questioning strategies asking higher level questions and offering some suggested activities that lead students to construct viable arguments and analyze the arguments of others. However, the materials lack suggestions or ideas that guide a teacher with setting up scenarios where students experiment with mathematics and based on those experiments construct and present ideas.

• In the sidebar of the teachers edition, the teacher is provided with many scaffolding questions. The beyond level questions do a great job of asking higher depth of knowledge level questions and provide supportive structures to analyze student arguments.
• In the sidebar of the teachers edition, there are suggested activities for teachers to use with students. Very often these suggested activities have students compare, critique, and analyze answers. For example, the "Pairs Check" in chapter 1, lesson 3, students work in pairs to complete a worksheet. Then they trade their solutions with another pair of students and discuss the differences.
• The Higher Order Thinking Problems in the student practice section of the materials incorporate some of the MPs that help students to construct viable arguments and analyze the arguments of others. Students are given opportunities to be persistent in their problem-solving, to express their reasoning, and apply mathematics to real-world situations. However, further guidance on how to promote this and support students in the development of these skills is not given. This is coupled with the fact that many students are rarely given authentic opportunities to develop the true intent of any of the MPs mentioned above.