8th Grade - Gateway 2

The instructional Materials for Course 3 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 real-world situations and multiple 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.

• In general, the activities were procedural in nature and did not enhance the student's ability to form a conceptual understanding of major work within the grade. There is a lack of concrete and/or visual representations when developing conceptual understanding and more of a reliance on algorithmic understanding. Students have ample opportunities for independent practice but it is not specifically indicated in the textbook if problems would be best completed within a group to encourage discussion and the ability to have to justify one’s answers. Looking at standards 8.EE.B, 8.F.A and 8.G.A, there are minimal opportunities for hands-on activities.

Each Unit contains activities and then each activity contains lessons. Within these there are essential questions at the beginning of every unit that a teacher could continue to refer back to in order to assist in developing conceptual understanding, and there is a "Teach" section on the wrap around teacher addition that has question numbers and is labeled 'Use Manipulatives, Visualization, Predict and Confirm, Think-Pair-Share, Look for a Pattern, Discussion Groups'. However, there ar no specific questions or statements that will guide a teacher towards building conceptual understanding in these sections and those suggestions are never fully developed or explained.

Examples where the material specifically relate to conceptual understanding and meet the requirements asked for in the Evidence Collection to support a partial rating include:

• 8.EE.B is taught in Unit 2, specifically activities 11-13 and is about understanding connections between proportional relationships, lines, and linear equations. .
  • Activity 11 builds on the concept of slope as the rate of change by creating tables, writing linear equations and plotting a linear graph. Also, by graphing proportional relationships, the slope and y-intercept are determined.
  • Continued development and practice can be found in Activities 12 and 13.
  • However, activity 11, page 136 problem 12 is the only opportunity for students to use similar triangles to explain why the slope is the same between any two distinct points.

• 8.F.A is taught in Unit 4, specifically activities 27-29 and is the introduction to functions.
  • Activity 27
    • Students are given a specific way, written in steps, to write ordered pairs
    • Problem 14 on page 261 asks students to justify their work which requires a higher depth of knowledge.
    • There are many problems that require the students to explain their thinking which can lead to having to defend their answers.
  • Activity 28
    • In this activity, there is a matching portion that can easily be completed in a group setting. This activity would really help students to understand tables, functions, and graphs because they have to match the three.

• 8.G.A is taught in Unit 3 and focuses on angles, transformations, reflections, rotations, and similar figures
  • There is ample practice with angles interspersed throughout the unit
  • This unit provides many opportunities to justify their argument/answer.

The instructional materials for Course 3 partially meet the expectation 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. Overall, the materials have multiple opportunities for application, but in many of the application-based activity problems, the activities are scaffolded in a manner that leads students to the desired outcome and contexts are often very similar, with few of them being non-routine.

• Materials for the major work of the grade did not adequately address the application portion of rigor. Many of the practice problems were procedural in nature and were not designed so that teachers and students spend time working with engaging applications, and did not require students to access their prior knowledge and think critically about the mathematics.
• Many activities have real-world application type problems included in them. However, most of these end up layered in a manner that the student is following given steps to lead them to the outcome wanted.

Examples where the material does not fully develop the student in engaging application of the mathematics include:

• 8.F.B - activity 11, lesson 11-1, page 13. The launching activity was layered in a way that significantly reduces the rigor of the problem. Application problems should be written in a manner that encourages multiple approaches. These multiple approaches will encourage students to think critically.
• 8.EE.C.8.C - activity 15, lesson 15-2, page 194. The subsequent activities do not lend to what one would consider being an application of mathematical concepts. The practice problems for this lesson were at a basic level and do not enhance the critical thinking skills of students. An application problem should encourage students to view the situation from multiple access points and students should be able to solve it using various methods.

The instructional materials reviewed for Course 3 partially meet the expectation that the materials balance all three aspects of rigor with the three aspects not always combined together nor are they always separate. Overall, the majority of the lessons focus on procedural skills and fluency with very few opportunities for students to discover and apply procedures for themselves.

The three aspects of rigor are not always treated together and are not always treated separately, meeting the expectations for this indicator. However, there is not a balance of the three aspects of rigor within the grade.

• All three aspects of rigor are present in the program material, but there is some under-emphasis on the conceptual understanding and application parts. Each leg of rigor was not adequately addressed in these materials because conceptual understanding was not enhanced during the early parts of each unit, many of the activities were procedural in nature and the application leg of rigor did not adequately challenge the students nor did it contain situations that encourage students to think critically about the mathematics.
• There isn't enough opportunities for students to make their own connections. Occasionally, they will ask students to make a reflection, but a majority of the lessons require memorized tasks and procedures without meaningful connections. There are several missed opportunities to challenge students to explore their own strategies and create opportunities for multiple solution pathways.
• The materials provide mostly procedural skills and this is the strongest aspect of the indicator. The application type problems are scaffolded in such a way to be more procedural in nature, however their use of real world helps with the balance between procedural skill and application.
• There is very little opportunity for the students to dig deep into the standards with application problems and the lack of opportunity for students to engage in applications and deep problem-solving with these real world situations was significantly noticeable.
• There were many missed opportunities to build from the fluency/procedural problems to move to having the students apply their knowledge.

The Instructional materials reviewed for Course 3 partially meet the expectation for identifying and using MPs to enrich mathematics content within and throughout each grade. While each practice is represented in this book, they are not often used in a way that would promote or enrich the mathematics content and are over identified in most of the units.

• All 8 MPs are evident throughout the materials, however, it was confusing when in some of the problems the MPs are bolded but there is no evident reason why they are bolded and others are not.
• The MPs could be used to enrich the mathematical content, however, there is no guide in the teachers edition as to how to use the problems to enrich the MPs.
• The only guidance that is given to teachers is found on page xii, with a paragraph discussing the integration of the practices.
• The MPs used are only found in the student text and are embedded within the questions asked in each lesson.
• As a whole, the MPs were identified the least in unit 5 with only being addressed 32 times. They were identified the most in unit 4 with being addressed 79 times.
• The summative assessments do test the MPs specifically and most often with above grade-level standards.

Examples where the material does meet the expectation for identifying and using MPs to enrich mathematics content include:

• Unit 1 - activity 4, page 56, question 7 asks students to "Critique the reasoning of others" and then poses a situation where a student incorrectly determines a calculator display.
• Unit 1 - activity 5, page 65, question 4 asks students to "Reason quantitatively" and asks students how to compare without using rational number approximations.
• Unit 3 - activity 18, page 241, question 1c asks student to "Reason abstractly" and then has students describe in their own words why the origin is the center of rotation for a specific rotation transformation.

Examples where the material does not meet the expectation for identifying and using MPs to enrich mathematics content include:

• Unit 1 - activity 5, page 61, question 4 asks student to "Use appropriate tools strategically" and then tells them to use a calculator to determine the values of the square roots, rounding to the nearest tenth.
• Unit 2 - activity 11, page 139, question 24 asks student to "Attend to precision" and then has students calculate the amount of money a student would earn in 52 weeks.
• Unit 3 - activity 21, page 281, question 3 asks student to "Use appropriate tools strategically" and then tells them to use a ruler to draw a dilated image.

The materials reviewed for Course 3 partially meet the expectation for appropriately prompting students to construct viable arguments and analyze the arguments of others. Materials occasionally prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards. However, there are very few opportunities for students to both construct arguments and analyze the arguments of others together.

• The problems are identified directly with a title of 'Construct Viable Arguments' and 'Analyze the Arguments of Others'.
• There are numerous examples throughout the text addressing this indicator. However, activities could have been embedded to elicit this behavior in a more meaningful way. When one considers MP 4, they think of how students used mathematical discourse with one another, not just looking at scenarios in the text.
• Students are asked to “explain” often, however that frequently falls short of the full meaning of the practice.

Examples of students having to justify, explain, or show their thinking:

• Activity 10, question 18, page 128 – students are asked to explain how equations can have no or many solutions.
• Activity 11, question 11, page 143 – students have to justify their thinking by writing a letter to explain which package Misty should use.
• Activity 12, question 8, page 153 – students have to justify their thinking as to which problem has the greatest rate of change.
• Activity 12, question 15, page 154 – students have to justify their reasoning as to which line is the steepest.

Examples of students having to evaluate someone else’s explanation, work or thinking:

• Activity 3, question 7, page 42 – students have to determine who solved the problem correctly and why.
• Activity 6, question 15, page 74 – students have to determine the mistake that Sebastian made.