8th Grade - Gateway 2

The instructional materials reviewed partially meet the expectation for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. Overall, the development of conceptual understanding is presented in a directed way so that students would not fully develop and refine their ability to reason mathematically.

All sections in the material begin with a one-day activity intended to build understanding of the concept within each section. Most of the activities are hands-on, but they are presented in a step-by-step manner, which leads all students to solve them in the same way and, in turn, produce the same results. This limits opportunities to explore and make connections between various solution paths. The following activities address conceptual development in an explicit way:

• The initial activity for Chapter 2, Lesson 6 has students find the perimeters and areas of various similar figures. After completing the chart, students have to describe the pattern and find the relationship between perimeter, area, and the original figure. The conceptual understanding of 8.G.4 is the primary goal.
• Activities in Chapter 4, Lesson 1 address 8.EE.5 by requiring students to find multiple ordered pairs that satisfy the equation, graph them, and then find and check additional ordered pairs. Students understand that in order to be a solution of an equation, a point has to lie on the line.
• Lessons attending to 8.F are found in Chapter 6. The 6.1 activities have students generate the outputs, or areas of the given two- and three-dimensional figures, using a given input for the length, and then students map these values in activity 1. In activity 2 students generate rules. In the activities for section 6.2, students write equations in problems 1 and 2, and students graph points from a given table to see if the given statements are true in problem 3. These connections are built in a procedural way until students are shown all three representations in Example 3 and the summary of section 6.2. Students are often expected to complete the representation rather than make connections between them.
• The activities found in Chapter 10, Lessons 1-4, align to 8.EE.1, and all use the expanded form of exponents to help students understand operations with powers and develop a rule using patterns.

On the second day, the lesson is presented through multiple examples that can be reviewed as a class and On Your Own examples that allow students to practice the lesson concept. There is usually one reasoning or logic problem per lesson in the exercises of each section. One problem in each of the lessons is explained at length in the Taking Math Deeper. Lesson notes for the teacher mainly focus on procedures and the steps necessary to solve the problems.

Additional features included in the material show an emphasis on conceptual development:

• The beginning of each exercise has a section called Vocabulary and Concept Check, and the end of each activity has a section called What is Your Answer? Both of these sections often expect students to explain and demonstrate conceptual understanding through reasoning and writing about the concepts.
• The online lesson plans also account for sections titled Start Thinking, and the student’s copy is located in the Resources by Chapter book. These questions assist in connecting to previous knowledge that students need in order to engage in a new task and also require explanations, justifications, or comparisons, which are important to conceptual understanding.

In the overall structure of the material, concepts are proceduralized in each section because the activities are accompanied by directed steps. Students are asked to make generalizations in the reasoning and logic items before engaging with the exercises. Communications between the students and teacher addressing conceptual understanding are not always stressed.

The materials reviewed partially 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 the grade. Overall, there were few opportunities for students to engage in non-routine problems, and in many cases, the items direct students in such a way that they do not have a chance to decide on alternate ways to solve them.

The following structures and resources sometimes attend to application, but many times are scaffolded for students with steps and given paths to the solution:

• Performance Tasks for each section are provided online for each grade-level standard. These provide opportunities for students to engage in the mathematics from the section in a new way, but these can sometimes be directive and rarely address more than one standard.
• Enrichment and Extension is found in Resources by Chapter, and there is one for each lesson. These are not included in the lesson plans provided online, but they are suggested as extra practice. There are questions in these activities that would meet the expectations of application, but a direct path for the answer is provided. For example, in lesson 1.4 students are asked to rewrite the formula for volume of a cylinder. They are given two methods for rewriting the formula with explicit steps for each and then asked which method they prefer.
• In Appendix A, there are additional opportunities for students to apply what they have learned over the course of the year. This section has four Big Idea Projects. Students look at examples of math in the real world and in a cross-curricular way. The projects contain scaffolding and guide students in how to complete the work.
• Each lesson exercise has one problem that applies the skill of the lesson, and it is titled Taking Math Deeper in the teacher edition. In the student materials, these problems do not have any special identification. This section provides an extended solution for the teacher for one specific item in the exercises and then provides a related project that the teacher can assign. For example, in Lesson 10.2, “Taking Math Deeper” explains the steps to solving exercise 31, where students show the number of pieces of mail sent by the United States Postal Service in 6 days as a product of powers expression. The related project involves researching the price of a postage stamp and finding the range in the cost.

There are some examples of items that allow students to engage in applying mathematics to the given situations.

• In Chapter 1 Lesson 2, Solving Multi-Step Equations, there are two activities that address 8.EE.7 on page 11. These problems require students to apply past and current concepts to solve a multistep problem. In both, students may work with a partner to find multiple unknowns.
• In Chapter 1 Lesson 3, Solving Equations with Variables on Both Sides, item 39 is an opportunity to apply 8.EE.7 in a real world context. Students must find the price of mailing DVDs by setting the two different companies’ rates equal to each other. Students are given a chart with the needed information to solve the problem.
• As students work with functions (8.F) in Chapter 6 (pages 254, 262), they often work with real-world problems requiring them to write an equation, graph the relationship, and then answer various questions using both the equation and graph. These questions vary as to content and encourage students to think and analyze the relationships.
• Activity 2 in Chapter 8, lesson 1 has students engage with 8.G.9 by asking students to use a defined amount of wax purchased for $20 to make eight candles of three different sizes and then decide on candle size and price using properties of cylinders. Students are not given guidance on this item, making this a very rigorous problem for students.

While opportunities for application are seen in several features and sections found in the material, evidence of proceduralizing opportunities for application was also found.

• In Chapter 4, lesson 3, Graphing and Comparing Proportional Relationships (8.EE.5,6), students are presented with two 2-variable relationships, one is in algebraic form and the other in graphical form. Students must write both equations and graph as well. Nearly all problems are application, i.e., cost in dollars/hour, growth in mm/year; however, all are presented in the same format so that students actually follow the procedures more than applying the mathematics.
• The initial activity for Chapter 5, Lesson 1 begins by asking students to write two equations for a real-world situation involving starting a bed-and-breakfast and finding when it would actually make money. Instead of leaving the problem open for students to solve, the students are walked through writing the equations and given a table to complete using the given equations. In the 5.1 lesson, students solve systems of equations by graphing and are asked to use the graphical representation to compare the relationships (8.EE.8a). Four out of 24 problems include a real world scenario. These are all routine problems that follow the steps given in example 2, Real Life Application, on page 205.

Overall, students are given multiple opportunities to solve mathematics in real world contexts, but given the way the material is structured, students can easily use the examples provided in the section to figure out which types of relationships or structures to apply to given situations. Problems are routine and occur in the labeled lesson. For example, all Pythagorean theorem problems appear in Chapter 7 in lessons where students are given worked examples and already know what procedures to apply.

The materials reviewed for Grade 8 partially meet the expectations for not always treating the three aspects of rigor together and not always treating them separately. Overall, there is not a balance of the three aspects of rigor within the grade.

• Most of the items require calculating a solution, and students are given step-by-step procedures to use when solving them. The activities sometimes offer opportunities for students to engage in procedures with connections to explore concepts, but they are given targeted and scaffolded paths to reach the desired understanding limiting opportunities for students to apply mathematics previously learned.
• According to the given online lesson plans, many of the application opportunities are not part of the regular program, but are offered as other opportunities.

The materials reviewed for Grade 8 partially meet the expectations for identifying the mathematical practices (MPs) and using them to enrich the mathematics content within and throughout the Grade 8 materials.

The MPs are clearly labeled in the teacher edition in the activity portion of each section. They can also be found in the introduction within Laurie’s Notes, where there is also an explanation of how they are connected to the lesson and what the teacher should expect out of the students.

• A Math Practice box is found in the student edition. It is not labeled with the specific MP for the student, but an explanation can be found in the teacher’s edition explaining the MP in which students are engaging.
• Sometimes MP1 and MP3 are split and labeled “MPa” and “MPb”, when the full MPs are not reached. For example, in section 4.2, MP1a is listed with a note that “drawing arrow diagrams will help students visualize the slope triangle.” There is no explanation of what the “a” means. In section 4.3, MP3a is listed, and the teacher is asked to use “volunteers to justify their procedures and explain why their procedure shows a proportional relationship.” An explanation of what the “a” means is not provided.
• The MPs are also identified in the online lesson plans. Each lesson has the specific MP stated in a box in the upper right hand corner, and within the lesson, there is a section that states the focus of the MP. This is generally an explanation of what to look for while students are engaging in the problems, but sometimes it offers a question to ask the students.

The MPs are identified for the student on page iv in the beginning of the student textbook as well as in a Math Practice box located in the activities. The Math Practice boxes cause students to think about the habits of mind to be used while solving problems, but sometimes it is not clear which MP is connected to the activity since the numbers 1 through 8, used to identify the MPs in the standards, are not used in the student textbook within the series.

• On page 19 in the student textbook, the Math Practice box is labeled “Use Operations” and asks students, “What properties of operations do you need to use in order to find the value of x?” This appears to be a prompt more than the identification of a math practice.
• On page 49, the box is labeled “Justify Conclusions,” connecting it to Math Practice 3, and asks students what information they need to make a conclusion.
• The box on page 203 is labeled Use Technology to Explore, and students are asked how they decided on the values they used in setting the calculator window.
• On page 355, students are asked how repeating calculations assist them in describing the pattern in the activity. The box is labeled Repeat Calculations.

It should also be noted that the Math Practices are only identified within lesson activities and class examples. They are not identified within the problem set. For example, teachers could be aware that part of MP3 is being addressed in item 30 on page 24 as the problem is labeled Error Analysis and MP6 is being addressed in item 35 on page 24 as the problem is labeled Precision, but these connections are not explicitly made in the materials.

The materials reviewed for Grade 8 partially meet the expectations for attending to the full meaning of each MP. The MPs are most frequently identified in notes where they are aligned to a particular practice activity or question item. Many times the note is guidance on what the teacher does or says rather than engaging students in the practice. Little evidence was found to show MPs used to enhance understanding of a standard in an intentional way, with the exception on MP8.

The MPs are often applied to problems where they could be beneficial. However, the depth of the MPs is often not met since teachers, not students, engage in the MPs as they show students how to solve the problems. Examples of how MPs are presented in the materials follow:

• On page 77 in Activity 4 of section 2.6, MP1 is identified. Students are given step-by-step directions on reaching the answer. The end of the problem then tells students that there are three other similar rectangles.
• In section 5.1, Activities 1 and 2, the materials identifies MP1 when students are “investigating” a given system of equations. The example provides all steps of representing the language of the problem with equations, moving to a table, and then finally a graph. Students would not have to formulate a plan in order to find “the break even point.”
• MP2 is used quantitatively on page T-148 in the teacher edition in Activity 1 as students work with a partner to find the slope using slope triangles in multiple places on a line. MP2 is identified as being used abstractly on page T-167 in Activity 2 when students arrive at slope-intercept form of the given graph of the relationship. In both instances, students are given explicit directions and explanations of how to engage with the mathematics of the problems.
• MP4 is rarely identified in situations where students are modeling a mathematical problem and making choices about that process. This MP is frequently identified in situations where a particular form of modeling is already chosen for students. In many situations, it is labeled with directions for how the teacher should “model” rather than guidance on experiences where students engage with mathematical modeling. Examples can be found in the following lessons:
  • Lesson 1.3 - The teacher is prompted to use algebra tiles “if students are familiar with algebra tiles.”
  • Lesson 2.3 “Set up a table …”
  • Lesson 3.3 “This table helps to organize data.”
  • Lesson 8.3 - The teacher is given guidance on different ways to measure a sphere and use it to make a net for a cylinder. Teacher/students are “modeling” the conceptual connection, but they are not mathematically modeling a problem situation. All activities are guided step-by-step.

• MP5 is rarely identified in a problem solving situation or in a situation where students must choose a tool. MP5 is frequently labeled when the materials suggest a specific tool for teachers to give to students. However, guiding questions provided to the student may help develop some aspects of MP5 (i.e., page 143 asks “What are some advantages and disadvantages of using a graphing calculator to graph a linear equation?”) Language from the teacher edition that illustrates how MP5 is typically presented in the materials follows:
  • Lesson 6.2 - “If available, provide square tiles to students.”
  • Lesson 10.5 - “In the first two activities, students will use calculators to multiply very large and very small numbers.”
  • Lesson 10.7 - “You may wish to give students access to calculators for this activity.”
  • Lesson 9.2 - “It is helpful to model this with a piece of spaghetti.”

• MP7 is identified on page T-27 in Activity 2. In this example, structure is used as the materials identifies the area of the base as B instead of using each specific area formula. This is explained in the materials for the teacher, but the students are not discussing or arriving at this use of structure.
• MP8 is used appropriately in many activities as students are completing tables of repeated examples to arrive at a conclusion. On page 76 in Activities 2 and 3, students experiment with the perimeters and areas of similar figures to discover how they change when the dimensions of the figures change.

The materials reviewed for Grade 8 partially 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. While the material prompts students to construct their own arguments, there are few times when they are asked to consider and critique the reasoning of their peers.

In many cases, students are asked to construct arguments and justify them, but there are limited opportunities for them to critique and provide feedback to another student or student groups. Prompts in the “Math Practice” box found in the activities in the student textbook include questions requiring explanations and/or justifications.

• In section 3.3, after completing Activity 2 with a partner, students are asked to analyze a given conjecture about the sum of the interior angles of any polygon, including convex polygons. They are told to “explain,” but there is no instruction or prompt aimed at considering the reasoning of others in order to build on knowledge or refine thinking.
• On pages T-2 and T-3, MP3 is identified when the teacher asks “What rule did you write for the sum of the angle measures of a triangle?” A critique of the problem is not required, and the students are not asked to communicate with other students. This also occurs on page 49 in activity 4 as the teacher asks “What conjectures can you make about the two rectangles?” This could lead to an in-depth mathematical discussion, but there is no guidance for teachers that could ensure students would discuss and critique each other appropriately.

There are many explain “how” prompts, but students are asked to explain “why” they are able to use a certain procedure in many cases, which limits the arguments students may write or verbalize.

• For example, on page 119, students are asked how they can find the sum of both interior and exterior angle measures of given polygons, but they are not asked to explain why it is possible to calculate them.
• In Section 3.2, Angles of Triangles, students are working to prove that the sum of all three interior angles in a triangle measures 180 degrees in Activity 1 using their prior knowledge of parallel lines and transversals. Both MP3 and MP6 are identified by the publisher for these connected activities. Both MPs are correctly identified and met if students are given the opportunity to share their arguments with other groups in order to critique the reasoning, but the tmaterial does not prompt this conversation. An explanation for the teacher is provided on page T-110.
• MP6 is often used to remind students about vocabulary and units. For example, on page T-56 discussion of appropriately saying (x, -y) is found. Students should say “the opposite of y” and not “negative y.” On page T-79 in example 3, the material reminds teachers to “make sure students include the correct units in their answers.”

Students are asked to engage in an Error Analysis in many of the lessons. While these problems provide an opportunity for students to “describe and correct the error,” most of the errors are based on procedures that are completed incorrectly instead of requiring students to use mathematical reasoning and conceptual knowledge to strengthen an argument. For example, in item 17 of section 7.5, students must identify the error in using the distance formula to calculate the distance between two points. In the given solution, the error was in subtracting, instead of adding, the area of the squares before finding the square root. Another example is problem 13 of section 8.2; the diameter was used instead of the radius when calculating the volume of a cone with the given formula. An exception is found in item 14 of Lesson 5.4 when students must find the error in the given response, which also includes a statement about the system of equations having infinitely many solutions.

The materials reviewed for Grade 8 partially 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.

The materials, especially the activities, encourage student collaboration and discussion. Students are engaged in constructing arguments in terms of explaining why and justifying their answers, but students are rarely engaged in analyzing the work and arguments of others and critiquing that specific work. Many of the activities would lend themselves to this kind of class discussion and debate, but the materials do not provide much support for teachers to use the given materials to engage students in meaningful discourse.

• In section 2.4, MP3 is often identified in Laurie’s Notes for the teacher on pages T-61 through T-64 in reference to certain activities or examples. On page T-62 in example 1, the teacher is told to “note that the relationships between the coordinates of the vertices of a figure before and after rotation are not given” and then asked to explore the relationship if students are ready. Based on this note, students may or may not explore the relationship, and if they do, the teacher is not provided specific support, such as possible student responses, to help guide the discussion.
• In extension 4.2, “students are asked to make a conjecture and then justify their answers.” The publisher does include a note about the importance of constructing arguments, but there is no guidance for the teacher on having them critique the reasoning of other students.
• In the Activities for Section 4.7, within the Laurie’s Notes, teachers are told to “take time for discussions and explanations so that students’ reasoning is revealed,” but there are no prompts or possible responses to help the teacher facilitate this discussion.
• MP3 is misidentified on page T-2 and T-3 in Lesson 1.1 when the teacher asks “What rule did you write for the sum of the angle measures of a triangle?” Students do not have to construct a viable argument to support their rule or critique the reasoning of others that supports their rules.

Few directions are provided for the teacher other than “have students share out,” “listen for _____ methods,” etc. A few places within the teacher notes include MP3 outlined with relevant questioning and prompting for students to make conjectures about completed work within lesson activities. (For example, pages T-49, T-55, T-61, T-85, and T-111). Overall, teachers are given opportunity to engage students with MP3, but are not provided much assistance.