8th Grade - Gateway 2

The instructional materials reviewed for The Utah Middle School Math Project Grade 8 meet the expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

Each chapter/section starts with an Anchor Problem which poses a mathematical situation that students will learn to solve. Many of these are conceptual in nature and also provide explicit connections to prior knowledge. For example, Chapter 3 Anchor Problem 3.0: Solutions to a Linear Equation specifically refer to prior learning from Chapter 1 involving writing and solving linear equations with one variable. Students then move on to more complex linear equations as the activity guides them toward problems with infinite solutions and plotting ordered pairs on the coordinate plane. The Teacher Notes say, “Project the grid on the board and ask students to come up and plot an ordered pair that is a solution to the equation. They will soon see that the ordered pairs follow a pattern. Some students may even come up with solutions that include fractions. If not, ask them if there are solutions that fall between integer ordered pairs. Begin filling in all of these solutions as well. Soon a line will start to appear because all of the fractional solutions will start to 'merge' together.” The Teacher Notes emphasize developing understanding.

Many Class Activity problems involve hands-on activities or models. In Chapter 10 Homework 10.2a students use grid paper to draw squares adjacent to the given triangle sides showing a proof of the Pythagorean theorem.

The Teacher Notes for each lesson describe the purpose of the lesson and how to guide students to develop their understanding of a concept. The notes include prompts and questions during instruction that lead to conceptual understanding.

Chapter 2 addresses 8.EE.B as students make the connection between proportional relationships, lines, and linear equations. Students explain, generalize, and connect ideas using supporting evidence; make and justify conjectures; compare information within or across data sets; and generalize patterns. Chapter 2 builds conceptual development as students make the connection between proportional relationships, lines and linear equations.

• The Section 2.1 Concepts and Skills to Master lists conceptual objectives for the students.
  • "Graph and write equations for a proportional relationship and identify the proportional constant or unit rate given a table, graph, equation, or context."
  • "Compare proportional relationships represented in different ways."
• Throughout Chapter 2 Class Activities and Homework, students are asked to identify correspondences between contexts, tables, graphs, and equations. Students explain, generalize, and connect ideas using supporting evidence (Question 10a, page 8WB2 - 13 and Question 2e-f, page 8WB2 - 46); compare information within or across data sets (Question 7, page 8WB2 - 19), and generalize patterns (2.2a Class Activity & Homework).
• A Teacher Note provides the following directive and explanation to assist teachers in engaging students in the conceptual understanding of the work: “Please refer to the Mathematical Textbook for Chapter 2, as this will help the teacher understand why it is important to approach Standard 8.EE.6 from the perspective that the slope is the same between any two distinct points on a line because of dilations. Transformational geometry is integrated with slope by understanding that a dilation produces figures with proportional parts. Right triangles that are formed from any two distinct points on a line are dilations of one another. Since they are dilations of one another they have corresponding parts that are proportional and parallel. This is why the rise/run ratio is the same from any of these triangles and thus the slope is the same between any two distinct points.” (page 8WB2 - 85)

Chapter 9 addresses 8.G.A.: Lessons develop conceptual understanding of translations, reflections, rotations, dilations, their properties, as well as their roles in determining if/when two figures are congruent, and if/when they are similar. Students examine preimages and images, use patty paper to perform reflections and rotations, use tables to record coordinates of images and preimages, describe transformations, draw figures congruent and/or similar to ones shown, and write coordinate rules for transformations shown. Properties of translations, reflections, rotations, and dilations are experimentally verified and compared/contrasted.

The instructional materials reviewed for The Utah Middle School Math Project Grade 8 meet the expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. Overall, when the intention is that procedural skill and fluency be developed, the materials offer opportunities for their development.

There are examples and repetition in practice in each lesson and homework.

• 8.EE.7: In Lessons 1.1a, 1.2a, 1.2b, and 1.2d students have opportunities to develop procedural skill and fluency in solving linear equations. Students work with rational numbers throughout. In the Homework students are tasked with problems that promote fluency, including being able to identify common mistakes (pages 8WB1 - 35 and 36). Throughout Chapter 1 and in other chapters in the grade (Chapters 3, 4, and 6), students are solving linear equations in one variable that includes rational coefficients. This leads them to fluency by the end of the year.
• 8.G.9: In the student content in Chapter 8, students are introduced to volume in a conceptual way, as they are asked to describe what volume is and its importance. Attention is given to 8.G.9 in Section 8.3. In Lessons 8.3b, 8.3c, and 8.3d, students find the volume of spheres, cones, and cylinders using the correct formulas. Students derive the equations for the volume of cones, cylinders, and spheres and practice problems related to them in Section 8.3. In Class Activity 8.3b Questions 7-12 and Homework Questions 1-6, students find the volume and/or missing measurement of each cylinder. In Class Activity 8.3c Questions 7-12 and Homework Questions 1-6, students find the volume and/or missing measurement of each cone. In Class Activity 8.3d8 - 13 and Homework 1-6 students are given the directions to find the volume and/or missing measurement of each sphere.

Note that there are no spiral reviews in Grade 8 to provide additional procedural skill and fluency practice.

The instructional materials reviewed for The Utah Middle School Math Project Grade 8 meet the expectation that teachers and students spend sufficient time working with engaging applications of mathematics, without losing focus on the major work of the grade. Overall, the materials have multiple opportunities for application.

The materials incorporate Anchor Problems at the beginning of each chapter which provide students multi-step questions where they solve problems by using a variety of paths.

• In Anchor Problem 1.0 students use their own assumptions to model several situations mathematically. Students are directed to “(c)onsider the following situations. Then answer the questions below. Include any pictures, models, or equations you used to solve the problem and clearly explain the strategy you used.” For example, students explore the following situation: “Two students, Theo and Lance, each have some chocolates. They know that they have the same number of chocolates. Theo has four full bags of chocolates and five loose chocolates. Lance has two full bags of chocolates and twenty-nine loose chocolates. Determine the number of chocolates in a bag. Determine the number of chocolates each child has.” (page 8WB1 - 7)
• Chapter 4’s Anchor Problem, “Chickens and Pigs,” has multiple ways to solve (trial and error, a table, pictures or symbolic representations, and equations and graphs). There is flexibility for students in this contextual problem to apply their mathematical understanding. “A farmer saw some chickens and pigs in a field. He counted 30 heads and 84 legs. Determine exactly how many chickens and pigs he saw. There are many different ways to solve this problem, and several strategies have been listed below. Solve the problem in as many different ways as you can and show your strategies below.“ (page 8WB4 – 6)

In Grade 8, a specific standard and cluster that include application are 8.EE.8c and 8.F.B. Examples of problems that address these standards include:

• Chapter 4 "Who will win the race?" Class Activity and Homework is a multi-step problem leading to two linear equations in two variables that encourage students to use their own methods of problem solving so that there are multiple paths of entry. (8.EE.8c)
• In Chapter 4 Section 4.2 students solve simultaneous linear equations that have one, no, or infinitely many solutions using algebraic methods. For example, Homework 4.2e Question 4 asks the following: “Sarah has $400 in her savings account, and she has to pay $15 each month to her parents for her cellphone. Darius has $50, and he saves $20 each month from his job walking dogs for his neighbor. At this rate, when will Sarah and Darius have the same amount of money? How much money will they each have?” (8.EE.8c)
• In Chapter 2 students calculate the cost for attending the state fair and riding various rides given the costs for the fair and each ride. Students determine the total number of rides that can be be purchased, given a specific amount, and create a table and graph to represent the situation. (8.F.B)
• Chapter 5 Section 5.3 includes real-world contexts such as a height vs. time function, pennies earned per day, and the half-life of Carbon-14. (8.F.B)

The instructional materials reviewed for The Utah Middle School Math Project Grade 8 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. Every chapter includes all three aspects of rigor. In some lessons the aspects of rigor are addressed separately, and in some lessons multiple aspects of rigor are addressed. Overall, the three aspects of rigor are balanced in this program.

There are lessons where the aspects of rigor are not combined.

• Homework 3.1a provides a variety of problems for students to practice their procedural skill of writing equations in slope-intercept form.
• In Class Activity 9.1a students develop their conceptual understanding of translations by answering a variety of questions designed to illicit similarities and differences between the properties of translations.

There are multiple lessons where two or all three of the aspects are interwoven.

• In Class Activity 4.2c (page 8WB4 - 44) students are first asked to solve, in any way they choose, a couple of contextual problems that can be modeled and solved using systems of linear equations. For example, “Carter and Sani each have the same number of marbles. Sani’s little sister comes in and takes some of Carter’s marbles and gives them to Sani. After she has done this, Sani has 18 marbles and Carter has 10 marbles. How many marbles did each of the boys start with? How many marbles did Sani’s sister take from Carter and give to Sani?” Students then determine the values of "shape-addends" that form two shape equation systems designed to match the contextual problems. This work is followed by more shape equation systems for students to represent algebraically and to solve using the elimination method. Students apply their understanding of the elimination method and develop procedural skills as they solve several systems of linear equations. In the final task students create a context for a given system, solve the system, and write the solution in a complete sentence.
• In Class Activity 1.1c students use models to solve linear equations by combining like terms. Students develop an understanding of like terms through the models leading to development of procedural skill. By the end of the lesson, students are asked to solve equations and verify their solutions.

The instructional materials reviewed for The Utah Middle School Math Project Grade 8 meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade.

The Standards for Mathematical Practices are identified in both the Teacher and Student Workbooks in most lessons. The MPs are listed in the beginning of the chapter and are also identified using an icon within the lessons where they occur.

Overall, the materials clearly identify the MPs and incorporate them into the lessons. All of the MPs are represented and attended to multiple times throughout the year, and MPs are used to enrich the content and are not taught as a separate lesson.

• Chapter 3 Homework 3.1g Questions 1-4 ask students to "reason quantitatively" as they write equations from graphs of linear representations and then tell the story that relates to the equation (MP2). For example, “The graph below shows a trip taken by a car where x is time (in hours) the car has driven and y is the distance (in miles) from Salt Lake City. Label the axes of the graph. Use your graph and equation to tell the story of this trip taken by the car.”
• Chapter 4 Class Activity 4.2c highlights MP1 as students make sense of problems. The problem states: “Ariana and Emily are both standing in line at Papa Joe’s Pizza. Ariana orders 4 large cheese pizzas and 1 order of breadsticks. Her total before tax is $34.46. Emily orders 2 large cheese pizzas and 1 order of breadsticks. Her total before tax is $18.48. Determine the cost of 1 large cheese pizza and 1 order of breadsticks. Explain the method you used for solving this problem..” The Teacher Note also emphasizes that “there are a variety of ways to solve this problem” and gives examples of methods which reinforce the goal of students making sense and persevering in solving the problem as students can solve the problem many different ways.
• Chapter 5 Class Activity 5.1b, “The Function Machine,” asks students to “attend to precision” by figuring out the rule from a table of values. Students rely upon accurately calculating the values to figure out the rules (MP6).

The instructional materials reviewed for The Utah Middle School Math Project Grade 8 meet the expectation for prompting students to construct viable arguments concerning grade-level mathematics detailed in the content standards.

In many cases, students are asked to construct arguments and justify their thinking,

• Throughout the materials students are asked to justify their thinking. For example, in Class Activity 1.2c Questions 9-13 students are asked to “plot each fraction on the number line. Fill in the blank with < , > or = . How do you know your answer is correct? Justify your answer.”
• There are instances where students are asked to make conjectures. For example, in Chapter 5 Class Activity 5.1a Questions 1-3 the directions state: “Make a conjecture (an educated guess) about what kind of relationship makes a function and what disqualifies a relation from being a function.”
• Students are asked to engage in Error Analysis in some of the lessons. For example, in Chapter 10 Homework 10.2c Question 15 students identify the error in using the Pythagorean Theorem Formula to calculate the leg between hypotenuse and other leg. In the given solution, the error was in subtracting, instead of adding the area of the squares before finding the square root.

The instructional materials reviewed for The Utah Middle School Math Project Grade 8 meet the expectation of 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. Many of the directions for MP3 are the same as those found written in the Student Workbook. Guidance is given on how to assist students in expressing arguments.

A few examples of guidance provided for teachers include:

• There is assistance for the teacher in engaging students in constructing viable arguments. Chapter 6 Class Activity 6.1a Question 1a asks students to “make some observations about the data shown in the dot plot.” The Teacher Notes state the following: “Listen to what students say. They may say things like, the average amount she makes is around $100. The data does not appear to be very spread out. The point 55 appears to be an outlier and may pull the average down. What could have caused this outlier? She can usually expect to make between $75 and $120 a day.”
• There is assistance for the teacher in engaging students in analyzing the arguments of others. For example, in Chapter 8 Class Activity 8.1d the Teacher Notes say,“When they are finished have them discuss their answers with a neighbor before moving on to a group discussion. Ask for people to come to the board to show and justify how they fixed the mistake in each statement.”
• There are some prompts for the teacher in the form of questions to ask or problems to present. For example, in Chapter 5 Class Activity 5.2b students are prompted to determine if given situations can be modeled with linear functions and to provide evidence to back their claim. Teachers are given this guidance in the notes: “For the answers below, students can provide various pieces of evidence (constant/changing rate of change; first difference is the table is constant/not constant; graph is/is not a line; form of the equation). Accept all valid explanations.”

The instructional materials reviewed for The Utah Middle School Math Project Grade 8 meet the expectation for attending to the specialized language of mathematics. Overall, the materials provide explicit instruction on how to communicate mathematical thinking using words, diagrams, and symbols. When students are introduced to new mathematical vocabulary, it is explained, and teachers are encouraged to tell students to use the new terms.

• Each chapter in the workbook begins with a vocabulary list of words used in the chapter that includes words from previous learning as well as new terms.
• Throughout the chapter, new terms are used in context during Class Activities, Homework, and Self-Assessments.
• Vocabulary is bold in the context of the lesson.
• Vocabulary is presented throughout the textbook, Mathematical Foundations, along with accurate definitions. For example, on page 8MF2 - 3: “Given two quantities x and y, they are said to be proportional if, whenever we multiply one by a factor r, the other is multiplied by the same factor, r. For example, if we double the variable x, then y also doubles.”
• Students are encouraged to use vocabulary appropriately. For example, Chapter 9 Class Activity 9.1 introduces the terms: translation, pre-image, image, and corresponding vertices. These terms are introduced, defined, and taught to the students. They are used throughout the chapter.
• At times the Teacher Notes give suggestions for using vocabulary in a lesson. For example, Chapter 5 Class Activity 5.1a says, “Talk to the students about the term unique and how it is used in mathematics, as they will see it in many definitions in the future.”
• The terminology that is used in the course is consistent with the terms in the standards.