8th Grade - Gateway 2

The materials reviewed for enVision Mathematics Grade 8 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. 

Materials include problems and questions that develop conceptual understanding throughout the grade level. According to the Teacher Resource Program Overview, “Problem-Based Learning The Solve & Discuss It in Step 1 of the lesson helps students connect what they know to new ideas embedded in the problem. When students make these connections, conceptual understanding takes seed. Visual Learning In Step 2 of the instructional model, teachers use the Visual Learning Bridge, either in print or online, to make important lesson concepts explicit by connecting them to students’ thinking and solutions from Step 1.” Examples from the materials include:

• Topic 1, Lesson 1-3, Practice & Problem Solving, Problem 10, students develop conceptual understanding by comparing real numbers in various forms. “Does \frac{1}{6}, -3, \sqrt{7}, -$$\frac{6}{5}$$, or 4.5 come first when the numbers are listed from least to greatest? Explain.” (8.NS.2)

• Topic 3, Lesson 3-1, Do You Understand?, Problem 3, students demonstrate conceptual understanding by generalizing about relations and functions. “Generalize Is a relation always a function? Is a function always a relation? Explain?” (8.F.1)

• Topic 6, Lesson 6-5, Solve and Discuss It!, students develop conceptual understanding by describing the transformations required to map one shape onto another. “Simone plays a video game in which she moves shapes into empty spaces. After several rounds, her next move must fit the blue piece into the dashed space. How can Simone move the blue piece to fit in the space?” (8.G.2 and 8.G.3)

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Practice & Problem Solving exercises found in the student materials provide opportunities for students to demonstrate conceptual understanding. Try It! provides problems that can be used as a formative assessment of conceptual understanding following Example problems. Do You Understand?/Do You Know How? Problems have students answer the Essential Question and determine students’ understanding of the concept. Examples from the materials include:

• Topic 2, Lesson 2-2, Do You Know How?, Item 4, students independently demonstrate conceptual understanding of solving equations with like terms on both sides of the equation. “Maria and Liam work in a banquet hall. Maria earns a 20% commission on her food sales. Liam earns a weekly salary of $625 plus a 10% commission on his food sales. What amount of food sales will result in Maria and Liam earning the same amount for the week?” (8.EE.7b)

• Topic 4, Lesson 4-2, Solve & Discuss It!, students analyze bivariate data and connect it to analyzing linear associations. “Angus has a big test coming up. Should he stay up and study or go to bed early the night before the test? Defend your recommendation.” An image of a sheet of paper is shown with data on Angus' bedtime and test scores, for example: Test #1 - went to bed at 9:15, got 80%. (8.F.3, 8.F.4, and 8.SP.2)

• Topic 6, Lesson 6-8, Solve & Discuss It!, students analyze the relationship among angles formed by a line intersecting two parallel lines. “Draw two parallel lines. Then draw a line that intersects both lines. Which angles have equal measures? (8.G.5)

The materials reviewed for enVision Mathematics Grade 8 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. 

The materials develop procedural skill and fluency throughout the grade level. According to the Teacher Resource Program Overview, “Students develop skill fluency when the procedures make sense to them. Students develop these skills in conjunction with understanding through careful learning progressions.” Try It! And Do You Know How? Provide opportunities for students to build procedural fluency from conceptual understanding. Examples include:

• Topic 1, Lesson 1-6, Try it!, students calculate the weight of an adult elephant using properties of exponents. “The local zoo welcomed a newborn African elephant that weighed 3$$^4$$ kg. It is expected that at adulthood, the newborn elephant will weigh approximately 3$$^4$$ times as much as its birth weight. What expression represents the expected adult weight of the newborn elephant?” (8.EE.1)

• Topic 2, Lesson 2-4, Practice & Problem Solving, Problem 22, students classify equations as having one solution, no solution, or infinitely many solutions. “Classify the equation 64x - 16 = 16 (4x -1) as having one solution, no solution, or infinitely many solutions.”(8.EE.7a) 

• Topic 4, Lesson 4-4, Do You Know How?, Problem 4, students complete a two-way frequency table based on a real-world scenario. “A basketball coach closely watches the shots of 60 players during basketball tryouts. Complete the two-way frequency table to show her observations.” A partially completed table of grade levels and basketball shots is provided. (8.SP.4)

The materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade level. Practice & Problem Solving exercises found in the student materials provide opportunities for students to independently demonstrate procedural skill and fluency. Additionally, at the end of each Topic is a Concepts and Skills Review which engages students in fluency activities. Examples include:

• Topic 1, Lesson 1-3, Practice & Problem Solving, Problem 8a, students approximate the square root of a number by using perfect squares. “Find the rational approximation of \sqrt{15}. a. Approximate using perfect squares. __<15<__  __<$$\sqrt{15}$$<__  __<$$\sqrt{15}$$<__” (8.NS.2)

• Topic 3, Lesson 3-4, Practice & Problem Solving, Problem 7, students calculate slope in order to write a linear equation. “A line passes through the points (4, 19) and (9, 24). Write a linear function in the form y = mx + b for this line.” (8.F.4)

• Topic 7, Lesson 7-1, Concepts and Skills Review, Problem 1, students find the length of a hypotenuse. “Find the length of the hypotenuse.” A triangle is shown with side lengths of 12 cm and 5 cm. (8.G.7)

The materials reviewed for enVision Mathematics Grade 8 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which mathematics is applied. 

The materials include multiple opportunities for students to independently engage in routine and non-routine application of mathematical skills and knowledge of the grade level. According to the Teacher Resource Program Overview, “3-Act Mathematical Modeling Lessons In each topic, students encounter a 3-Act Mathematical Modeling lesson, a rich, real-world situation for which students look to apply not just math content, but math practices to solve the problem presented.” Additionally, each Topic provides a STEM project that presents a situation that addresses real social, economic, and environmental issues, along with applied practice problems for each lesson. For example:

• Topic 1, STEM Project, Going, Going, Gone, students use real numbers such as rational and irrational numbers to show the depletion rate of a natural resource. "Natural resource depletion is an important issue facing the world. Suppose a natural resource is being depleted at the rate of 1.333% per year. If there were 300 million tons of this resource in 2005, and there are no new discoveries, how much will be left in the year 2045? You and your classmates will explore the depletion of this resource over time.” (8.NS.1, 8.NS.2, 8.EE.1 and 8.EE.2)

• Topic 5, 3-Act Mathematical Modeling: Up and Downs, Problem 12, students develop a mathematical model to represent and answer the question which route is faster? “Model with Math Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?” (8.EE.8, 8.F.4, and 8.SP.3)

• Topic 8, 3-Act Mathematical Modeling: Measure Up, Problem 15,  students determine whether the liquid in one container will fit into a container with a different shape. “Generalize Suppose you have a graduated cylinder half the height of the one in the video. How wide does the cylinder need to be to hold the liquid in the flask?" (8.G.9)

The materials provide opportunities for students to independently demonstrate the use of mathematics flexibly in a variety of contexts. Pick a Project is found in each Topic and students select from a group of projects that provide open-ended rich tasks that enhance mathematical thinking and provide choice. Additionally, Practice & Problem Solving exercises found in the student materials provide opportunities for students to independently demonstrate mathematical flexibility in a variety of contexts. For example:

• Topic 3, Lesson 3-4, Practice & Problem Solving, Problem 15, students interpret the rate of change and initial value of a linear function.  “Reasoning The graph shows the relationship between the number of cubic yards of mulch ordered and the total cost of the mulch delivered. a. What is the constant rate of change? What does it represent? b. What is the initial value? What might that represent?” (8.F.4)

• Topic 4, Lesson 4-3, Practice & Problem Solving, Problem 10, students make predictions using the slope of a trend line. “Higher Order Thinking The graph shows the temperature y, in a freezer x minutes after it was turned on. Five minutes after being turned on, the temperature was actually three degrees from what the trend line shows. What values could the actual temperature be after the freezer was on for five minutes?” (8.SP.3)

• Topic 7, Pick a Project 7A, students find the distances in a coordinate plane they mapped of their community route. “Another century-ride option is the metric century ride. Research the number of miles in a metric century ride and how long it would take to complete one. On a coordinate grid, map out a metric bike route through your community. Increase at least five stops. Use at least three diagonal line segments to represent different parts of your route. Calculate the distance between the stops. Include a paragraph with your map explaining how you calculated each distance on your route.” (8.G.8)

The materials reviewed for enVision Mathematics Grade 8 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade.

All three aspects of rigor are present independently throughout the program materials. Examples, where materials attend to conceptual understanding, procedural skill and fluency, and application, include:

• Topic 2, Lesson 2-3, Concepts and Skills Review, Problem 1, students develop procedural skills and fluency as they solve multistep equations. “Solve each equation for x. 1. (4x + 4) + 2x = 52” (8.EE.7b)

• Topic 6, Lesson 6-1, Solve & Discuss It!, students develop conceptual understanding of translating two dimensional figures. “Ashanti draws a trapezoid on the coordinate plane and labels it Figure 1. Then she draws Figure 2. How can she determine whether the figures have the same side lengths and the same angle measure?” (8.G.1 and 8.G.3)

• Topic 7, Lesson 7-4, Practice & Problem Solving, Problem 11, students use application of the Pythagorean Theorem to calculate distance on the coordinate plane. “Suppose a park is located 3.6 miles east of your home. The library is 4.8 miles north of the park. What is the shortest distance between your home and the library?” (8.G.8)

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. Examples include:

• Topic 2, Lesson 2-1, Do You Know How?, Problem 4, students solve equations in real-world context while developing procedural skill and fluency with solving linear equations with rational number coefficients. “Henry is following the recipe card (shown) to make a cake. He has 95 cups of flour. How many cakes can Henry make?” (8.EE.7b)

• Topic 4, Lesson 4-1, Practice & Problem Solving, Problem 9, students develop conceptual understanding and application as they identify and interpret clusters, gaps, and outliers on a scatter plot. “The table shows the number of painters and sculptures enrolled in seven art schools. Jashar makes an incorrect scatter plot to represent the data. a. What error did Jashar likely make? b. Explain the relationship between the number of painters and sculptors enrolled in the art schools. c. Reasoning Jashar’s scatter plot shows two possible outliers. Identify them and explain why they are outliers.” This question develops conceptual understanding and application of 8.SP.1, construct and interpret scatter plots for bivariate data to investigate patterns of association between two quantities. (8.SP.1)

• Topic 7, Mid-Topic Performance Task, Part A, students develop conceptual understanding as they apply the Pythagorean Theorem to measure the height of a tree.“Javier is standing near a palm tree. He holds an electronic tape measure near his eyes and finds the three distances shown. Part A. Javier says that he can now use the Pythagorean Theorem to find the height of the tree. Explain. Use vocabulary terms in your explanation. (8.G.7)

The materials reviewed for enVision Mathematics Grade 8 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them, and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Some examples where the materials support the intentional development of MP1 are:

• Topic 2, Lesson 2-5, Practice & Problem Solving, Problem 6, students must make sense of the different representations of data to decide who cycled faster. “Sam and Bobby want to know who cycled faster. The table shows the total miles. Sam traveled over time. The graph shows the same relationship for Bobby. Who cycled​ faster?”

• Topic 7, Performance Task, Problem 3, students solve a real-world problem by making sense of the given diagram to understand how to find the unknown length. “Cameron decides to make the center part of the tabletop out of inlaid wood. He sketches the design shown, where four trapezoids form a square with 6-inch sides in the middle of the table. What is the length, d, of the side of each trapezoid? Round to the nearest tenth of an inch. Explain.

• Topic 8, Lesson 8-2, Practice & Problem Solving, Problem 13, students make sense of the problem and persevere in solving it as they explain how the radius would change if the height would change but the volume would remain the same. “The cylinder shown has a volume of 885 cubic inches. a. What is the radius of the​ cylinder? Use 3.14 for \pi. b. Reasoning If the height of the cylinder is​ changed, but the volume stays the​ same, then how will the radius​ change? Explain.

Some examples where the materials support the intentional development of MP2 are:

• Topic 1, Lesson 1-1, How do you know?, Problem 2, students reason abstractly and quantitatively as students explain why certain multiplication must occur to change a repeating decimal to a rational number. Use Structure Why do you multiply by a power of 10 when writing a repeating decimal as a rational number?

• Topic 5, Lesson 5-2, Explore It!, students reason abstractly and quantitatively when they interpret graphs of linear systems of equations and make meaning by understanding that the solution is the intersection point(s). “Beth and Dante pass by the library as they walk home using separate straight paths. A. Model with Math The point on the graph represents the location of the library. Draw and label lines on the graph to show each possible path to the library. B. Write a system of equations that represents the paths taken by Beth and Dante. Reasoning What does the point of intersection of the lines represent in this situation?”

• Topic 8, Lesson 8-4, Focus on math practices, students relate the volume of a sphere and the volume of a cone. “Reasoning How are the volume of a sphere and the volume of a cone related? What must be true about the radius and the height measurements for this relationship to be valid?”

The materials reviewed for enVision Mathematics Grade 8 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Student materials consistently prompt students to construct viable arguments. These opportunities are found in the following activities: Solve & Discuss It!, Explain It!, Explore It!, Practice & Problem Solving, Do You Understand?, and Performance Tasks. Examples include:

• Topic 2, Lesson 2-5, Solve & Discuss It!, students use their understanding of proportional relationships to construct arguments and support their response. “Mei Li is going apple picking. She is choosing between two places. The cost of a crate of apples at each place is shown. Where should Mei Li go to pick her apples? Explain.”

• Topic 4, Lesson 4-2, Do You Understand?, Problem 3, students construct arguments as they explain the difference between linear and nonlinear association. “Construct Arguments How does the scatter plot of a nonlinear association differ from that of a linear association?”

• Topic 6, Lesson 6-2, Practice & Problem Solving, Problem 10, students use their understanding of reflections to construct arguments. “ Construct Arguments Your friend incorrectly says that the reflection of \triangleEFG to its image \triangleE’F’G’ is a reflection across the x-axis. a. What is your friend’s mistake? b. What is the correct description of the reflection?” 

Student materials consistently prompt students to analyze the arguments of others. These opportunities are found in the following activities: Solve & Discuss It!, Explain It!, Explore It!, Practice & Problem Solving, Do You Understand?, and Performance Tasks. Examples include:

• Topic 4, Lesson 4-5, Do You Understand?, Problem 3, students analyze the arguments of others as they find the relative frequency from a two-way table. “ Critique Reasoning Maryann says that if 100 people are surveyed, the frequency table will provide the same information as a total relative frequency table. Do you agree? Explain why or why not.”

• Topic 5, Lesson 5-3, Explain It!, students analyze the arguments of others as they graph a system of equations to determine the most cost-effective cab company. “Jackson needs a taxi to take him to a destination that is a little over 4 miles away. He has a graph that shows the rates for two companies. Jackson says that because the slope of the line that represents the rates for On Time Cabs is less than the slope of the line that represents Speedy Cab Co., the cab ride from On Time Cabs will cost less. A. Do you agree with Jackson? Explain. B. Which taxi service company should Jackson call? Explain your reasoning.”

• Topic 8, Lesson 8-1, Do You Understand?, Problem 3, students analyze the arguments of others as they use formulas for polygons. “Construct Arguments Aaron says that all cones with a base circumference of 8$$\pi$$ inches will have the same surface area. Is Aaron correct? Explain.”

The materials reviewed for enVision Mathematics Grade 8 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Use appropriate tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials allow for the intentional development of MP4 to meet its full intent in connection to grade-level content. Examples of this include:

• Topic 2, Lesson 2-1, Try It!, students work with equations to model the purchases made from a store. “Selena spends $53.94 to buy a necklace and bracelet set for each of her friends. Each necklace costs $9.99, and each bracelet costs $7.99. How many necklace and bracelet sets, s, did Selena buy? Convince Me! Suppose the equation is 9.99s + 7.99s + 4.6 = 53.94. Can you combine the s terms and 4.6? Explain.”

• Topic 6, Lesson 6-4, Practice & Problem Solving, Problem 8, students apply what they know about transformations and model how to move a table.  "Model with Math A family moves a table, shown as rectangle EFGH, by translating it 3 units left and 3 units down followed by a 90° rotation about the origin. Graph E’ F’ G’ H’ to show the new location of the table." 

• Topic 7, 3-Act Mathematical Modeling: Go With the Flow, students investigate the proof of the Pythagorean Theorem by modeling a situation involving sand falling from squares representing the legs of the triangle squared into one larger square representing the hypotenuse squared. 

The materials allow for the intentional development of MP5 to meet its full intent in connection to grade-level content. Examples of this include:

• Topic 1, Lesson 1-3, Lesson Quiz, Problem 3, students compare and order a set of numbers from least to greatest. They have to choose the best approach to doing this, one approach is to use the tool of a number line as demonstrated in the lesson. “Compare and order the numbers below from least to greatest. 4.6, 2.$$\overline{8}$$, \pi, \sqrt{17}, \sqrt{7}

• Topic 3, Lesson 3-2, Do You Understand?, Problem 2, students explain how to use a graph strategically to determine when a relationship is not a function. “Use Appropriate Tools How can you use a graph to determine that a relationship is NOT a function?” 

• Topic 5, 3-Act Mathematical Modeling: Up and Downs, Problem 6, students determine the fastest route between taking an elevator or the stairs in a multistory building. “Use Appropriate Tools What tools can you use to solve the problem? Explain how you would use them strategically?”

The materials reviewed for enVision Mathematics Grade 8 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students are encouraged to attend to the specialized language of mathematics throughout the materials. A chart in the Topic Planner lists the vocabulary being introduced for each lesson in the Topic. As new words are introduced in a Lesson they are highlighted in yellow and students are encouraged to utilize the Vocabulary Glossary in the back of the text (with an animated version online in both English and Spanish) to find both definitions and examples where relevant. Lesson Practice includes questions that reinforce vocabulary comprehension and the teacher's side notes provide specific information about what math language and vocabulary are pertinent for each section.

Examples where students are attending to the full intent of MP6 and/or attend to the specialized language of mathematics include:

• Topic 1, Lesson 1-5, Practice & Problem Solving, Problem 20, students attend to precision as they evaluate an expression and write their answer as an integer. “Evaluate \sqrt[3]{-512}. a. Write your answer as an integer. b. Explain how you can check that your result is correct.”

• Topic 4, Topic Review, Use Vocabulary in Writing, students attend to the specialized language of mathematics as they describe a given scatter plot using mathematical terms.   “Describe the scatter plot at the right. Use vocabulary terms in your description.” Students are provided a word bank containing, “categorical data, outlier(s), cluster(s), relative frequency, measurement data, and trend line.” 

• Topic 6, Lesson 6-1, Practice & Problem Solving, Problem 11, students attend to precision as they accurately graph the image using the translation information provided. “Graph the image of the given triangle after a translation 3 units right and 2 units up.”

• Topic 8, Mid-Topic Checkpoint, Problem 1, students attend to the specialized language of mathematics as they select statements that describe either surface area and volume. “Vocabulary Select all the statements that describe surface area and volume.” A) Surface area is the sum of the areas of all the surfaces of a figure. B) Volume is the distance around a figure. C) Surface area is a three-dimensional measure. D) Volume is the amount of space a figure occupies. E) Volume is a three-dimensional measure.”

The materials reviewed for enVision Mathematics Grade 8 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards.

Students are encouraged to look for and make use of structure as they work throughout the materials, both with the instructor's guidance and independently. Examples of where there is intentional development of MP7 include:

• Topic 2, Lesson 2-3, Do You Understand?, Problem 3, students use the structure of the order of operations to explain simplifying using the distributive property. “Use Structure How can you use the order of operations to explain why you cannot combine the variable terms before using the Distributive Property when solving the equation 7(x + 5) - x = 42?”

• Topic 7, Lesson 7-4, Practice & Problem Solving, Problem 12, students use the structure of ordered pairs and the Pythagorean Theorem to find the distance between two points. “Use Structure Point B has coordinates (2, 1). The x-coordinate of coordinate A is -10. The distance between point A and point B is 15 units. What are the possible coordinates of point A?”

• Topic 8, Lesson 8-2, Do You Understand?, Problem 2, students analyze the structure of a cylinder to identify which two measurements are needed for the volume. “Use Structure What two measurements do you need to know to find the volume of a cylinder?” 

Students look for and express regularity in repeated reasoning as they are engaged in the course materials. Examples of intentional development of MP8 include: 

• Topic 4, Lesson 4-1, Practice & Problem Solving, Problem 7b, students demonstrate an ability to identify related reasoning in bi-variate data, by looking at the repeated clusters of the scatter plot. “Generalize How does the scatter plot show the relationship between the data points? Explain”

• Topic 5, Lesson 5-2, Do You Understand?, Problem 2, students use their knowledge of repeatedly graphing systems of equations to make a generalization about lines of a system of no solution. “Reasoning If a system has no solution what do you know about the lines being graphed?” 

• Topic 7, Lesson 7-4, Convince Me!, students make a generalization about finding the distance between two points on a coordinate plane. “Why do you need to use the Pythagorean Theorem to find the distance between points A and B.” An image of a coordinate plane is given with points A and B labeled on it.