8th Grade - Gateway 2

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

According to the Course Guide, About These Materials, Design Principles, “Each unit begins with a pre-assessment that helps teachers gauge what students know about both prerequisite and upcoming concepts and skills, so that teachers can gauge where students are and make adjustments accordingly. The initial lesson in a unit is designed to activate prior knowledge and provide an easy entry to point to new concepts, so that students at different levels of both mathematical and English language proficiency can engage productively in the work. As the unit progresses, students are systematically introduced to representations, contexts, concepts, language and notation. As their learning progresses, they make connections between different representations and strategies, consolidating their conceptual understanding, and see and understand more efficient methods of solving problems, supporting the shift towards procedural fluency. The distributed practice problems give students ongoing practice, which also supports developing procedural proficiency.” 

Materials develop conceptual understanding throughout the grade level. Examples include:

• Unit 1: Rigid Transformations and Congruence, Section C: Congruence, Lesson 12: Congruent Polygons, Activity 1: Congruent Pairs (Part 1), students develop conceptual understanding of transformations and congruent figures. “For each of the following pairs of shapes, decide whether or not they are congruent. Explain your reasoning.” Students are shown 4 sets of figures with 2 or more transformations on a square grid. (8.G.2)

• Unit 2: Dilations, Similarity, and Introducing Slope, Section C: Slope, Lesson 10: Meet Slope, Activity 1: Similar Triangles on the Same Line, Problem 1, students develop conceptual understanding of why certain triangles with one side along the same line are similar. “The figure shows three right triangles, each with its longest side on the same line. Your teacher will assign you two triangles. Explain why the two triangles are similar.” Activity 2: Multiple Lines with the Same Slope, students develop conceptual understanding of two lines with similar slopes are parallel. “a. Draw two lines with slope 3. What do you notice about the two lines? b. Draw two lines with slope \frac{1}{2}. What do you notice about the two lines?” Students use an applet in presentation mode. (8.EE.6)

• Unit 5: Functions and Volume, Section B: Representing and Interpreting Functions, Lesson 4: Tables, Equations, and Graphs of Functions, Activity 2: Running Around a Track, Problem 1,students develop conceptual understanding of functions and non-functions by interpreting coordinates on graphs. “Kiran was running around the track. The graph shows the time, t, he took to run various distances, d. The table shows his time in seconds after every three meters. a. How long did it take Kiran to run 6 meters? b. How far had he gone after 6 seconds? c. Estimate when he had run 19.5 meters. d. Estimate how far he ran in 4 seconds. e. Is Kiran’s time a function of the distance he has run? Explain how you know.” Students are shown a graph and a table with distance values ranging from 0 to 27 and time values ranging from 0 to 9. (8.F.1)

Materials allow students to demonstrate conceptual understanding independently throughout the grade level. Examples include:

• Unit 1: Rigid Transformations and Congruence, Section A: Rigid Transformations, Lesson 6: Describing Transformations, Cool Down: Describing a Sequence of Transformations, students independently develop conceptual understanding of information required to make a translation or reflection. “Jada applies two transformations to a polygon in the coordinate plane. One of the transformations is a translation and the other is a reflection. What information does Jada need to provide to communicate the transformations she has used?” (8.G.1 and 8.G.3).

• Unit 3: Linear Relationships, Section B: Representing Linear Relationships, Lesson 6: More Linear Relationships. Activity 2: Summer Reading, students independently develop conceptual understanding of the meaning of slope and the y-intercept. “Lin has a summer reading assignment. After reading the first 30 pages of the book, she plans to read 40 pages each day until she finishes. Lin makes the graph shown here to track how many total pages she’ll read over the next few days. After day 1, Lin reaches page 70, which matches point (1,70) she made on her graph. After day 4, Lin reaches page 190, which does not match the point (4, 160) she made on her graph. Lin is not sure what went wrong since she knows she followed her reading plan. a. Sketch a line showing Lin’s original plan on the axes. b. What does the vertical intercept mean in this situation? How do the vertical intercepts of the two lines compare? c. What does the slope mean in this situation? How do the slopes of the two lines compare?” (8.EE.B)

• Unit 5: Functions and Volume, Section C: Linear Functions and Rate of Change, Lesson 8: Linear Functions, Cool Down: Beginning to See Daylight, students independently develop conceptual understanding that linear functions can be represented by an equation in y = mx + b format. “In a certain city in France, they gain 2 minutes of daylight each day after the spring equinox (usually in March), but after the autumnal equinox (usually in September) they lose 2 minutes of daylight each day. a. Which of the graphs is most likely to represent the graph of daylight for the month after the spring equinox? b. Which of the graphs is most likely to represent the graph of daylight for the month after the autumnal equinox? c. Why are the other graphs not likely to represent either month?” Four graphs are pictured. (8.F.3 and 8.F.4)

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

According to the Course Guide sections “About These Materials” and “Design Principles,” “Each unit begins with a pre-assessment that helps teachers gauge what students know about both prerequisite and upcoming concepts and skills, so that teachers can gauge where students are and make adjustments accordingly. The initial lesson in a unit is designed to activate prior knowledge and provide an easy entry to point to new concepts, so that students at different levels of both mathematical and English language proficiency can engage productively in the work. As the unit progresses, students are systematically introduced to representations, contexts, concepts, language and notation. As their learning progresses, they make connections between different representations and strategies, consolidating their conceptual understanding, and see and understand more efficient methods of solving problems, supporting the shift towards procedural fluency. The distributed practice problems give students ongoing practice, which also supports developing procedural proficiency.” Materials develop procedural skills and fluency throughout the grade level. Examples include:

• Unit 4: Linear Equations and Linear Systems, Section B: Linear Equations in One Variable, Lesson 4, More Balanced Moves, Activity 2: Make Your Own Steps, students develop procedural skill and fluency in solving linear equations. “Solve these equations for x. a. \frac{12+6x}{3} = \frac{5-9}{2} b. x - 4 = \frac{1}{3}(6x - 54) c. -(3x - 12) = 9x - 4” (8.EE.7).

• Unit 5: Functions and Volume, Section D: Cylinders and Cones, Lesson 13: The Volume of a Cylinder, Activity 3: A Cylinder’s Volume, Problem 1, students develop procedural skill and fluency of finding volume of cylinders. “Here is a cylinder with height 4 units and diameter 10 units. a. Shade the cylinder’s base. b. What is the area of the cylinder’s base? Express your answer in terms of \pi. c. What is the volume of this cylinder? Express your answer in terms of \pi.” (8.G.9)

• Unit 7: Exponents and Scientific Notation, Section B: Exponent Rules, Lesson 2: Multiplying Powers of Ten, Activity 2: Multiplying Powers of Ten, Problem 1, students develop procedural skill and fluency in multiplying exponents with the same base. “Complete the table to explore patterns in the exponents when multiplying powers of 10. You may skip a single box in the table, but if you do, be prepared to explain why you skipped it.” Students fill in a table with columns labeled expression, expanded, and single power of 10. For example, {10}^2 \cdot {10}^3 = (10 \cdot 10) (10 \cdot 10 \cdot 10) = {10}^5. (8.EE.1)

Materials provide opportunities for students to independently demonstrate procedural skills and fluency throughout the grade level. Examples include:

• Unit 4: Linear Equations and Linear Systems, Section B: Linear Equations in One Variable, Lesson 6: Strategic Solving, Practice Problems, Problem 1, students independently demonstrate procedural skill and fluency of solving linear equations in one variable. “Solve each of these equations. Explain or show your reasoning. a. 2b + 9 - 5b + 3 = -13 + 8b - 5; b. 2x + 7 - 5x + 8 = 3(5 + 6x); c. 2c - 3 = 2(6 - c) + 7c.” (8.EE.7)

• Unit 4, Linear Equation and Linear Systems, Section C: Systems of Linear Equations, Lesson 12: Systems of Equations, student independently demonstrate procedural skill and fluency as they solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Student Work Time, “A stack of n small cups has a height, h, in centimeters of h = 1.5n + 6. A stack of n large cups has a height, h, in centimeters of h = 1.5n + 9. a. Graph the equations for each cup on the same set of axes. Make sure to label the axes and decide on an appropriate scale. b. For what number of cups will the two stacks have the same height?” (8.EE.8b)

• Unit 7, Exponents and Scientific Notation, Section C: Scientific Notation, Lesson 14, Multiplying, Dividing, and Estimating with Scientific Notation, Activity 1, Biomass, Student Work Time students independently demonstrate procedural skill and fluency as they “use scientific notation as a tool for working with small and large numbers—to describe quantities, make estimates, and make comparisons.” “Use the table to answer questions about different creatures on the planet. Be prepared to explain your reasoning.” Students are presented with a table with three columns: creature, number, mass of one individual (kg). Students answer the following questions: “a. Which creature is least numerous? Estimate how many times more ants there are. b. Which creature is the least massive? Estimate how many times more massive a human is. c. Which is more massive, the total mass of all the humans or the total mass of all the ants? About how many times more massive is it? d. Which is more massive, the total mass of all the krill or the total mass of all the blue whales? About how many times more massive is it?” (8.EE.1)

The materials reviewed for Open Up Resources Grade 8 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

According to the Course Guide sections “About These Materials” and “Design Principles,” “Students have opportunities to make connections to real-world contexts throughout the materials. Frequently, carefully-chosen anchor contexts are used to motivate new mathematical concepts, and students have many opportunities to make connections between contexts and the concepts they are learning. Additionally, most units include a real-world application lesson at the end. In some cases, students spend more time developing mathematical concepts before tackling more complex application problems, and the focus is on mathematical contexts.”

Materials include multiple routine and non-routine applications of the mathematics throughout the grade level. Examples include:

• Unit 1: Rigid Transformations and Congruence, Section E: Let’s Put it to Work, Lesson 17: Rotate and Tessellate, Activity 1: Tessellate This, students engage in a non-routine application problem using congruence and similarity to create their own tessellation. “a. Design your own tessellation. You will need to decide which shapes you want to use and make copies. Remember that a tessellation is a repeating pattern that goes on forever to fill up the entire plane. b. Find a partner and trade pictures. Describe a transformation of your partner’s picture that takes the pattern to itself. How many different transformations can you find that take the pattern to itself? Consider translations, reflections, and rotations. If there’s time, color and decorate your tessellation.” (8.G.A)

• Unit 4: Linear Equations and Linear Systems, Section C: Systems of Linear Equations, Lesson 15: Writing Systems of Equations, Activity 1: Situations and Systems, students engage in a routine application problem creating systems of equations. “For each situation: Create a system of equations. Then, without solving, interpret what the solution to the system would tell you about the situation. a. Lin’s family is out for a bike ride when her dad stops to take a picture of the scenery. He tells the rest of the family to keep going and that he’ll catch up. Lin’s dad spends 5 minutes taking the photo and then rides at 0.24 miles per minute until he meets up with the rest of the family further along the bike path. Lin and the rest were riding at 0.18 miles per minute. b. Noah is planning a kayaking trip. Kayak Rental A charges a base fee of $15 plus $4.50 per hour. Kayak Rental B charges a base fee of $12.50 plus $5 per hour. c. Diego is making a large batch of pastries. The recipe calls for 3 strawberries for every apple. Diego used 52 fruits all together. d. Flour costs $0.80 per pound and sugar costs $0.50 per pound. An order of flour and sugar weighs 15 pounds and costs $9.00.” (8.EE.8)

• Unit 5: Functions and Volume, Section D: Cylinders and Cones, Lesson 11: Filling Containers, Activity 2: What is the Shape?, Problem 1, students engage in a non-routine application problem using functions to model relationships. “The graph shows the height vs. volume function of an unknown container. What shape could this container have? Explain how you know and draw a possible container.” Students are given a graph to analyze. (8.F.B) 

Materials provide opportunities for students to independently demonstrate multiple routine and non-routine applications of the mathematics throughout the grade level. Examples include:

• Unit 4: Linear Equations and Linear Systems, Section C: Systems of Linear Equations, Lesson 16: Solving Problems with Systems of Equations, Activity 1: Cycling, Fundraising, Working, and ___? students independently engage in a non-routine application problem as they create a problem about situations involving two different relationships between the same two quantities. “Solve each problem. Explain or show your reasoning. a. Two friends live 7 miles apart. One Saturday, the two friends set out on their bikes at 8 am and started riding towards each other. One rides at 0.2 miles per minute, and the other rides at 0.15 miles per minute. At what time will the two friends meet? b. Students are selling grapefruits and nuts for a fundraiser. The grapefruits cost $1 each and a bag of nuts cost $10 each. They sold 100 items and made $307. How many grapefruits did they sell? c. Jada earns $7 per hour mowing her neighbors’ lawns. Andre gets paid $5 per hour for the first hour of babysitting and $8 per hour for any additional hours he babysits. What is the number of hours they both can work so that they get paid the same amount? d. Pause here so your teacher can review your work. Then, invent another problem that is like one of these, but with different numbers. Solve your problem. e. Create a visual display that includes: The new problem you wrote, without the solution. Enough work space for someone to show a solution. f. Trade your display with another group, and solve each other’s new problem. Make sure that you explain your solution carefully. Be prepared to share this solution with the class. g. When the group that got the problem you invented shares their solution, check that their answer is correct.” (8.EE.8)

• Unit 5: Functions and Volume, Section C: Linear Functions and Rate of Change, Lesson 9: Linear Models, Cool Down: Board Game Sales, Student Work Time, students independently engage in a routine application problem determining if a single linear model can be used. “A small company is selling a new board game, and they need to know how many to produce in the future. After 12 months, they sold 4 thousand games; after 18 months, they sold 7 thousand games; and after 36 months, they sold 15 thousand games. a. Could this information be reasonably estimated using a single linear model? b. If so, use the model to estimate the number of games sold after 48 months, If not explain your reasoning.” (8.F.B)

• Unit 8: Pythagorean Theorem and Irrational Numbers, Section B: The Pythagorean Theorem, Lesson 10: Applications of the Pythagorean Theorem, Cool Down: Jib Sail, students independently engage in a routine application problem using the Pythagorean Theorem and its converse. “Sails come in many shapes and sizes. The sail on the right is a triangle. Is it a right triangle? Explain your reasoning.” A picture of a sailboat with sail dimensions 97.5 m, 10.24 m, and 3.45 m is shown. (8.G.7)

The materials reviewed for Open Up Resources 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 each grade level. Examples include:

• Unit 1: Rigid Transformations and Congruence, Section A: Rigid Transformations, Lesson 6: Describing Transformations, Activity 1: Info Gap: Transformation Information, Are You Ready For More? Students demonstrate conceptual understanding as they communicate precise information about transformations on the coordinate plane. “Sometimes two transformations, one performed after the other, have a nice description as a single transformation. For example, instead of translating 2 units up followed by translating 3 units up, we could simply translate 5 units up. Instead of rotating 20 degrees counterclockwise around the origin followed by rotating 80 degrees clockwise around the origin, we could simply rotate 60 degrees clockwise around the origin.Can you find a simple description of reflecting across the x-axis followed by reflecting across the y-axis? (8.G.3) 

• Unit 3: Linear Relationships, Section A: Proportional Relationships, Lesson 1: Understanding Proportional Relationships, Practice Problems, Problem 2, students apply their understanding of proportional relationships as they graph a situation in context. “A you-pick blueberry farm offers 6 lbs of blueberries for $16.50. Sketch a graph of the relationship between cost and pounds of blueberries.” (8.EE.5)

• Unit 5: Functions and Volume, Section C: Linear Functions and Rates of Change, Lesson 8: Linear Functions, Practice Problems, Problem 2, students develop procedural skill and fluency as they represent linear functions with equations. “Two car services offer to pick you up and take you to your destination. Service A charges 40 cents to pick you up and 30 cents for each mile of your trip. Service B charges $1.10 to pick you up and charges c cents each mile of your trip. a. Match each service to the lines l and m. A. Service A, B. Service B, 1. Line l, 2. Line m b. For Service B, is the additional charge per mile greater or less than 30 cents per mile of the trip. Explain your reasoning.” (8.F.4) 

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

• Unit 4: Linear Equations and Linear Systems, Section C: Systems of Linear Equations, Lesson 10: On or Off the Line?, Practice Problems, Problem 3, students develop procedural skill and fluency as they apply their understanding of a point simultaneously satisfying two relationships. “Mai earns $7 per hour mowing her neighbor’s lawn. She also earned $14 for hauling away bags of recyclables for some neighbors. Priya babysits her neighbors’ children. The table shows the amount of money m she earns in h hours. Priya and Mai have agreed to go to the movies the weekend after they have earned the same amount of money for the same number of work hours. a. How many hours do they each have to work before they go to the movies? b. How much will each of them have earned? c. Explain where the solution can be seen in the tables of values, graphs, and equations that represent Priya’s and Mai’s earnings.” (8.EE.8) 

• Unit 5: Functions and Volume, Section C: Linear Functions and Rate of Change, Lesson 9: Linear Models, Practice Problems, Problem 2, students demonstrate conceptual understanding of linear models as they apply their knowledge of volume. “In science class, Jada uses a graduated cylinder with water in it to measure the volume of some marbles. After dropping in 4 marbles so they are all under water, the water in the cylinder is at a height of 10 milliliters. After dropping in 6 marbles so they are all under water, the water in the cylinder is at a height of 11 milliliters. a. What is the volume of 1 marble? b. How much water was in the cylinder before any marbles were dropped in? c. What should be the height of the water after 13 marbles are dropped in? d. Is the volume of water a linear relationship with the number of marbles dropped in the graduated cylinder? If so, what does the slope of the line mean? If not, explain your reasoning.” (8.F.4)

• Unit 7: Exponents and Scientific Notation, Section B: Exponent Rules, Lesson 7: Practice with Rational Bases, Cool Down: Working with Exponents, Problem 2, students build conceptual understanding and develop procedural skill and fluency of exponent rules. “Diego wrote 6^4 \cdot 8^3 = {48}^7. Explain what Diego’s mistake was and how you know the equation is not true.” (8.EE.1)

The materials reviewed for Open Up Resources 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. 

Students have opportunities to engage with the Math Practices across the year. Course Guide, How to Use the Materials, Noticing and Assessing Student Progress in Mathematical Practices, The Standards for Mathematical Practices Chart, “The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain Mathematical Practices. Some units, due to their size or the nature of their content, may have fewer predicted chances for students to engage in a particular Mathematical Practice. A dash in the chart indicates that there may not be enough opportunities to reliably look for this Mathematical Practice in the unit. One primary place Mathematical Practice 4 is tagged is the optional modeling lesson at the end of each unit. Aside from these lessons, optional activities and lessons are not included in this chart.” The Mathematical Practices are also identified in Lesson Preparation Narratives and Lesson Activities Narratives for some lessons.

There is intentional development of MP1 to meet its full intent in connection to grade-level content. Students make sense of problems and persevere in solving them as they work with the teacher's support and independently throughout the units. Examples include:

• Unit 2: Dilations, Similarity, and Introducing Slope, Section A: Dilations, Lesson 3, Dilations with no Grid, Lesson Narrative, “Performing dilations without a grid engages students in MP1 as they think about the meaning of dilation in terms of the given information (center, scale factor, point being dilated).” Warm Up: Points on a Ray, students use a variety of strategies to find a way to measure a ray without a grid. “a. Find and label a point C on the ray whose distance from A is twice the distance from B to A. b. FInd and label a point D on the ray whose distance from A is half the distance from B to A.”

• Unit 6: Associations in Data, Section C: Associations in Categorical Data, Lesson 9: Looking for Associations, Lesson Narrative, “In this lesson, students study categorical data displayed in two-way tables, bar graphs, and segmented bar graphs. The different graphical representations help students visualize the frequencies and relative frequencies, aiding them in making judgment about whether there is evidence of an association or not in the next lesson. While the concepts and structures in this lesson are not very complex, there are many new terms and representations, and students are given the opportunity to study them carefully so that they can make sense of them (MP1).” Activity 1: Matching Representations Card Sort, students determine if their answers make sense as they match data tables and graphs. “Your teacher will hand out some cards. Some cards show two-way tables like this: Some cards show bar graphs like this: Some cards show segmented bar graphs like this: The bar graphs and segmented bar graphs have their labels removed. a. Put all the cards that describe the same situation in the same group. b. One of the groups does not have a two way table. Make a two way table for the situation described by the graphs in the group. c. Label the bar graphs and segmented bar graphs so that the categories represented by each bar are indicated. d. Describe in your own words the kind of information shown by a segmented bar graph.” Cards have two-way tables, double bar graphs, or segmented bar graphs with data showing numbers of children by age groups who have and do not have cell phones.

• Unit 7: Lesson 15: Adding and Subtracting with Scientific Notation, Lesson Narrative, “Students add and subtract with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Students must make sense and use quantitative reasoning when making comparisons, for example, when comparing whether 5 planets side by side are wider than the Sun (MP1, MP2).” Activity 2: A Celestial Dance, students determine if their answers make sense as they attend to place value to add numbers in scientific notation. “a. When you add the distances of Mercury, Venus, Earth, and Mars from the Sun, would you reach as far as Jupiter? b. Add all the diameters of all the planets except the Sun. Which is wider, all of these objects side by side, or the Sun? Draw a picture that is close to scale.” A table shows the diameter of each planet and the sun and their distance from the sun in km.

There is intentional development of MP2 to meet its full intent in connection to grade-level content. Students reason abstractly and quantitatively as they work with the teacher's support and independently throughout the units. Examples include:

• Unit 3: Linear Relationships, Section E: Let’s Put it to Work, Lesson 14: Using Linear Relationships to Solve Problems, Instructional Routine, “Students represent a scenario with an equation and use the equation to find solutions. They create a graph (either with a table of values or by using two intercepts), interpret points on the graph, and interpret points not on the graph (MP2).” Activity 2: Fabulous Fish, students understand the relationships between problem scenarios and mathematical representations as they write and solve equations. “The Fabulous FIsh Market orders tilapia, which costs $3 per pound, and salmon, which cost $5 per pound. The market budgets $210 to spend on this order each day. a, What are five different combinations of salmon and tilapia that the market can order? b. Define variables and write an equation representing the relationship between the amount of each fish bought and how much the market spends. c. Sketch a graph of the relationship. Label your axes. d. On your graph, plot and label the combinations A-F. Which of these combinations can the market order? Explain or show your reasoning.” 

• Unit 4: Linear Equations and Linear Systems, Section C: Systems of Linear Equations, Lesson 12: System of Equations, Instructional Routine, “Students explore a system of equations with no solutions in the familiar context of cup stacking. The context reinforces a discussion about what it means for a system of equations to have no solutions, both in terms of a graph and in terms of the equations (MP2).” Activity 2: Stacks of Cups, students explain what numbers in an equation represent as they graph systems of equations to find solutions. “A stack of n small cups has a height, h, in centimeters of h = 1.5n + 6. A stack of n large cups has a height, h, in centimeters of h = 1.5n + 9. a. Graph the equations for each cup on the same set of axes. Make sure to label the axes and decide on an appropriate scale. b. For what number of cups will the two stacks have the same height?”

• Unit 7: Exponents and Scientific Notation, Section C: Scientific Notation, Lesson 12: Applications of Arithmetic with Powers of 10, Instructional Routine, “Students use numbers and exponents flexibly and interpret their results in context (MP2).” Activity 2, students attend to the meaning of quantities as they compare large quantities. “In 2016, the Burj Khalifa was the tallest building in the world. It was very expensive to build. Consider the following question: Which is taller, the Burj Khalifa or a stack of the money it cost to build the Burj Khalifa? a. What information would you need to be able to solve the problem? b. Record the information your teacher shares with the class. c. Answer the question, ‘Which is taller, the Burj Khalifa or a stack of money it cost to build the Burj Khalifa?’ and explain or show your reasoning. d. Decide what power of 10 to use to label the rightmost tick mark of the number line, and plot the height of the stack of the money and the height of the Burj Khalifa. e. Which has more mass, the Burj Khalifa or the mass of the pennies it cost to build the Burj Khalifa? What information do you need to answer this? f. Decide what power of 10 to use to label the rightmost tick mark of the number line, and plot the mass of the Burj Khalifa and the mass of the pennies it cost to build the Burj Khalifa.”

The materials reviewed for Open Up Resources 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.

Students have opportunities to engage with the Math Practices across the year. Course Guide, How to Use the Materials, Noticing and Assessing Student Progress in Mathematical Practices, The Standards for Mathematical Practices Chart, “The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain Mathematical Practices. Some units, due to their size or the nature of their content, may have fewer predicted chances for students to engage in a particular Mathematical Practice. A dash in the chart indicates that there may not be enough opportunities to reliably look for this Mathematical Practice in the unit. One primary place Mathematical Practice 4 is tagged is the optional modeling lesson at the end of each unit. Aside from these lessons, optional activities and lessons are not included in this chart.” The Mathematical Practices are also identified in Lesson Preparation Narratives and Lesson Activities Narratives for some lessons.

Students construct viable arguments in connection to grade-level content as they work with the teacher's support and independently throughout the units. Examples include:

• Unit 1: Rigid Transformations and Congruence, Section C: Congruence, Lesson 13: Congruence, Instructional Routines, “There are two likely strategies for identifying corresponding points on the two corresponding figures: Looking for corresponding parts of the figures such as the line segments. Performing rigid motions with tracing paper to match the figures up. Both are important. Watch for students using each technique and invite them to share during the discussion.” Activity 2: Corresponding Points in Congruence, “Here are two congruent shapes with some corresponding points labeled. a. Draw the points corresponding to B, D, and E, and label them B’, D’, and E’. b. Draw line segments. AD and A’D’ and measure them. Do the same for segments BC and B’C’ and for segments AE and A’E’. What do you notice? c. Do you think there could be a pair of corresponding segments with different lengths? Explain.” 

• Unit 2: Dilatons, Similarity, and Slope, Section A: Dilations, Lesson 2: Circular Grid, Instructional Routines, “The purpose of this activity is to begin to think of a dilation with a scale factor as a rule or operation on points in the plane. Students work on a circular grid with a center of dilation at the center of the grid. They examine what happens to different points on a given circle when the dilation is applied and observe that these points all map to another circle whose radius is scaled by the scale factor of the dilation. For example, if the scale factor is 3 and the points lie on a circle whose radius is 2 grid units, then the dilated points will all lie on a circle whose radius is 6 units. Students need to explain their reasoning for finding the scale factor (MP3).” Activity 1: A Droplet on the Surface, Problem 3, “The center of dilation is point P. What is the scale factor that takes the smaller circle to the larger circle? Explain your reasoning.”

• Unit 3: Linear Relationships, Section A: Proportional Relationships, Lesson 3: Representing Proportional Relationships, Instructional Routines, “The purpose of this activity is for students to graph a proportional relationship when given a blank pair of axes. They will need to label and scale the axes appropriately before adding the line representing the given relationship. In each problem, students are given two representations and asked to create two more representations so that each relationship has a description, graph, table, and equation. Then, they explain how they know these are different representations of the same situation (MP3). In the next lesson, students will use these skills to compare two proportional relationships represented in different ways.” Activity 1: Representations of Proportional Relationships, Problem 1, “Here are two ways to represent a situation. Description: Jada and Naoh counted the number of steps they took to walk a set distance. To walk the same distance, Jada took 8 steps. Noah took 10 steps. Equation: Let x represent the number of steps Jada takes and let y represent the number of steps Noah takes. y = \frac{5}{4}x. Then they found that when Noah took 15 steps, Jada took 12 steps. a. Create a table that represents this situation with at least 2 pairs of values. b. Graph this relationship and label the axes. c. How can you see or calculate the constant of proportionality in each representation? What does it mean? d. Explain how you can tell the equation, description, graph, and table all represent the same situation.” 

Students critique others' reasoning concerning grade-level content as they work with the teacher's support and independently throughout the units. Examples include:

• Unit 2: Dilations, Similarity, and Introducing Slope, Section B: Similarity, Lesson 7: Similar Polygons, Instructional Routines, “In the previous lesson, students saw that figures are similar when there is a sequence of translations, rotations, reflections, and dilations that map one figure onto the other. This activity focuses on some common misconceptions about similar figures, and students have an opportunity to critique the reasoning of others (MP3).” Activity 1: Are They Similar, Problem 1, “Let’s look at a square and a rhombus. Priya says, “These polygons are similar because their side lengths are all the same.” Clare says, “These polygons are not similar because the angles are different.” Do you agree with either Priya or Clare? Explain your reasoning.”

• Unit 4: Linear Equations and Linear Systems, Section B: Linear Equations in One Variable, Lesson 3: Balanced Moves, Lesson Narrative, “In this lesson students move from using hangers to using equations in order to represent a problem. In the Warm Up they match a series of hangers with the corresponding series of equations. They see how moves that maintain the balance of a hanger correspond to moves that maintain the equality of an equation, such as halving the value of each side or subtracting the same unknown value from each side. In the next activity students match pairs of equations with the corresponding equation move—performing the same operation on each side—that produces the second from the first. In the activity after that, they compare different choices of moves that lead to the same solution. In this activity the solution is negative, which would not have been representable with hangers. Students can check that it is a solution by substituting into the equation, reinforcing the idea that a solution is a number that makes the equality in an equation true, and that different moves maintain the equality. As students reason about why the steps in solving an equation maintain the equality and compare different solution methods, they engage in MP3.” Activity 2: Keeping Equality, Problem 1, “Noah and Lin both solved the equation 14a = 2(a - 3). Do you agree with either of them? Why? Noah’s solution: 14a = 2(a - 3), 14a = 2a - 6, 12a = -6, a = -12. Lin’s solution: 14a = 2(a - 3), 71 = a - 3, 6a = -3, a = 0 - \frac{1}{2}."

• Unit 7: Exponents and Scientific Notation, Section B: Exponent Rules, Lesson 4: Powers of 10, Instructional Routines, “Students extend exponent rules to discover why it makes sense to define {10}^0 as 1. Students create viable arguments and critique the reasoning of others when discussing Noah’s argument that {10}^0 should equal 0 (MP3).” Activity 2: Zero Exponent, Problem 4, “Noah says, ‘If I try to write {10}^0 expanded, it should have zero factors that are 10, so it must be equal to 0.’ Do you agree? Discuss with your partner.”

The materials reviewed for Open Up Resources 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.

Students have opportunities to engage with the Math Practices across the year. Course Guide, How to Use the Materials, Noticing and Assessing Student Progress in Mathematical Practices, The Standards for Mathematical Practices Chart, “The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain Mathematical Practices. Some units, due to their size or the nature of their content, may have fewer predicted chances for students to engage in a particular Mathematical Practice. A dash in the chart indicates that there may not be enough opportunities to reliably look for this Mathematical Practice in the unit. One primary place Mathematical Practice 4 is tagged is the optional modeling lesson at the end of each unit. Aside from these lessons, optional activities and lessons are not included in this chart.” The Mathematical Practices are also identified in Lesson Preparation Narratives and Lesson Activities’ Narratives for some lessons.

There is intentional development of MP4 to meet its full intent in connection to grade-level content. Students make sense of problems and persevere in solving them as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 2: Dilations, Similarity, and Introducing Slope, Section D: Let’s Put It to Work, Lesson 13: The Shadow Knows, Lesson Narrative, “This lesson involves modeling (MP4), not only because students interpret real-world data (both the given heights and shadow lengths and the measurements that they take themselves) but also because they need to make simplifying assumptions in order to justify why the relationship is proportional.” Activity 3: The Height of a Tall Object, students put a situation in their own words and identify important information as they measure the perpendicular height of objects outside. “1. Head Outside. Make sure that it is a sunny day and you take a measuring device (like a tape measure or meter stick) as well as a pencil and some paper. 2. Choose an object whose height is too large to measure directly. Your teacher may assign you an object. 3. Use what you have learned to figure out the height of the object! Explain or show your reasoning.” 

• Unit 6: Associations in Data, Section B: Associations in Numerical Data, Lesson 6: The Slope of a Fitted Line, Instructional Routine, “This activity returns to scatter plots without linear models given. Students determine whether the data seems to have a linear association or not. If it does, students are asked to decide whether the variables have a positive or negative association (MP4).” Activity 3: Positive or Negative, students use the math they know to determine if scatter plots have a positive, negative, or no association. “Problem 1: For each of the scatter plots, decide whether it makes sense to fit a linear model to the data. If it does, would the graph of the model have a positive slope, a negative slope, or a slope of zero? Problem 2: Which of these scatter plots show evidence of a positive association between the variables? Of a negative association? Which do not appear to show an association?” 

• Unit 8: Pythagorean Theorem and Irrational Numbers, Section E: Let’s Put it to Work, Lesson 16: When Is the Same Size Not the Same Size?, Lesson Narrative, “There is an element of mathematical modeling (MP4) in the last activity, because in order to quantify the screens’ sizes to compare them, students need to refine the question that is asked.” Activity 2: The Screen is the Same Size…Or Is it?, students appropriately model the situation as they use ratios and the Pythagorean Theorem to compare screen sizes. “Before 2017, a smart phone manufacturer’s phones had a diagonal length of 5.8 inches and an aspect ratio of 16 : 9. In 2017, they released a new phone that also had a 5.8-inch diagonal length, but an aspect ratio of 18.5 : 9. Some customers complained that the new phones had a smaller screen. Were they correct? If so, how much smaller was the new screen compared to the old screen?” 

There is intentional development of MP5 to meet its full intent in connection to grade-level content. Students use appropriate tools strategically as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 1: Rigid Transformations and Congruence, Section A: Rigid Transformations, Lesson 1: Moving in the Plane, Lesson Narrative, “In all of the lessons in this unit, students should have access to their geometry toolkits, which should contain tracing paper, graph paper, colored pencils, scissors, ruler, protractor, and an index card. For this unit, access to tracing paper and a straight edge are particularly important. Students may not need all (or even any) of these tools to solve a particular problem. However, to make strategic choices about when to use which tools (MP5), students need to have opportunities to make those choices. Apps and simulations should supplement rather than replace physical tools.” Activity 1: Triangle Square Dance, students use appropriate models and strategies as they describe translations and rotations. “Your teacher will give you three pictures. Each shows a different set of dance moves. 1. Arrange the three pictures so one of you can see them right way up. Choose who will start the game. The starting player chooses one of the three applets below and describes the dance to the other player. The other player identifies which dance is being talked about: A, B, or C. 2. After one round, trade roles. When you have described all three dances, come to an agreement on the words you use to describe the moves in each dance. 3. With your partner, write a description of the moves in each dance.” 

• Unit 2: Dilations, Similarity, and Introducing Slope, Section A: Dilations, Lesson 2: Circular Grid, Lesson Narrative, “A circular grid is an effective tool for performing a dilation. A circular grid has circles with radius 1 unit, 2 units, and so on all sharing the same center. Students experiment with dilations on a circular grid, where the center of dilation is the common center of the circles. By using the structure of the grid, they make several important discoveries about the images of figures after a dilation including: Each grid circle maps to a grid circle. Line segments map to line segments and, in particular, the image of a polygon is a scaled copy of the polygon. The next several lessons will examine dilations on a rectangular grid and with no grid, solidifying student understanding of the relationship between a polygon and its dilated image. This echoes similar work in the previous unit investigating the relationship between a figure and its image under a rigid transformation. As with previous geometry lessons, students should have access to geometry toolkits so they can make strategic choices about which tools to use (MP5).” Activity 1: A Droplet on the Surface, Problem 1, students use appropriate tools and strategies as they dilate circles. “The larger circle d is a dilation of the smaller circle c. P is the center of dilation. a. Draw four points on the smaller circle using the Point on Object tool. b. Draw the rays from P through each of those four points. Select the Ray tool, then point P, and then the second point. c. Mark the intersection points of the rays and circle d by selecting the Intersect tool and clicking on the point of intersection.” 

• Unit 4: Linear Equations and Linear Situations, Section A: Puzzle Problems, Lesson 1: Number Puzzles, Instructional Routines, “The purpose of this activity is for students to solve number puzzles using any representation they choose. Students then make sense of other representations for the same problems, starting with those of a partner. The whole-class discussion should focus on the strengths and weaknesses of different representations (MP5). For example, tape diagrams only work for problems with all positive values, so you could use one for the distance puzzle, but a tape diagram would not work for the temperature puzzle.” Activity1: Telling Temperatures, students use appropriate tools and strategies as they solve number puzzles. “Solve each puzzle. Show your thinking. Organize it so it can be followed by others. 1. The temperature was very cold. Then the temperature doubled. Then the temperature dropped by 10 degrees. Then the temperature increased by 40 degrees. The temperature is now 16 degrees. What was the starting temperature? 2. Lin ran twice as far as Diego. Diego ran 300 m farther than Jada. Jada ran \frac{1}{3} the distance that Noah ran. Noah ran 1200 m. How far did Lin run?”

The materials reviewed for Open Up Resources 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 have opportunities to engage with the Math Practices across the year. Course Guide, How to Use the Materials, Noticing and Assessing Student Progress in Mathematical Practices, The Standards for Mathematical Practices Chart, “The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain Mathematical Practices. Some units, due to their size or the nature of their content, may have fewer predicted chances for students to engage in a particular Mathematical Practice. A dash in the chart indicates that there may not be enough opportunities to reliably look for this Mathematical Practice in the unit. One primary place Mathematical Practice 4 is tagged is the optional modeling lesson at the end of each unit. Aside from these lessons, optional activities and lessons are not included in this chart.” The Mathematical Practices are also identified in Lesson Preparation Narratives and Lesson Activities Narratives for some lessons.

There is intentional development of MP6 to meet its full intent in connection to grade-level content. Students attend to precision as they work with the teacher's support and independently throughout the units. Examples include:

• Unit 6: Associations in Data, Section B: Associations in Numerical Data, Lesson 3: What a Point in a Scatter Plot Means, Lesson Narrative, “In this lesson, students continue their investigation of scatter plots. They interpret points in a scatter plot in terms of a context, and add points to a scatter plot given information about an individual in the population. They compare individuals represented by different points and informally discuss trends in the data. There are two levels of analysis needed to successfully make sense of scatter plots: what is happening for a particular individual, and what is happening at a global level for the entire population. The ability to move between these two zoom levels develops over time. In this lesson, students spend a lot of time looking at the details of a scatter plot, naming the quantities represented in a scatter plot, and focusing on the meaning of individual points (MP6).” Activity 2: Coat Sales, students attend to precision as they analyze average temperatures and coat sales. “ a. What does the point (15, 680) represent? b. For the month with the lowest average temperature, estimate the total amount made from coat sales. Explain how you used the table to find this information. c. For the month with the smallest coat sales, estimate the average monthly temperature. Explain how you used the scatter plot to find this information. d. If there were a point at (0, A) what would it represent? Use the scatter plot to estimate a value for A. e. What would a point at (B, 0) represent? Use the scatter plot to estimate a value for B. f. Would it make sense to use this trend to estimate the value of sales when the average monthly temperature is 60 degrees Celsius? Explain your reasoning.” 

• Unit 7: Exponents and Scientific Notation, Section C: Scientific Notation, Lesson 9: Describing Large and Small Numbers Using Powers of 10. Lesson Narrative, “This lesson serves as a prelude to scientific notation and builds on work students have done in previous grades with numbers in base ten. Students use base-ten diagrams to represent different powers of 10 and review how multiplying and dividing by 10 affect the decimal representation of numbers. They use their understanding of base-ten structure as they express very large and very small numbers using exponents. Students also practice communicating—describing and writing—very large and small numbers in an activity, which requires attending to precision (MP6). This leads to a discussion about how powers of 10 can be used to more easily communicate such numbers.” Activity 1: Base-ten Representations Matching, Problem 2, students attend to precision as they work with the place value structure of scientific notation. “a. Write an expression to describe the base-ten diagram if each small square represents {10}^{-4}. What is the value of this expression? b. How does changing the value of the small square change the value of the expression? Explain or show your thinking. c. Select at least two different powers of 10 for the small square, and write the corresponding expressions to describe the base-ten diagram. What is the value of each of your expressions?”

• Unit 8: Pythagorean Theorem and Irrational Numbers, Section A: Side Lengths and Areas of Squares, Lesson 3: Rational and Irrational Numbers, Lesson Narrative, “In this lesson, students build on their work with square roots to learn about a new mathematical idea, irrational numbers. Students recall the definition of rational numbers (MP6) and use this definition to search for a rational number x such that x^2 = 2. Students should not be left with the impression that looking for and failing to find a rational number whose square is 2 is a proof that \sqrt{2} is irrational; this exercise is simply meant to reinforce what it means to be irrational and to provide some plausibility for the claim. Students are not expected to prove that \sqrt{2} is irrational in grade 8, and so ultimately must just accept it as a fact for now.” Activity 2: Looking for a Solution, students attend to precision as they investigate the value of \sqrt{2}. “Are any of these numbers a solution to the equation x^2 = 2? Explain your reasoning. a. 1 b. \frac{1}{2} c. \frac{3}{2} d. \frac{7}{5}.”

Students attend to the specialized language of mathematics as they work with the teacher's support and independently throughout the units. Examples include:

• Unit 1: Rigid Transformations, and Congruence, Section A: Rigid Transformations, Lesson 2: Naming the Moves, Lesson Narrative, “In this lesson, students begin to describe a given translation, rotation, or reflection with greater precision and are introduced to the terms translation, rotation, and reflection. The collective terms ‘transformation’ and ‘rigid transformation’ are not used until later lessons. Students are introduced to the terms clockwise and counterclockwise. Students then use this language to identify the individual moves on various figures. Students engage in MP6 as they experiment with ways to describe moves precisely enough for another to understand their meaning.” Cool Down: Is it a Reflection? students use the specialized language of mathematics as they describe a figure’s transformation. “What type of move takes Figure A to Figure B? Explain your reasoning.”

• Unit 2: Dilations, Similarity and Introducing Slope, Section A: Dilations, Lesson 4: Dilations on a Square Grid, Lesson Narrative, “Students continue to find dilations of polygons, providing additional evidence that dilations map line segments to line segments and hence polygons to polygons. The scale factor of the dilation determines the factor by which the length of those segments increases or decreases. Using coordinates to describe points in the plane helps students develop language for precisely communicating figures in the plane and their images under dilations (MP6).” Activity 2: Card Sort: Matching Dilations on a Coordinate Grid, students use the specialized language of mathematics as they dilate figures and describe the center of dilation and vertices. “Your teacher will give you some cards. Each of Cards 1 through 6 shows a figure in the coordinate plane and describes a dilation. Each of Cards A through E describes the image of the dilation for one of the numbered cards. Match number cards with letter cards. One of the number cards will not have a match. For this card, you’ll need to draw an image.” 

• Unit 5: Functions and Volume, Section B: Representing and Interpreting Functions, Lesson 6: Even More Graphs of Functions, Lesson Narrative, “In the following activity, students identify independent and dependent variables from contexts and select an appropriate graph to match their choices. Different choices are possible, so students must be precise about which choice they are making and explain how the choice relates to the graph (MP6).” Activity 2: Sketching a Story About a Boy and a Bike, students use the specialized language of mathematics as they define independent and dependent variables as they sketch a graph to represent a situation. “Your teacher will give you tools for creating a visual display. With your group, create a display that shows your response to each question. Here is a story: ‘Noah was at home. He got on his bike and rode to his friend’s house and stayed there for a while. Then he rode home again. Then he rode to the park. Then he rode home again.’ a. Sketch a graph of this story. b. What are the two quantities? Label the axes with their names and units of measure. (For example, if this were a story about pouring water into a pitcher, one of your labels might say ‘volume (liters).’) c. Which quantity is a function of which? Explain your reasoning. d. Based on your graph, is his friend’s house or the park closer to Noah’s home? Explain how you know. e. Read the story and all your responses again. Does everything make sense? If not, make changes to your work.”

The materials reviewed for Open Up Resources 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 have opportunities to engage with the Math Practices across the year. Course Guide, How to Use the Materials, Noticing and Assessing Student Progress in Mathematical Practices, The Standards for Mathematical Practices Chart, “The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain Mathematical Practices. Some units, due to their size or the nature of their content, may have fewer predicted chances for students to engage in a particular Mathematical Practice. A dash in the chart indicates that there may not be enough opportunities to reliably look for this Mathematical Practice in the unit. One primary place Mathematical Practice 4 is tagged is the optional modeling lesson at the end of each unit. Aside from these lessons, optional activities and lessons are not included in this chart.” The Mathematical Practices are also identified in Lesson Preparation Narratives and Lesson Activities’ Narratives for some lessons.

There is intentional development of MP7 to meet its full intent in connection to grade-level content. Students look for and use the structure as they work with the teacher's support and independently throughout the units. Examples include:

• Unit 1: Rigid transformations and Congruence, Section A: Rigid Transformations, Lesson 3: Grid Moves, Lesson Narrative, “Prior to this lesson, students have learned the names for the basic moves (translation, rotation, and reflection) and have learned how to identify them in pictures. In this lesson, they apply translations, rotations, and reflections to figures. They also label the image of a point P as P’. While not essential, this practice helps show the structural relationship (MP7) between a figure and its image. Students also encounter the isometric grid (one made of equilateral triangles with 6 meeting at each vertex). They perform translations, rotations, and reflections both on a square grid and on an isometric grid. Expect a variety of approaches, mainly making use of tracing paper (MP5) but students may also begin to notice how the structure of the different grids helps draw images resulting from certain moves (MP7).” Activity 1: Transformation Information, Problem 1, students look for patterns or structures as they interpret necessary information to perform a transformation and draw an image resulting from the transformation. “Follow the directions below each statement to tell GeoGebra how you want the figure to move. It is important to notice that GeoGebra uses vectors to show translations. A vector is a quantity that has magnitude (size) and direction. It is usually represented by an arrow. These applets are sensitive to clicks. Be sure to make one quick click, otherwise it may count a double-click. After each example, click the reset button, and then move the slider over for the next question. Translate triangle ABC so that A goes to A’. a. Select the Vector tool. b. Click on the original point A and A’ then the new point . You should see a vector. c. Select the Translate by Vector tool. d. Click on the figure to translate, and then click on the vector. Translate triangle ABC so that C goes to C’. Rotate triangle ABC 90° counterclockwise using center O. a. Select the Rotate around Point tool. b. Click on the figure to rotate, and then click on the center point. c. A dialog box will open; type the angle by which to rotate and select the direction of rotation. d. Click on ok. Reflect triangle ABC using line l. a. Select the Reflect about Line tool. b. Click on the figure to reflect, and then click on the line of reflection.” 

• Unit 2: Dilations, Similarity, and Introducing Slope, Section A: Dilations, Lesson 4: Dilations on a Square Grid, Lesson Narrative, “Students continue to find dilations of polygons, providing additional evidence that dilations map line segments to line segments and hence polygons to polygons. The scale factor of the dilation determines the factor by which the length of those segments increases or decreases. Using coordinates to describe points in the plane helps students develop language for precisely communicating figures in the plane and their images under dilations (MP6). Strategically using coordinates to perform and describe dilations is also a good example of MP7.” Cool Down: A Dilated Image, students look for and explain the structure within mathematical representations as they apply a dilation to a polygon where the center of dilation is on the interior of the figure. “Draw the image of rectangle ABCD under dilation using center P and scale factor \frac{1}{2}.” 

• Unit 4: Linear Equations and Linear Systems, Section B: Linear Equations in One Variable, Lesson 8: How Many Solutions?, Lesson Narrative, “In the previous lesson, students learned that sometimes an equation has one solution, sometimes no solution, and sometimes infinitely many solutions. The purpose of this lesson is to help students identify structural features of an equation that tell them which of these outcomes will occur when they solve it. They also learn to stop solving an equation when they have reached a point where it is clear which of the outcomes will occur, for example when they reach an equation like 6x + 2 = 6x + 5 (no solution) or 6x + 2 = 6x + 2 (infinitely many solutions). When students monitor their progress in solving an equation by paying attention to the structure at each step, they engage in MP7.” Activity 2: Make Use of Structure, Problem 1, students compare the structure of equations that have no solution, one solution, and infinitely many solutions. “For each equation, determine whether it has no solutions, exactly one solution, or is true for all values of x (and has infinitely many solutions). If an equation has one solution, solve to find the value of x that makes the statement true. a. 6x + 8 = 7x + 12 b. 6x + 8 = 2(3x + 4) c. 6x + 8 = 6x + 13”

There is intentional development of MP8 to meet its full intent in connection to grade-level content. Students look for and express regularity in repeated reasoning as they work independently with the teacher's support throughout the units. Examples include:

• Unit 3: Linear Relationships, Section B: Representing Linear Relationships, Lesson 5: Introduction to Linear Relationships, Instructional Routines, “In this task, students are presented with a situation that leads to a linear relationship that is not proportional because there is a non-zero starting amount. By trying to answer the question, ‘How many cups are needed to get to a height of 50 cm?’ the students explore the rate of change, which is the increase per cup after the first cup. The rate of change can be seen in the graph as the slope. Students use representations and ideas from previous lessons on proportional relationships. They use tables and graphs to represent the situation and make deductions by generalizing from repeated reasoning (MP8), arguing that each additional cup increases the height of the stack by the same amount. As students are working, suggest that students make a graph or a table if they are stuck or if they have trouble explaining their reasoning. Students should be prepared to share their strategies with the class.” Activity 1: Stacking Cups, students make generalizations about rate of change as they describe a linear relationship. “We have two stacks of styrofoam cups. One stack has 6 cups, and its height is 15 cm. The other stack has 12 cups, and its height is 23 cm. How many cups are needed for a stack with a height of 50 cm?” 

• Unit 7: Exponents and Scientific Notation, Section B: Exponent Rules, Lesson 5: Negative Exponents with Powers of 10, Lesson Narrative, “In analogy to positive powers of 10 that describe repeated multiplication by 10, this lesson presents negative powers of 10 as repeated multiplication by \frac{1}{10}, leading ultimately to the rule 10^{-n} = \frac{1}{10n}. Students use repeated reasoning to generalize about negative exponents (MP8). Students create viable arguments and critique the reasoning of others when comparing and contrasting, for example, (10^{-2})^3 and (10^{2})^{-3} (MP3). With this understanding of negative exponents, all of the exponent rules created so far are seen to be valid for any integer exponents.” Activity 1: Negative Exponent Table, students use the general idea exponent rules to include negative exponents. “a. Complete the table to explore what negative exponents mean. b. As you move toward the left, each number is being multiplied by 10. What is the multiplier as you move right? c. How does a multiplier of 10 affect the placement of the decimal in the product?  How does the other multiplier affect the placement of the decimal in the product?  d. Use the patterns you found in the table to write 10^{-7} as a fraction. e. Use the patterns you found in the table to write 10^{-5} as a decimal. f. Write \frac{1}{100,000,000} using a single exponent. g. Use the patterns in the table to write 10^{-n} as a fraction.”

• Unit 8: Pythagorean Theorem and Irrational Numbers, Section B: The Pythagorean Theorem, Lesson 6: Finding Side Lengths of Triangles, Lesson Narrative, “In the Warm Up for this lesson, students notice and wonder about 4 triangles. While there is a lot to notice, one important aspect is whether the triangle is a right triangle or not. This primes them to notice patterns of right and non-right triangles in the other activities in the lesson. In the next two activities, students systematically look at the side lengths of right and non-right triangles for patterns (MP8).” Activity 2: Meet the Pythagorean Theorem, Problem 1, students make generalizations about triangles as they use squares or count grid units to find side lengths and check whether the Pythagorean identity a^2 + b^2 = c^2 holds or not. “Find the missing side lengths. Be prepared to explain your reasoning. Find the missing side lengths. Be prepared to explain your reasoning.” Students are shown three different triangles on grid paper with side measurements labeled.