8th Grade - Gateway 2

The materials reviewed for Math Nation Grade 8 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. Materials develop conceptual understanding throughout the grade level and materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade level. There are opportunities for students to develop their conceptual understanding in the various parts of each lesson: Warm-up, Exploration Activities, Lesson Synthesis, Cool Down, Check Your Understanding, and Practice Problems. Additionally, students’ conceptual understanding was assessed on Mid-Unit Assessments and End-of-Unit Assessments.

Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. Course Guide, “Concepts Develop from Concrete to Abstract, Mathematical concepts are introduced simply, concretely, and repeatedly, with complexity and abstraction developing over time. Students begin with concrete examples, and transition to diagrams and tables before relying exclusively on symbols to represent the mathematics they encounter.” Examples include:

• Unit 3, Lesson 4, Cool-Down, students develop conceptual understanding by comparing two mixtures, represented in different ways, and determine which mixture is saltier (8.EE.5). “Here are recipes for two mixtures of salt and water that taste different. Mixture A: (A table has the following columns) salt (teaspoons) 4, 7, 9 and water(cups) 5, 8\frac{3}{4}, 11\frac{1}{4} Mixture B is defined by the equation y=2.5x, where x is the number of teaspoons of salt and y is the number of cups of water. 1. If you used 10 cups of water, which mixture would use more salt? How much more? Explain or show your reasoning. 2. Which mixture tastes saltier? Explain how you know.”

• Unit 5, Lesson 1, 5.1.1 Warm-Up, students develop conceptual understanding as they work to figure out what value makes a statement true (8.F.1). “Study the statement carefully. 12\div3=4 because 12=4\cdot3 6\div0=x because 6=x\cdot0 What value can be used in place of x to create true statements? Explain your reasoning.”

• Unit 8, Lesson 7, 8.7.2 Exploration Activity, students develop conceptual understanding as they prove the Pythagorean Theorem using area (8.G.6). “Both figures shown here are squares with a side length of a+b. Notice that the first figure is divided into two squares and two rectangles.  The second figure is divided into a square and four right triangles with legs of lengths a and b. Let’s call the hypotenuse of these triangles c. 1. What is the total area of each figure? 2. Find the area of the 9 smaller regions shown in the figures and label them. 3. Add up the area of the four regions in Figure F and set this expression equal to the sum of the areas of the five regions in Figure G. If you rewrite this equation using as few terms as possible, what do you have?” 

The materials provide students with opportunities to engage independently with concrete and semi-concrete representations while developing conceptual understanding. Examples include:

• Unit 1, Lesson 5, 1.5.2 Exploration Activity, students build conceptual understanding as they explain the effects of translations and reflections of points on a coordinate plane (8.G.3). “1. Five points are plotted on the coordinate plane. a. Using the Pen tool or the Text tool, label each with its coordinates. b. Using the x-axis as the line of reflection, plot the image of each point. c. Label the image of each point with its coordinates. d. Include a label using a letter. For example, the image of point A should be labeled A'. 2. If the point (13, 10) were reflected using the x-axis as the line of reflection, what would be the coordinates of the image? What about (13, -20)? (13, 570)? Explain how you know. 3. The point R has coordinates (3, 2). a. Without graphing, predict the coordinates of the image of point R if point R was reflected using the x-axis as the line of reflection. b. Check your answer by finding the image of R on the graph. c. Label the image of point R as R'. d. What are the coordinates of R'? 4. Suppose you reflect a point using the y-axis as line of reflection. How would you describe its image?”

• Unit 2, Lesson 10, 2.10.7 Practice Problems, Question 1, students develop conceptual understanding by assigning the correct slope to the correct line by comparing them (8.EE.6).  “Of the three lines in the graph, one has slope 1, one has slope 2, and one has slope \frac{1}{5}. Label each line with its slope.” 

• Unit 8, Lesson 7, 8.7.7 Practice Problems, Question 2, students develop conceptual understanding by explaining a proof of the Pythagorean Theorem using area (8.G.6). “Use the areas of the two identical squares to explain why 5^2+12^2=13^2 without doing any calculations.” 

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

There are opportunities for students to develop their procedural skills and fluency throughout the grade levels in each lesson, these opportunities can be found in the Warm-up, Exploration Activities, and Practice Problems. Examples include: Examples include:

• Unit 4, Lesson 4, 4.4.3 Exploration Activity, students develop procedural fluency and skills as they solve linear equations involving the distributive property (8.EE.7). “1. \frac{12+6x}{3}=\frac{5-9}{2}; 2. x-4=\frac{1}{3}(6x-54); 3. -(3x-12)=9x-4.” 

• Unit 7, Lesson 14, 7.14.7 Practice Problems, Question 1, students develop procedural skills and fluency as they solve problems with scientific notation (8.EE.4). “Evaluate each expression. Use scientific notation to express your answer. 1. (1.5 10^2 )(5 10^10); 2. \frac{4.8\times10^{-8}}{3\times10^{-3}}; 3. (5\times10^8)(4\times10^3); 4. (7.2\times10^3)\div(1.2\times10^5)”. 

• Unit 8, Lesson 13, 8.13.1 Warm-Up, students develop procedural skills and fluency as they solve problems with roots (8.EE.2). “Decide if each statement is true or false. (\sqrt[3]{5})^3=5; (\sqrt[3]{27})^3=3; 7=(\sqrt[3]{7})^3; (\sqrt[3]{10})^3=1,000; (\sqrt[3]{64})=2^3.” 

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

• Unit 5, Test Yourself! Practice Tool, students develop procedural skills as they find the volume of a cylinder (8.G.9). “Fill in the required values to find the volume of the cylinder. V=\pi(\frac{d}{2})^2h; V=\pi(\frac{8}{2})^2; V=\pi\cdot16\cdot10; V=_____\pi.” 

• Unit 7, Lesson 7, Cool-Down, Question 1, students independently demonstrate procedural skills and fluency by rewriting expressions using the properties of integer exponents (8.EE.1). “1. Rewrite each expression using a single, positive exponent: a. \frac{9^3}{9^9}; b. 14^{-3}\cdot14^{12}” 

• Unit 8, End-of-Unit Assessment (A), Question 3, students convert a decimal expansion which repeats into a rational number (8.NS.1). “Which of these is equal to 0.\overline{13}? A. \frac{13}{99}; B. \frac{12}{90}; C. \frac{1}{3}; D. 1\frac{1}{3}”

The materials reviewed for Math Nation Grade 8 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Materials provide opportunities for students to independently demonstrate multiple routine and non-routine applications of mathematics throughout the grade level. Applications of mathematics occur throughout a lesson in the exploration activities, practice problems, and assessments.

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

• Unit 3, Lesson 4, 3.4.1 Warm-up, students write a description of a situation based on a given equation and decide what the variables represent mean in the context of the situation  (8.EE.B). “The equation y=4.2x could represent a variety of different situations. 1. Write a description of a situation represented by this equation. Decide what quantities x and y represent in your situation.” 

• Unit 5, Lesson 20, 5.20.5 Practice Problems, Question 6, students solve routine word problems using systems of equations (8.EE.8). “While conducting an inventory in their bicycle shop, the owner noticed the number of bicycles is 2 fewer than 10 times the number of tricycles.  They also know there are 410 wheels on all the bicycles and tricycles in the store. Write and solve a system of equations to find the number of bicycles in the store.” 

• Unit 8, End-of-Unit Assessment (A), Question 7, students apply the Pythagorean Theorem to solve a word problem involving surface area and volume (8.G.7 and 8.G.9). “Elena wonders how much water it would take to fill her cup. She drops her pencil in her cup and notices that it just fits diagonally. (See the diagram.) The pencil is 17 cm long and the cup is 15 cm tall. How much water can the cup hold? Explain or show your reasoning.”(The surface area and volume of a cylinder formulas are given.)

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

• Unit 3, Lesson 7, 3.7.2 Exploration Activity, students use proportional relationships to figure out how many identical solid objects fit within a cylinder (8.EE.B). “1. What is the volume, V, in the cylinder after you add: a. 3 objects? b. 7 objects? c. x objects? Explain your reasoning. 2. If you wanted to make the water reach the highest mark on the cylinder, how many objects would you need? 3. Plot and label points that show your measurements from the experiment. 4. Plot and label a point that shows the depth of the water before you added any objects. 5. The points should fall on a line. Use the Line tool to draw this line. 6. Compute the slope of the line using several different triangles. Does it matter which triangle you use to compute the slope? Why or why not? 7. The equation of the line in the experiment has two numbers and two variables. What physical quantities do the two numbers represent? What does V represent and what does x represent?”

• Unit 4, Lesson 12, 4.12.5 Practice Problems, Question 4, students independently solve routine problems using linear equations (8.EE.8). “The temperature in degrees Fahrenheit, F, is related to the temperature in degrees Celsius, C, by the equation F=\frac{9}{5}C+32. A. In the Sahara desert, temperatures often reach 50 degrees Celsius. How many degrees Fahrenheit is this? B. In parts of Alaska, the temperatures can reach -60 degrees Fahrenheit. How many degrees Celsius is this? C. There is one temperature where the degrees Fahrenheit and degrees Celsius are the same, so that C=F. Use the expression from the equation, where F is expressed in terms of C, to solve for this temperature.” 

• Unit 8, Lesson 10, 8.1.7 Practice Problems, Question 2, students solve non-routine world problems by using the Pythagorean theorem (8.G.7). “At a restaurant, a trash can's opening is rectangular and measures 7 inches by 9 inches. The restaurant serves food on trays that measure 12 inches by 16 inches. Jada says it is impossible for the tray to accidentally fall through the trash can opening because the shortest side of the tray is longer than either edge of the opening. Do you agree or disagree with Jada's explanation? Explain your reasoning.”

The materials reviewed for Math Nation Grade 8 meet expectations 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 level.

All three aspects of rigor are present independently throughout each grade level. Examples include:

• Unit 2, Lesson 6, 2.6.4 Exploration Activity, students develop their conceptual understanding of sketching similar figures by using the set of transformations given to create a similar figure (8.G.4). “Sketch figures similar to figure A that use only the transformations listed to show similarity. 1. A translation and a reflection. Label your sketch Figure B. Pause here so that your teacher can check your work. 2. A reflection and a dilation with scale factor greater than 1. Label your sketch Figure C. 3. A rotation and a reflection. Label your sketch Figure D. 4. A dilation with scale factor less than 1 and a translation. Label your sketch Figure E.”

• Unit 3, Lesson 8, 3.8.6 Practice Problems, Question 1, students develop procedural skills and fluency as they identify graphs with the same y-intercept (8.EE.B). “Select all equations that have graphs with the same y-intercept. 1. y=3x-8; 2. y=3x-9;  3. y=3x+8; 4. y=5x-8; 5. y=2x-8; 6. y = \frac{1}{3}x - 8”

• Unit 4, Lesson 2, 4.2.6 Practice Problems, Question 4, students apply their understanding of equations to solve an age puzzle (8.EE.C). “Andre came up with the following puzzle. ‘I am three years younger than my brother, and I am 2 years older than my sister. My mom's age is one less than three times my brother's age. When you add all our ages, you get 87. What are our ages?’ A. Try to solve the puzzle. B. Jada writes this equation for the sum of the ages: (x)+(x+3)+(x-2)+3(x+3)-1=87. Explain the meaning of the variable and each term of the equation. C. Write the equation with fewer terms. D. Solve the puzzle if you haven't already.” 

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

• Unit 4, Lesson 13, 4.13.2 Exploration Activity, students develop procedural skills and fluency, and conceptual understanding by graphing and solving systems of equations (8.EE.8c). “Here are three systems of equations graphed on a coordinate plane: 1. Match each figure to one of the systems of equations shown here. A: y=3x+5 ; y=-2x+20 ; B. y=2x-10; y=4x-1; C. y=0.5x+12; y=2x+27. 2. Find the solution to each system and check that your solution is reasonable based on the graph.” 

• Unit 8, Lesson 4, 8.4.6 Practice Problems, Question 2, students develop conceptual understanding and procedural skills and fluency by comparing the size of rational and irrational numbers by plotting them on the x-axis (8.NS.2). “Plot each number on the x -axis: \sqrt{16}, \sqrt{35}, \sqrt{66}. Consider using the grid to help.” 

• Unit 8, End-of-Unit Assessment (B), Question 7, students develop procedural skills and fluency as they apply their understanding of the Pythagorean Theorem to solve real-world word problems about volume (8.G.7 and 8.G.9). “Clare has a \frac{1}{2}-liter bottle full of water. A cone-shaped paper cup has diameter 10 cm and slant height 13 cm as shown. Can she pour all the water into one paper cup, or will it overflow? Explain your reasoning.” (The volume of a cylinder formula and the conversion from liters to cubic centimeters are given.)

The materials reviewed for Math Nation 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. Mathematical practices are identified in the Teacher Edition, Lesson Narrative, and Teacher Edition, Activities Narrative. Explanations are provided in terms of how the mathematical practices “come into play” within the lesson and activity narratives (Grade 8, Course Guide).  

There is intentional development of MP1 to meet its full intent in connection to grade-level content. Throughout the grade level, students are required to analyze and make sense of problems, work to understand the information in problems, and use a variety of strategies to make sense of problems. Examples include:

• Unit 1, Lesson 4, 1.4.6 Practice Problems, Question 1, students describe rigid motions that will take one polygon onto another. “For each pair of polygons, describe a sequence of translations, rotations, and reflections that takes Polygon P to Polygon Q.” Three different pairs of polygons of various shapes are displayed. This problem attends to the full intent of MP1 as students make sense of transformations on a grid by describing a sequence of possible translations, rotations, and/or reflections that transform one shape onto another. 

• Unit 4, Lesson 7, 4.7.6 Practice Problems, Question 4, students explain why based on one equation having infinitely many solutions another equation also would have infinitely many solutions. “Here is an equation that is true for all values of x: 5(x+2)=5x+10. Elena saw this equation and says she can tell 20(x+2)+31=4(5x+10)+31 is also true for any value of x. How can she tell? Explain your reasoning.” This problem attends to the full intent of MP1 as students need to make sense of the structure of the two equations as they work to identify why an equation has infinitely many solutions. 

• Unit 8, Lesson 8, 8.8.1 Warm-up, students look at a group of equations and remove the one(s) that does not belong. “Which one doesn’t belong? A. 3^2+b^2=5^2 B. b^2=5^2-3^2 C. 3^2+5^2=b^2 D. 3^2+4^2=5^2" This Warm-Up attends to the full intent of MP1 as students need to make sense of the equations to pick the one that does not belong.

There is intentional development of MP2 to meet its full intent in connection to grade-level content. Throughout the grade level, students are required to represent situations symbolically, attend to the meaning of quantities, and understand relationships between problem scenarios and mathematical representations. Examples include:

• Unit 3, Lesson 1, 3.1.6 Practice Problems, Question 4, students find an equation that relates x and y based on other points on the same line. “The points (2,-4), (x,y), A, and B all lie on the line. Find the equation relating x and y." This activity attends to the full intent of MP2 as students reason about a graph of a linear relationship to develop their understanding of proportional relationships and slope.

• Unit 4, Lesson 9, Cool-down, students reason about quantities in an equation and the mean of the equation solution. “To own and operate a home printer, it costs $100 for the printer and an additional $0.05 per page for ink. To print out pages at an office store, it costs $0.25 per page. Let p represent number of pages. 1. What does the equation 100+0.05p=0.25p represent? 2. The solution to that equation is p=500. What does the solution mean?” This Cool-down attends to the full intent of MP2 as students reason abstractly and quantitatively about the representation of the equation and solution.

• Unit 7, Lesson 2, 7.2.1 Warm-up, students look at a given picture and decide which student representation they agree with. “Clare said she sees 100. Tyler says he sees 1. Mai says she sees \frac{1}{100}. Who do you agree with?” The materials show a picture of a 100-square grid. This activity attends to the full intent of MP2 as students reason about the different ways to represent a 100-square grid.

The materials reviewed for Math Nation 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. The materials provide opportunities for students to construct viable arguments and critique the reasoning of others in whole class and small group settings (i.e. exploration activities) and independent work settings (i.e. practice problems and assessments). 

There is intentional development of MP3 to meet its full intent in connection to grade-level content. Throughout the grade level, students are required to construct mathematical arguments by explaining/justifying their strategies and thinking, performing error analysis of provided student work/solutions, listening to the arguments of others and deciding if it makes sense, asking useful questions to better understand, and critiquing the reasoning of others. Examples include:

• Unit 4, Lesson 14, 4.14.3 Exploration Activity, students analyze a system of equations to make sense of a student’s answer regarding how many solutions a given system of equations has. “Tyler was looking at this system of equations: x+y=5 and x+y=7. He said, ‘Just looking at the system, I can see it has no solution. If you add two numbers, that sum can’t be equal to two different numbers.’ Do you agree with Tyler?” This activity attends to the full intent of MP3 as students critique the reasoning of another student while constructing viable arguments.

• Unit 6, Lesson 10, 6.10.1 Warm-Up, students complete a relative frequency table and answer questions about information in the table. “For a survey, students in a class answered these questions: Do you play a sport? Do you play a musical instrument? 1. Here is a two-way table that gives some results from the survey. Complete the table, assuming that all students answered both questions. 2. To the nearest percentage point, what percentage of students who play a sport don't play a musical instrument? 3. To the nearest percentage point, what percentage of students who don't play a sport also don't play a musical instrument?” Full Lesson Plan, Teacher Guidance: “Ask students to share the missing information they found for the table… Ask the rest of the class if they agree or disagree with the strategies and give time for any questions they have.” This activity attends to the full intent of MP3 as students critique the reasoning of their classmates while constructing viable arguments. 

• Unit 8, Lesson 9, 8.9.7 Check Your Understanding, students analyze another student's work to see where they made an error or if they are correct. “A triangle has side lengths 12, 16, and 20. Audrija is trying to determine whether the triangle is a right triangle. Her work is shown in the table. Which is the true statement about Audrija’s work? 1. Audrija made her first error in step 1.  The work should be 12^2+16^2=400. 2. Audrija made her first error in step 2. The work should be 16^2=32. 3. Audrija made her first error in step 1. The work should be 12^2+20^2=16^2. 4. Audrija made no errors. Her work is correct." This activity attends to the full intent of MP3 as students critique the reasoning of their classmates while constructing viable arguments.

The materials reviewed for Math Nation 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. Mathematical practices are identified in the Teacher Edition, Lesson Narrative and Teacher Edition, Activities Narrative. Explanations are provided in terms of how the mathematical practices “come into play” within the lesson and activity narratives (Grade 8, Course Guide). 

Examples where and how the materials use MPs 4 and/or 5 to enrich the mathematical content and demonstrate the full intent of the mathematical practices include:

• Unit 1, Lesson 13, 1.13.1 Warm-Up, students identify, draw and label corresponding parts of congruent figures.“Trapezoids ABCD and A′B′C′D′ are congruent. Draw and label the points on A′B′C′D′ that correspond to E and F. Draw and label the points on ABCD that correspond to G′ and H′. Draw and label at least three more pairs of corresponding points.” This activity intentionally develops (MP4) as students use the mathematical model to label corresponding points, and (MP5) as students are given access to tracing paper or a grid to do the activity. 

• Unit 3, Lesson 3, 3.3.2 Exploration Activity, Question 1, students create a table of values, graph the ordered pairs, identify/calculate the constant of proportionality, and synthesize the results. “Here are two ways to represent a situation. Description: Jada and Noah counted the number of steps they took to walk a set distance. To walk the same distance, Jada took 8 steps while Noah took 10 steps. Then they found that when Noah took 15 steps, Jada took 12 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; a. Create a table that represents this situation with at least  3 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 that the equation, description, graph, and table all represent the same situation.” This activity intentionally develops (MP4) as students model the given scenario in various ways, and (MP5) are prompted to choose appropriate tools and/or strategies throughout the activity. 

• Unit 4, Lesson 2, 4.2.6 Practice Problems, Question 3, students find the weight of an object on a hanger diagram given only one known value.“What is the weight of a square if a triangle weighs 4 grams? Explain your reasoning.” A hangar diagram is given that has one triangle and two squares on one side and three triangles and one square on the next side. This activity intentionally develops MP4 as students create an equation for the hangar diagram in order to solve it.

• Unit 5, Lesson 9, 5.9.3 Exploration Activity, students sketch linear functions based on data provided and analyze the resulting graphs. “In an earlier lesson, we saw this graph that shows the percentage of all garbage in the U.S. that was recycled between 1991 and 2013. (Scatterplot is given to students.) 1. Sketch a linear function that models the change in the percentage of garbage that was recycled between 1991 and 1995. For which years is the model good at predicting the percentage of garbage that is produced? For which years is it not as good? 2. Pick another time period to model with a sketch of a linear function. For which years is the model good at making predictions? For which years is it not very good?” This activity intentionally develops (MP4) as students see how a model relates to a problem situation and check the reasonableness of the model, and (MP5) as students choose which tool to use to graph the linear function.

The materials reviewed for Math Nation 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.Mathematical practices are identified in the Teacher Edition, Lesson Narrative and Teacher Edition, Activities Narrative. Explanations are provided in terms of how the mathematical practices “come into play” within the lesson and activity narratives (Grade 8, Course Guide). 

There is intentional development of MP6 to meet its full intent in connection to grade-level content and the instructional materials attend to the specialized language of mathematics. Students communicate using grade-level appropriate vocabulary and conventions and formulate clear explanations when engaging with course materials. Students must calculate accurately and efficiently, specify units of measure, and use and label tables and graphs appropriately when engaging with course materials. Teacher guidance very clearly develops the specialized language of mathematics as teachers are explicitly prompted when to introduce content-related vocabulary and use accurate definitions when communicating mathematically. Examples include:

• Unit 1, Lesson 2, 1.2.5 Practice Problems, Question 2, students describe and identify a series of transformations that will make one shape look like another. “The five frames show a shape's different positions. Describe how the shape moves to get from its position in each frame to the next. Frame 1 to 2: Frame 2 to 3: Frame 3 to 4: Frame 4 to 5:” This activity attends to the full intent of MP6 as students use specific mathematical language to precisely describe the geometric transformations. 

• Unit 4, Lesson 6, 4.6.6 Practice Problems, Question 1, students solve two-sided linear equations. “Solve each of these equations. Explain or show your reasoning. a. 2b+8-5b+3=-13+8b-5 b. 2x+7-5x+8=3(5+6x)-12x c. 2c-3=2(6-c)+7c”  This activity intentionally develops MP6 as students must be accurate in their calculations in order to get the right answer and attends to the specialized language of mathematics as students must use the correct wording in order to clearly explain how they solved the problems. 

• Unit 7, Lesson 12, 7.12.6 Practice Problems, Question 1, students perform a calculation using numbers expressed as power of 10. “Which is larger: the number of meters across the Milky Way, or the number of cells in all humans? Explain or show your reasoning. Some useful information: The Milky Way is about 100,000 light years across. There are about 37 trillion cells in a human body. One light year is about 10^{16} meters. The world population is about 7 billion.” This activity attends to the full intent of MP6 as students must perform the calculations accurately and precisely.

• Unit 8, Lesson 3, Cool-Down, students define rational and irrational numbers in their own words and give examples for each. “1. In your own words, say what a rational number is. Give at least three different examples of rational numbers. 2. In your own words, say what an irrational number is. Give at least two examples,” This activity attends to the full intent of MP6 and the specialized language of mathematics as students must attend to precision when giving the examples and use the language of mathematics when they are saying the definition.

The materials reviewed for Math Nation 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. Mathematical practices are identified in the Teacher Edition, Lesson Narrative and Teacher Edition, Activities Narrative. Explanations are provided in terms of how the mathematical practices “come into play” within the lesson and activity narratives (Grade 8, Course Guide). 

There is intentional development of MP7 to meet its full intent in connection to grade-level content.  Materials provide opportunities for students to look for patterns or structures to make generalizations and solve problems, and look for and explain the structure of mathematical representations, look at and decompose “complicated” into “simpler.”  Examples include:

• Unit 4, Lesson 6, 4.6.2 Exploration Activity, students look for structure of equations to determine whether a solution will be positive, negative, or zero. “Without solving, identify whether these equations have a solution that is positive, negative, or zero. 1. \frac{x}{6}=\frac{3x}{4}; 2. 7x=3.25; 3. 7x=32.5; 4. 3x+11=11; 5. 9-4x=4; 6. -8+5x=-20; 7. -\frac{1}{2}(-8+5x)=-20. This activity attends to the full intent of MP7 as students use the structure of the equation to determine whether the solution will be positive, negative or zero.

• Unit 6, Lesson 6, 6.6.2 Exploration Activity, students match three scatterplots with three statements describing a relationship between the variables in the scatterplots. “For each scatter plot, decide if there is an association between the two variables, and describe the situation using one of these sentences: For these data, as ________________ increases, ________________ tends to increase. For these data, as ________________ increases, ________________ tends to decrease. For these data, ________________ and ________________ do not appear to be related.” This activity intentionally develops MP7 as students look at how the graphs are structured in order to correctly match the scatterplot with the questions.

• Unit 8, Lesson 11, 8.11.7 Practice Problems, Question 2, students calculate the distance between ordered pairs. “Find the distance between each pair of points. If you get stuck, try plotting the points on graph paper. b. M=(0,-11) and P=(0,2)  c. A=(0,0) and B=(-3,-4) d. C=(8,0) and D=(0,-6)” This activity attends to the full intent of MP7 as students can use the structure of the coordinate points to find the distance.

There is intentional development of MP8 to meet its full intent in connection to grade-level content. Materials provide opportunities for students to notice repeated calculations to understand algorithms and make generalizations or create shortcuts, evaluate the reasonableness of their answers and their thinking, and create, describe, or explain a general method/formula/process/algorithm. Examples include:

• Unit 3, Lesson 10, 3.10.2 Exploration Activity, students plot two points, draw a line and identify whether the slope will be negative or positive. Then students calculate the slope of the line. “1. Plot the points (1, 11) and (8, 2), and use a ruler to draw the line that passes through them. 2. Without calculating, do you expect the slope of the line through (1, 11) and (8, 2) to be positive or negative? How can you tell? 3. Calculate the slope of this line.” This activity attends to the full intent of MP8, look for and express regularity in repeated reasoning, as students use the repeated reasoning of positive slope and negative slope graphs to answer the question.

• Unit 7, Lesson 2, 7.2.2 Exploration Activity, students use repeated reasoning about multiplication, to answer questions about the power of 10. “In the diagram, the medium rectangle is made up of 10 small squares. The large square is made up of 10 medium rectangles. 1. How could you represent the large square as a power of 10? 2. If each small square represents 10^2, then what does the medium rectangle represent? The large square? 3. If the medium rectangle represents 10^5, then what does the large square represent? The small square? 4. If the large square represents 10^{100}, then what does the medium rectangle represent? The small square?” This activity attends to the full intent of MP8 as students do repeated calculations of powers of 10 to understand algorithms and make generalizations.

• Unit 8, Lesson 15, 8.15.2 Exploration Activity, students answer problems involving decimal expansion and look for patterns. “1. The cards show Noah's work calculating the fraction representation of 0.\overline{485}. Arrange these in order to see how he figured out that 0.\overline{485}=\frac{481}{990} without needing a calculator. 2. Use Noah's method to calculate the fraction representation of: a. 0.\overline{186} b. 0.\overline{788}.” This activity attends to the full intent of MP8 as students notice repeated calculations to understand algorithms and create general methods.