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. $V=\pi(\frac{d}{2})^2h$; $V=\pi(\frac{8}{2})^2$; $V=\pi\cdot16\cdot10$; .” 

• 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.

The materials reviewed for Math Nation Grade 8 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students to guide their mathematical development. 

Examples of where and how the materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials include:

• Course Overview: A Course Overview (Unit 0) is found at the beginning of each course.  Within each Course Overview there is a Course Narrative, which contains a summary of the mathematical content contained in each course, and a Course Guide. The Course Guide contains the following sections: Introduction, About These Materials, How to Use These Materials, Assessment Overview, Scope and Sequence, Required Resources, Corrections, and Cool-Down Guidance. Each of these sections contains specific guidance for teachers on implementing lesson instruction. For example, in the About These Materials section, teachers can find an outline of and detailed information about the components of a typical lesson, including Warm-Up, Classroom Activities, Lesson Synthesis, and Cool-Down. The How to Use These Materials section contains guidance about the three phases of classroom activities (Launch-Work-Synthesize) and utilizing instructional routines. In the Scope and Sequence section, teachers will find a Pacing Guide which contains time estimates for coverage of each of the units.

• Teacher Edition: There is a Teacher Edition section for each unit that contains a unit introduction, unit assessments, and unit-level downloads. The Unit Introduction contains a summary of the mathematical content to be found in the unit. The Assessment component contains downloads for multiple types of assessments (Check Your Readiness, Mid-Unit, and End-of-Unit Assessment). Unit Level Downloads include: Student Task Statements Cool-downs, Practice Problems, Blackline Masters, and My Reflections all of which provide support for teacher planning. Each lesson has a Teacher Edition component that contains guidance for Lesson Preparation, Cool-down Guidance, and a Lesson Narrative. The Lesson Preparation component includes a Teacher Prep Video, Learning Goal(s), Required Material(s), and Full Lesson Plan downloads. Cool-down Guidance provides teachers with guidance on what to look for or emphasize over the next several lessons to support students in advancing their current understanding. The Lesson Narrative provides specific guidance about how students can work with the lesson activities.

• Full Lesson Plan: Within each Teacher Edition lesson component, teachers can find a Full Lesson Plan that contains lesson learning goals and targets, a lesson narrative, and specific guidance for implementing each of the lesson activities. The Lesson Narrative contains the purpose of the lesson, standards and mathematical practices alignments, specific instructional routines, and required materials related to the lesson. Teachers are given guidance for implementing these routines as a way of introducing students to the learning targets. There is also teacher guidance for launching lesson activities, such as suggestions for grouping students, working with a partner, or whole group discussion. The planning section identifies possible student errors and misconceptions that could occur. There is also guidance on how to support English Language Learners and Students with Disabilities.

Materials include sufficient and useful annotations and suggestions that are presented within the context of specific learning objectives. Preparation and lesson narratives within the Course Guide, Lesson Plans, Lesson Narratives, Overviews, and Warm-up provide useful annotations. Examples include:

• Course Guide, Assessments Overview, “Pre-Unit Diagnostic Assessments At the start of each unit is a pre-unit diagnostic assessment. These assessments vary in length. Most of the problems in the pre-unit diagnostic assessment address prerequisite concepts and skills for the unit. Teachers can use these problems to identify students with particular below-grade needs, or topics to carefully address during the unit. Teachers are encouraged to address below-grade skills while continuing to work through the on-grade tasks and concepts of each unit, instead of abandoning the current work in favor of material that only addresses below-grade skills…What if a large number of students can’t do the same pre-unit assessment problem? Look for opportunities within the upcoming unit where the target skill could be addressed in context…What if all students do really well on the pre-unit diagnostic assessment? Great! That means that they are ready for the work ahead, and special attention likely doesn’t need to be paid to below-grade skills.”

• Unit 1, Lesson 4,  Full Lesson Plan, 1.4.3 Exploration Activity, “Anticipated Misconceptions Students may struggle drawing the image under transformation from the quick flashes of the image because they are trying to count the number of spaces each vertex moves. Encourage these students to use the line in the image to help them reflect the image.”

• Unit 2, Lesson 5, Full Lesson Plan, “Lesson NarrativeIn previous lessons, students learned what a dilation is and practice dilating points and figures on a circular grid, on a square grid, on a coordinate grid, and with no grid. In this lesson, they work on a coordinate grid and use the coordinates to communicate precisely the information needed to perform a dilation. Students use the info gap structure. The student with the problem card needs to dilate a polygon on the coordinate grid. In order to do so, they need to request the coordinates of the polygon's vertices and the center of dilation as well as the scale factor. After obtaining all of this information from the partner with the data card, the student performs the dilation. The focus here is on deciding what information is needed and communicating clearly to request the information and explain why it is needed. One important use of coordinates in geometry is to facilitate precise and concise communication about the location of points (MP6). This allows students to indicate where the center of the dilation is and also to communicate the vertices of the polygon that is dilated.”

The materials reviewed for Math Nation Grade 8 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.

The Course Guide, About These Materials sections, states the following note about standards alignment, “There are three kinds of alignments to standards in these materials: building on, addressing, and building towards. Oftentimes a particular standard requires weeks, months, or years to achieve, in many cases building on work in prior grade-levels. When an activity reflects the work of prior grades but is being used to bridge to a grade-level standard, alignments are indicated as ‘building on.’ When an activity is laying the foundation for a grade-level standard but has not yet reached the level of the standard, the alignment is indicated as ‘building towards.’ When a task is focused on the grade-level work, the alignment is indicated as ‘addressing.’” All lessons in the materials have this correlation information. An example:

• Unit 7, Lesson 4, Full Lesson Plan, Lesson Standards Alignment, Building on 5.NF.5b; Addressing 8.EE.1; Building Towards 8.EE.1.

Explanations of the role of the specific grade-level mathematics in the context of the series can be found throughout the materials including but not limited to the Course Guide, Scope and Sequence section, the Course Overview, Unit Introduction, Lesson Narrative and Full Lesson Plan. Examples include:

• Course Guide, Scope and Sequence, Unit 1: Rigid Transformations and Congruence, “Work with transformations of plane figures in grade 8 draws on earlier work with geometry and geometric measurement. Students began to learn about two- and three-dimensional shapes in kindergarten, and continued this work in grades 1 and 2, composing, decomposing, and identifying shapes. Students' work with geometric measurement began with length and continued with area. Students learned to "structure two-dimensional space," that is, to see a rectangle with whole-number side lengths as composed of an array of unit squares or composed of iterated rows or iterated columns of unit squares. In grade 3, students distinguished between perimeter and area…In grade 6, students combined their knowledge of geometry and geometric measurement to produce formulas for the areas of parallelograms and triangles, using these formulas to find surface areas of polyhedra. In grade 7, students worked with scaled copies and scale drawings, learning that angle measures are preserved in scaled copies, but areas increase or decrease proportionally to the square of the scale factor…”

• Course Guide, Scope and Sequence, Unit 3: Linear Relationships, “Work with linear relationships in grade 8 builds on earlier work with rates and proportional relationships in grade 7, and grade 8 work with geometry. At the end of the previous unit on dilations, students learned the terms ‘slope’ and ‘slope triangle,’ used the similarity of slope triangles on the same line to understand that any two distinct points on a line determine the same slope, and found an equation for a line with a positive slope and vertical intercept…A proportional relationship is a collection of equivalent ratios. In high school-after their study of ratios, rates, and proportional relationships-students discard the term ‘unit rate,’referring to a to b, a:b, and ab as ‘ratios.’...”

The materials reviewed for Math Nation Grade 8 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. Instructional approaches of the program and identification of the research-based strategies can be found throughout the materials, but particularly in the Course Guide, About These Materials, and How to Use These Materials sections. 

The About These Materials section states the following about the instructional approach of the program, “What is a Problem Based Curriculum? In a problem-based curriculum, students work on carefully crafted and sequenced mathematics problems during most of the instructional time. Teachers help students understand the problems and guide discussions to ensure the mathematical takeaways are clear to all. Some concepts and procedures follow from definitions and prior knowledge so students can, with appropriately constructed problems, see this for themselves. In the process, they explain their ideas and reasoning and learn to communicate mathematical ideas. The goal is to give students just enough background and tools to solve initial problems successfully, and then set them to increasingly sophisticated problems as their expertise increases. However, not all mathematical knowledge can be discovered, so direct instruction is sometimes appropriate. A problem-based approach may require a significant realignment of the way math class is understood by all stakeholders in a student's education. Families, students, teachers, and administrators may need support making this shift. The materials are designed with these supports in mind. Family materials are included for each unit and assist with the big mathematical ideas within the unit. Lesson and activity narratives, Anticipated Misconceptions, and instructional supports provide professional learning opportunities for teachers and leaders. The value of a problem-based approach is that students spend most of their time in math class doing mathematics: making sense of problems, estimating, trying different approaches, selecting and using appropriate tools, evaluating the reasonableness of their answers, interpreting the significance of their answers, noticing patterns and making generalizations, explaining their reasoning verbally and in writing, listening to the reasoning of others, and building their understanding. Mathematics is not a spectator sport.”

Examples of materials including and referencing research-based strategies include:

• “The Five Practices Selected activities are structured using Five Practices for Orchestrating Productive Mathematical Discussions (Smith & Stein, 2011), also described in Principles to Actions: Ensuring Mathematical Success for All (NCTM, 2014), and Intentional Talk: How to Structure and Lead Productive Mathematical Discussions (Kazemi & Hintz, 2014). These activities include a presentation of a task or problem (may be print or other media) where student approaches are anticipated ahead of time. Students first engage in independent think-time followed by partner or small-group work on the problem…”

• “Supporting English Language Learners This curriculum builds on foundational principles for supporting language development for all students. This section aims to provide guidance to help teachers recognize and support students' language development in the context of mathematical sense-making. Embedded within the curriculum are instructional supports and practices to help teachers address the specialized academic language demands in math when planning and delivering lessons, including the demands of reading, writing, speaking, listening, conversing, and representing in math (Aguirre & Bunch, 2012).”

• “Instructional Routines … Some of the instructional routines, known as Mathematical Language Routines (MLR), were developed by the Stanford University UL/SCALE team…”

Within the Course Guide, How to Use These Materials, a Reference section is included. 

The materials reviewed for Math Nation Grade 8 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. Comprehensive lists of supplies needed to support the instructional activities can be found in Course Guides (Required Materials), Teacher Editions, for each lesson, under Lesson Preparation (Required Material(s)), and in Teacher Guides for specific lessons. Examples include:

• Unit 1, Lesson 1, Lesson Preparation, Required Materials: “Blackline master for Activity 1.2, Cool-down, copies of blackline master, geometry toolkits (tracing paper, graph paper, colored pencils, scissors, and an index card)”

• Unit 5, Lesson 13, Lesson Preparation, Required Materials: “Cool-down, colored pencils”

• Unit 8, Lesson 4, Lesson Preparation, Required Materials: “Cool-down, compasses, four-function calculators, tracing paper”

The materials reviewed for Math Nation Grade 8 meet expectations for including an assessment system that provides multiple opportunities throughout the grade, course, and/or series to determine students’ learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

Student sample responses are provided for all assessments. Rubrics are provided for scoring restricted constructed response and extended response questions on the Mid-Unit Assessments and End-of-Unit Assessments. Mid-Unit Assessments and End-of-Unit Assessments include notes that provide guidance for teachers to interpret student understanding and make sense of students’ correct/incorrect responses. 

Suggestions to teachers for following up with students are provided throughout the materials via the Check-Your-Readiness, Mid-Unit, and End-of-Unit Teacher Guides, and each lesson provides a Cool-down Guidance that details how to support student learning.

Examples of the assessment system providing multiple opportunities to determine students' learning and sufficient guidance to teachers for interpreting student performance include:

• Course Guide, Assessments Overview states the following: “Rubrics for Evaluating Students Answers Restricted constructed response and extended response items have rubrics that can be used to evaluate the level of student responses. 

  • Restricted Constructed Response

    • Tier 1 response: Work is complete and correct.

    • Tier 2 response: Work shows General conceptual understanding and mastery, with some errors.

    • Tier 3 response: Significant errors in work demonstrate lack of conceptual understanding or mastery. Two or more error types from Tier 2 response can be given as the reason for a Tier 3 response instead of listing combinations.

  • Extended Response

    • Tier 1 response: Work is complete and correct, with complete explanation or justification. 

    • Tier 2 response: Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification. 

    • Tier 3 response: Work shows a developing but incomplete conceptual understanding, with significant errors. 

    • Tier 4 response: Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.”

• Unit 8, End-of-Unit Assessment (A), Question 4, “Plot these numbers on the number line: $\sqrt{2}$, $\sqrt{5}$, $\sqrt[3]{8}$, $\sqrt{9}$, $\sqrt{15}$, $\sqrt[3]{25}$ (A number line is shown going start at 0, 1, 2, 3, 4, and 5 are labeled) Solution $\sqrt{2}$, $\sqrt[3]{8}$, $\sqrt{5}$, $\sqrt[3]{25}$, $\sqrt{9}$, $\sqrt{15}$ (on the number line) Minimal Tier 1 response: Work is complete and correct. Sample: See number line. Tier 2 response: Work shows general conceptual understanding and mastery, with some errors. Sample errors: some of the irrational numbers are placed in the correct unit interval but not within the correct half of the interval; one or two points are plotted completely incorrectly; all points are correct but at least two are not labeled (so it is not possible to tell which point represents which number). Tier 3 response: Significant errors in work demonstrate lack of conceptual understanding or mastery. Sample errors: three or more points are placed completely incorrectly. ”

Examples of the assessment system providing multiple opportunities to determine students' learning and suggestions to teachers for following up with students include:

• Course Guide, Cool-Down Guidance, states the following: “Each cool-down is placed into one of three support levels: 1. More chances. This is often associated with lessons that are exploring or playing with a new concept. Unfinished learning for these cool-downs is expected and no modifications need to be made for upcoming lessons. 2. Points to emphasize. For cool-downs on this level of support, no major accommodations should be made, but it will help to emphasize related content in upcoming lessons. Monitor the student who have unfinished learning throughout the next few lessons and work with them to become more familiar with parts of the lesson associated with this cool-down. Perhaps add a few minutes to the following class to address related practice problems, directly discuss the cool-down in the launch or synthesis of the warm-up of the next lesson, or strategically select students to share their thinking about related topics in the upcoming lessons. 3. Press pause. This advises a small pause before continuing movement through the curriculum to make sure the base is strong. Often, upcoming lessons rely on student understanding of the ideas from this cool-down, so some time should be used to address any unfinished learning before moving on to the next lesson.”

• Unit 1, Check-Your-Readiness (B), Question 2, “The content assessed in this problem is first encountered in Lesson 8: Rotation Patterns. Students identify parallel and perpendicular lines. If most students struggle with this item, plan to use Lesson 3 Activity 1 to review the term parallel using the isometric grid paper. Lesson 5 Activity 3 provides an opportunity to review the erm perpendicular.” 

• Unit 4, Lesson 4, Cool-down Guidance, “Support Level 1. More Chances. Notes Students will have more opportunities to develop procedural fluency in solving multistep equations. The card sort in the following lesson provides a great opportunity to reinforce concepts from this cool-down.”

The materials reviewed for Math Nation Grade 8 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of the course-level standards and practices across the series.

All assessments regularly demonstrate the full intent of grade-level content and practice standards through a variety of item types such as multiple choice, short answer, extended response prompts, graphing, mistake analysis, and constructed response items. Assessments are to be downloaded as Word documents or PDFs and designed to be printed and administered in-classroom. Examples Include:

• Unit 1, Mid-Unit Assessment (B), Question 6, demonstrates the full intent of 8.G.1 and MP1. “Describe a sequence of transformations that takes Figure Q to Figure P.” A coordinate plane is provided with two figures labeled Q and P. 

• Unit 2, End-of-Unit Assessment (B), Question 6, demonstrates the full intent of 8.EE.6.“All of the points in the picture are on the same line. 1. Find the slope of the line. Explain or show your reasoning. 2. Write an equation for the line. 3. What is the value of c? Explain or show your reasoning. 4. Is the point (0, -2) on this line? Explain how you know.”  A picture is shown of a line with the following points on the line labeled: (2, 4), (c, 10) and (5, 13).

• Unit 6, End-of-Unit Assessment (A), Question 4, demonstrates the full intent of 8.SP.1 and MP4.“1. Draw a scatter plot that shows a negative, linear association and has one clear outlier.  Circle the outlier. 2. Draw a scatter plot  that shows a positive association that is not linear.”

The materials reviewed for Math Nation Grade 8 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level mathematics.

Course Guide, How to Use These Materials, Supporting Students with Disabilities sections states the following: “The philosophical stance that guided the creation of these materials is the belief that with proper structures, accommodations, and supports, all children can learn mathematics. Lessons are designed to maximize access for all students, and include additional suggested supports to meet the varying needs of individual students. While the suggested supports are designed for students with disabilities, they are also appropriate for many children who struggle to access rigorous, grade-level content. Teachers should use their professional judgment about which supports to use and when, based on their knowledge of the individual needs of students in their classroom.” Suggested supports are identified for teachers in the Full Lesson Plan to support learners of all levels. Lesson and activity-level supports, identified as “Support for Students with Disabilities,” are aligned to an area of cognitive functioning and are paired with a suggested strategy aimed to increase access and eliminate barriers. Supports are classified under the following categories: eliminate barriers, processing time, peer tutors, assistive technology, visual aids, graphic organizers, and brain breaks. Examples include:

• Assistive Technology: “Assistive technology can be a vital tool for students with learning disabilities, visual spatial needs, sensory integration, and students with autism. Assistive technology supports suggested in the materials are designed to either enhance or support learning, or to bypass unnecessary barriers. Physical manipulatives help students make connections between concrete ideas and abstract representations. Often, students with disabilities benefit from hands-on activities, which allow them to make sense of the problem at hand and communicate their own mathematical ideas and solutions.” Unit 1, Lesson 3, Full Lesson Plan, 1.3.2 Exploration Activity, “Support for Students with Disabilities…Assistive Technology. Provide access to the digital version of this activity.”

• Graphic Organizers: “Word webs, Venn diagrams, tables, and other metacognitive visual supports provide structures that illustrate relationships between mathematical facts, concepts, words, or ideas. Graphic organizers can be used to support students with organizing thoughts and ideas, planning problem solving approaches, visualizing ideas, sequencing information, or comparing and contrasting ideas.” Unit 2, Lesson 11, Full Lesson Plan, 2.11.3 Exploration Activity, “Support for Students with Disabilities Executive Functioning: Graphic Organizers. Provide a Venn diagram with which to compare the between lines k and l.”

• Visual Aids: “Visual aids such as images, diagrams, vocabulary anchor charts, color coding, or physical demonstrations, are suggested throughout the materials to support conceptual processing and language development. Many students with disabilities have working memory and processing challenges. Keeping visual aids visible on the board allows students to access them as needed, so that they can solve problems independently. Leaving visual aids on the board especially benefits students who struggle with working or short term memory issues.” Unit 5, Lesson 7, Full Lesson Plan, 5.7.1 Warm Up, “Support for Students with Disabilities Visual-Spatial Processing: Visual Aids. Provide handouts of the representations for students to draw on or highlight.”

There are several accessibility options (accessed via the wrench icon in the lower left-hand corner of the screen) available to help students navigate the materials. Examples include:

• Tools Menu allow students to change the language, and access a Demos Scientific and Graphing Calculator.

• Accessibility Menu allows students to change the language, page zoom, font style, background and font color, and enable/disable the following features: text highlighter, notes, screen reader support. 

• UserWay, allows students to adjust the following: Change contrast (4 settings), Highlight links, Enlarge text (5 settings), Adjust text spacing (4 settings), Hide images, Dyslexia Friendly, Enlarge the cursor, show a reading mask, show a reading line, Adjust line height (4 settings), Text align (5 settings), Saturation (4 settings). 

Additionally, differentiated videos explaining course content - varying from review to in-depth levels of explanation - are resources available for each lesson to support students.

The materials reviewed for Math Nation Grade 8 meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.

Course Guide, How to Use These Materials, Are You Ready For More? section states the following: “Select classroom activities include an opportunity for differentiation for students ready for more of a challenge. Every extension problem is made available to all students with the heading ‘Are You Ready for More?’ These problems go deeper into grade-level mathematics and often make connections between the topic at hand and other concepts at grade level or that are outside of the standard K-12 curriculum. They are not routine or procedural, and intended to be used on an opt-in basis by students if they finish the main class activity early or want to do more mathematics on their own. It is not expected that an entire class engages in Are You Ready for More? problems and it is not expected that any student works on all of them. Are You Ready for More? problems may also be good fodder for a Problem of the Week or similar structure.” If individual students would complete these optional activities, then they might be doing more assignments than their classmates.

Examples of opportunities for advanced students to investigate grade-level mathematics content at a higher level of complexity include:

• Unit 3, Lesson 2, 3.2.4 Exploration Extension: Are you Ready for More?, “A giant tortoise travels at 0.17 miles per hour and an arctic hare travels at 37 miles per hour. 1. Draw separate graphs that show the relationship between time elapsed, in hours, and distance traveled, in miles, for both the tortoise and the hare. 2. Would it be helpful to try to put both graphs on the same pair of axes? Why or why not? 3. The tortoise and the hare start out together and after half an hour the hare stops to take a rest. How long does it take the tortoise to catch up?” 

• Unit 4, Lesson 14, 4.14.4 Exploration Extension: Are you Ready for More?, “In rectangle 𝐴𝐵𝐶𝐷, side 𝐴𝐵 is 8 centimeters and side 𝐵𝐶 is 6 centimeters. 𝐹 is a point on 𝐵𝐶 and 𝐸 is a point on 𝐴𝐵. The area of triangle 𝐷𝐹𝐶 is 20 square centimeters, and the area of triangle 𝐷𝐸𝐹 is 16 square centimeters. What is the area of triangle 𝐴𝐸𝐷?”

• Unit 7, Lesson 3, 7.4.4 Exploration Extension: Are you Ready for More?, "$2^{12}=4096$. How many other whole numbers can you raise to a power and get 4,096? Explain or show your reasoning.”

The materials reviewed for Math Nation Grade 8 meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

The Course Guide, How to Use These Materials section states the following: “The framework for supporting English language learners (ELLs) in this curriculum includes four design principles for promoting mathematical language use and development in curriculum and instruction. The design principles and related routines work to make language development an integral part of planning and delivering instruction while guiding teachers to amplify the most important language that students are expected to bring to bear on the central mathematical ideas of each unit.” The four design principles are, support sense-making, optimize output, cultivate conversation, and maximize meta-awareness. Each design principle has an explanation that goes into more detail about how teachers can use it to support students. The routines are the Mathematical Language Routines (MLRs), the materials state, “The mathematical language routines (MLRs) were selected because they are effective and practical for simultaneously learning mathematical practices, content, and language. A mathematical language routine is a structured but adaptable format for amplifying, assessing, and developing students' language. The routines emphasize uses of language that is meaningful and purposeful, rather than about just getting answers. These routines can be adapted and incorporated across lessons in each unit to fit the mathematical work wherever there are productive opportunities to support students in using and improving their English and disciplinary language use. Each MLR facilitates attention to student language in ways that support in-the-moment teacher-, peer-, and self-assessment for all learners. The feedback enabled by these routines will help students revise and refine not only the way they organize and communicate their own ideas, but also ask questions to clarify their understandings of others' ideas.” These design principles and routines are referenced under Instructional Routines, in the Full Lesson Plan for lesson, to assist teachers with lesson planning. The “Supports for English Language Learners” section within the Full Lesson Plan contains explanations of how to implement the MLRs. 

Examples where the materials provide strategies and supports for students who read, write, and/or speak in a language other than English include:

• Unit 2, Lesson 6, Full Lesson Plan, 2.6.3 Exploration Activity, “Support for English Language Learners Speaking, Listening: MLR 7 Compare and Connect. Ask students to prepare a visual display of their figures that are similar to Figure A. As students investigate each other’s work, ask students to share what transformations are especially clear in the display of similar figures. Listen for and amplify any comments about what might make the transformations clearer in the display. Then encourage students to make connections between the words ‘translation,’ ‘rotation,’ ‘reflection,’ and ‘dilation’ and how they affect the figure. Listen for and amplify language students use to describe what happens to figures under different kinds of transformations. This will foster students’ meta-awareness and support constructive conversations as they compare images of the same figure and make connections between transformations and their effects on figures. Design Principle(s): Cultivate conversation; Maximize meta-awareness”

• Unit 4, Lesson 14, Full Lesson Plan, 4.14.2 Exploration Activity, “Support for English Language Learners Representing, Speaking, Listening: MLR 2 Collect and Display. As students discuss which systems they thought would be easiest to solve and which would be hardest, create a table with the headings ‘least difficult’ and ‘most difficult’ in the two columns. Circulate through the groups and record student language in the appropriate column. Look for phrases such as ‘different variables on the same side,’ ‘variables already isolated,’ and ‘various terms.’ Invite students to share strategies they can use to address the features that make these systems of equations more difficult to solve. This will help students begin to generalize and make sense of the structures of equations for substitution. Design Principle(s): Support sense-making”

• Unit 6, Lesson 3, Full Lesson Plan, 6.3.3 Exploration Activity, “Support for English Language Learners Writing, Speaking: MLR 1 Stronger and Clearer Each Time. Use this routine to give students a structured opportunity to revise and refine their response to the last question. Ask each student to meet with 2–3 other partners in a row for feedback. Provide students with prompts for feedback that will help students strengthen their ideas and clarify their language (e.g., ‘Can you give an example?’, ‘Why do you think…?’, ‘How do you know…?’, etc.). Students can borrow ideas and language from each partner to strengthen their final version. Design Principle(s): Optimize output (for explanation)”

The materials reviewed for Math Nation Grade 8 meet expectations for providing manipulatives, both virtual and physical, that are representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

Virtual and physical manipulatives support student understanding throughout the materials. Examples include:

• Unit 1, Lesson 15, 1.15.3 Exploration Activity, students manipulate paper angles to try and form a triangle. “Your teacher will give you a page with three sets of angles and a blank space. Cut out each set of three angles. Can you make a triangle from each set that has these same three angles?” 

• Unit 3, Lesson 9, 3.9.3 Exploration Activity, students use an applet of a graph to answer questions and write a question about the information represented. “Here is a graph that shows the amount on Han’s fare card for every day of last July. 1. Describe what happened with the amount on Han’s fare card in July. 2. Plot and label 3 different points on the line. 3. Write an equation that represents the amount on the card in July, y, after x days. 4. What value makes sense for the slope of the line that represents the amounts on Han’s fare card in July?” A GeoGebra applet is provided that shows the amount on Han’s fare card, students are able to use the applet to put additional points on the line and draw additional lines.

• Unit 5, Lesson 11, 5.11.2 Exploration Activity, students use an applet to investigate the relationship between the height and volume of water in a cylinder. “Use the applet to investigate the height of water in the cylinder as a function of the water volume. 1. Before you get started, make a prediction about the shape of the graph. 2. Check Reset and set the radius and height of the graduated cylinder to values you choose. 3. Fill the cylinder with different amounts of water and record the data in the table. 4. Create a graph that shows the height of the water as a function of the water volume. 5. Choose a point on the graph and explain its meaning in the context of the situation.” Two applets are provided, one is a Geogebra applet that allows students to change the height and radius of the cylinder and fill it with water. Students can pause and fill at any time and the volume, height and diameter are displayed. The second is a Desmos applet for students to use to create the graph of the function.