## Narrative

Students begin grade 8 with transformational geometry. They study rigid transformations and congruence, then dilations and similarity (this provides background for understanding the slope of a line in the coordinate plane). Next, they build on their understanding of proportional relationships from grade 7 to study linear relationships. They express linear relationships using equations, tables, and graphs, and make connections across these representations. They expand their ability to work with linear equations in one and two variables. Building on their understanding of a solution to an equation in one or two variables, they understand what is meant by a solution to a system of equations in two variables. They learn that linear relationships are an example of a special kind of relationship called a function. They apply their understanding of linear relationships and functions to contexts involving data with variability. They extend the definition of exponents to include all integers, and in the process codify the properties of exponents. They learn about orders of magnitude and scientific notation in order to represent and compute with very large and very small quantities. They encounter irrational numbers for the first time and informally extend the rational number system to the real number system, motivated by their work with the Pythagorean Theorem.

### 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. They connected rectangle area with multiplication, understanding why (for whole-number side lengths) multiplying the side lengths of a rectangle yields the number of unit squares that tile the rectangle. They used area diagrams to represent instances of the distributive property. In grade 4, students applied area and perimeter formulas for rectangles to solve real-world and mathematical problems, and learned to use protractors. In grade 5, students extended the formula for the area of rectangles to rectangles with fractional side lengths. 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. Their study of scaled copies was limited to pairs of figures with the same rotation and mirror orientation. Viewed from the perspective of grade 8, a scaled copy is a dilation and translation, not a rotation or reflection, of another figure.

In grade 8, students extend their reasoning to plane figures with different rotation and mirror orientations.

Through activities designed and sequenced to allow students to make sense of problems and persevere in solving them (MP1), students use and extend their knowledge of geometry and geometric measurement. They begin the unit by looking at pairs of cartoons, each of which illustrates a translation, rotation, or reflection. Students describe in their own words how to move one cartoon figure onto another. As the unit progresses, they solidify their understanding of these transformations, increase the precision of their descriptions (MP6), and begin to use associated terminology, recognizing what determines each type of transformation, for example, two points determine a translation.

In the first few lessons, students encounter examples of transformations in the plane, without the added structure of a grid or coordinates. The reason for this choice is to avoid limiting students’ schema by showing the least restrictive examples of transformations. Specifically, students see examples of translations in any direction, rotations by any angle, and reflections over any arbitrary line. Through these examples, they begin to understand the features of these transformations without having their understanding limited to, for example, horizontal or vertical translations or rotations only by 90 or 180 degrees. Also, through the use of transparencies, students’ initial understanding of transformations involves moving the entire plane, rather than just moving a given figure. Since all transformations are transformations of the plane, it is preferable for students to first encounter examples that involve moving the entire plane.

They identify and describe translations, rotations, and reflections, and sequences of these. In describing images of figures under rigid transformations on and off square grids and the coordinate plane, students use the terms “corresponding points,” “corresponding sides,” and “image.” Students learn that angles and distances are preserved by any sequence of translations, rotations, and reflections, and that such a sequence is called a “rigid transformation.” They learn the definition of “congruent”: two figures are said to be congruent if there is a rigid transformation that takes one figure to the other. Students experimentally verify the properties of translations, rotations, and reflections, and use these properties to reason about plane figures, understanding informal arguments showing that the alternate interior angles cut by a transversal have the same measure and that the sum of the angles in a triangle is $$180^\circ$$. The latter will be used in a subsequent grade 8 unit on similarity and dilations. Throughout the unit, students discuss their mathematical ideas and respond to the ideas of others (MP3, MP6).

Many of the lessons in this unit ask students to work on geometric figures that are not set in a real-world context. This design choice respects the significant intellectual work of reasoning about area. Tasks set in real-world contexts are sometimes contrived and hinder rather than help understanding. Moreover, mathematical contexts are legitimate contexts that are worthy of study. Students do have opportunities in the unit to tackle real-world applications. In the culminating activity of the unit, students examine and create different patterns formed by plane figures. This is an opportunity for them to apply what they have learned in the unit (MP4).

In this unit, several lesson plans suggest that each student have access to a geometry toolkit. These contain tracing paper, graph paper, colored pencils, scissors, ruler, protractor, and an index card to use as a straightedge or to mark right angles, giving students opportunities to develop their abilities to select appropriate tools and use them strategically to solve problems (MP5). Note that even students in a digitally enhanced classroom should have access to such tools; apps and simulations should be considered additions to their toolkits, not replacements for physical tools.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as describing, generalizing, and justifying. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Describe

• movements of figures (Lessons 1 and 2)
• observations about transforming parallel lines (Lesson 9)
• transformations using corresponding points, line segments, and angles (Lesson 10)
• observations about angle measurements (Lesson 16)
• transformations found in tessellations and in designs with rotational symmetry (Lesson 17)

Generalize

• about categories for movement (Lesson 2)
• about rotating line segments $$180^\circ$$ (Lesson 8)
• about the relationship between vertical angles (Lesson 9)
• about transformations and congruence (Lesson 12)
• about corresponding segments and length (Lesson 13)
• about alternate interior angles (Lesson 14)
• about the sum of angles in a triangle (Lesson 16)

Justify

• whether or not rigid transformations could produce an image (Lesson 7)
• whether or not shapes are congruent (Lesson 11)
• whether or not polygons are congruent (Lesson 12)
• whether or not ovals are congruent (Lesson 13)
• whether or not triangles can be created from given angle measurements (Lesson 15)

In addition, students are expected to explain and interpret directions for transforming figures and how to apply transformations to find specific images. Students are also asked to use language to compare rotations of a line segment and compare perimeters and areas of rectangles. Over the course of the unit, teachers can support students’ mathematical understandings by amplifying (not simplifying) language used for all of these purposes as students demonstrate and develop ideas.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow the one in which it was first introduced.

lesson new terminology
receptive productive
8.1.1 vertex
plane
measure
direction
slide
turn
8.1.2 clockwise
counterclockwise
reflection
rotation
translation
opposite
8.1.3 image
angle of rotation
center (of rotation)
line of reflection
vertex
8.1.4 sequence of transformations
distance
clockwise
counterclockwise

reflect
rotate
translate
8.1.5 coordinate plane
point
segment
coordinates
$$x$$-axis
$$y$$-axis
8.1.6 polygon angle of rotation
center (of rotation)
line of reflection
8.1.7 rigid transformation
corresponding

measurements
preserve
reflection
rotation
translation

measure
point
8.1.8 midpoint segment
8.1.9 vertical angles
parallel
intersect
distance
8.1.10 image
rigid transformation

midpoint
parallel
8.1.11 congruent
perimeter
area
8.1.12 right angle
$$x$$-axis
$$y$$-axis
area
8.1.13 corresponding
8.1.14 alternate interior angles
transversal
vertical angles
congruent

supplementary angles
8.1.15 straight angle
8.1.16 alternate interior angles
transversal
straight angle
8.1.17 tessellation
symmetry

### Unit 2: Dilations, Similarity, and Introducing Slope

Work with transformations of plane figures in grade 8 builds on earlier work with geometry and geometric measurement, using students’ familiarity with geometric figures, their knowledge of formulas for the areas of rectangles, parallelograms, and triangles, and their abilities to use rulers and protractors. Grade 7 work with scaled copies is especially relevant. This work was limited to pairs of figures with the same rotation and mirror orientations (i.e. that are not rotations or reflections of each other). In grade 8, students study pairs of scaled copies that have different rotation or mirror orientations, examining how one member of the pair can be transformed into the other, and describing these transformations. Initially, they view transformations as moving one figure in the plane onto another figure in the plane. As the unit progresses, they come to view transformations as moving the entire plane.

Through activities designed and sequenced to allow students to make sense of problems and persevere in solving them (MP1), students use and extend their knowledge of geometry and geometric measurement. Students begin the first lesson of the unit by looking at cut-out figures, first comparing them visually to determine if they are scaled copies of each other, then representing the figures in a diagram, and finally representing them on a circular grid with radial lines. They encounter the term “scale factor” (familiar from grade 7) and the new terms “dilation” and “center of dilation.” In the next lesson, students again use a circular grid with radial lines to understand that under a dilation the image of a circle is a circle and the image of a line is a line parallel to the original. During the rest of the unit, students draw images of figures under dilations on and off square grids and the coordinate plane. In describing correspondences between a figure and its dilation, they use the terms “corresponding points,” “corresponding sides,” and “image.” Students learn that angle measures are preserved under a dilation, but lengths in the image are multiplied by the scale factor. They learn the definition of “similar”: two figures are said to be similar if there is a sequence of translations, rotations, reflections, and dilations that takes one figure to the other. They use the definition of “similar” and properties of similar figures to justify claims of similarity or non-similarity and to reason about similar figures (MP3). Using these properties, students conclude that if two triangles have two angles in common, then the triangles must be similar. Students also conclude that the quotient of a pair of side lengths in a triangle is equal to the quotient of the corresponding side lengths in a similar triangle. This conclusion is used in the lesson that follows: students learn the terms “slope” and “slope triangle,” and use the similarity of slope triangles on the same line to understand that any two distinct points on a line determine the same slope (MP7). In the following lesson, students use their knowledge of slope to find an equation for a line. They will build on this initial work with slope in a subsequent grade 8 unit on linear relationships. Throughout the unit, students discuss their mathematical ideas and respond to the ideas of others (MP3, MP6).

Many of the lessons in this unit ask students to work on geometric figures that are not set in a real-world context. This design choice respects the significant intellectual work of reasoning about area. Tasks set in real-world contexts are sometimes contrived and hinder rather than help understanding. Moreover, mathematical contexts are legitimate contexts that are worthy of study. Students do have opportunities in the unit to tackle real-world applications. In the culminating activity of the unit, students examine shadows cast by objects in the Sun. This is an opportunity for them to apply what they have learned about similar triangles (MP4).

In this unit, several lesson plans suggest that each student have access to a geometry toolkit. Each toolkit contains tracing paper, graph paper, colored pencils, scissors, ruler, protractor, and an index card to use as a straightedge or to mark right angles, giving students opportunities to develop their abilities to select appropriate tools and use them strategically to solve problems (MP5). Note that even students in a digitally enhanced classroom should have access to such tools; apps and simulations should be considered additions to their toolkits, not replacements for physical tools.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as describing, explaining, representing, and justifying. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Describe

• observations about scaled rectangles (Lesson 1)
• observations about dilated points, circles, and polygons (Lesson 2)
• sequences of transformations (Lesson 6)
• observations about side lengths in similar triangles (Lesson 9)

Explain

• how to apply dilations to find specific images (Lesson 5)
• how to determine whether triangles are congruent, similar, or neither (Lesson 8)
• strategies for finding missing side lengths (Lesson 9)
• how to apply dilations to find specific images of points (Lesson 12)
• reasoning for a conjecture (Lesson 13)

Represent

• dilations using given scale factors and coordinates (Lesson 4)
• figures using specific transformations (Lesson 6)
• graphs of lines using equations (Lesson 12)

In addition, students are expected to use language to interpret directions for dilating figures and for creating triangles; compare dilated polygons and methods for determining similarity; critique reasoning about angles, sides, and similarity; justify whether polygons are similar; and generalize about points on a line and similar triangles.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow the lesson in which it was first introduced.

lesson new terminology
receptive productive
8.2.1 scale factor
scaled copy
scaling
8.2.2 dilation
center of a dilation

dilate
8.2.4 center of a dilation
scale factor
8.2.6 similar dilate
8.2.7 dilation
8.2.9 quotient
8.2.10 similar slope
slope triangle
8.2.11 similarity
$$x$$-coordinate
$$y$$-coordinate
equation of a line
quotient
8.2.13 estimate
approximate / approximately

### 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. In this unit, students gain experience with linear relationships and their representations as graphs, tables, and equations through activities designed and sequenced to allow them to make sense of problems and persevere in solving them (MP1). Because of this dependency, this unit and the previous one should be done in order.

The unit begins by revisiting different representations of proportional relationships (graphs, tables, and equations), and the role of the constant of proportionality in each representation and how it may be interpreted in context (MP2).

Next, students analyze the relationship between number of cups in a given stack of cups and the height of the stack—a relationship that is linear but not proportional—in order to answer the question “How many cups are needed to get to a height of 50 cm?” They are not asked to solve this problem in a specific way, giving them an opportunity to choose and use strategically (MP5) representations that appeared earlier in this unit (table, equation, graph) or in the previous unit (equation, graph). Students are introduced to “rate of change” as a way to describe the rate per 1 in a linear relationship and note that its numerical value is the same as that of the slope of the line that represents the relationship. Students analyze another linear relationship (height of water in a cylinder vs number of cubes in the cylinder) and establish a way to compute the slope of a line from any two distinct points on the line via repeated reasoning (MP8). They learn a third way to obtain an equation for a linear relationship by viewing the graph of a line in the coordinate plane as the vertical translation of a proportional relationship (MP7).

So far, the unit has involved only lines with positive slopes and $$y$$-intercepts. Students next consider the graph of a line with a negative $$y$$-intercept and equations that might represent it. They consider situations represented by linear relationships with negative rates of change, graph these (MP4), and interpret coordinates of points on the graphs in context (MP2).

The unit concludes with two lessons that involve graphing two equations in two unknowns and finding and interpreting their solutions (MP2). Doing this involves considering correspondences among different representations (MP1), in particular, what it means for a pair of values to be a solution for an equation and the correspondence between coordinates of points on a graph and solutions of an equation.

In this unit, several lesson plans suggest that each student have access to a geometry toolkit. Each toolkit contains tracing paper, graph paper, colored pencils, scissors, ruler, protractor, and an index card to use as a straightedge or to mark right angles, giving students opportunities to select appropriate tools and use them strategically to solve problems (MP5). Note that even students in a digitally enhanced classroom should have access to such tools; apps and simulations should be considered additions to their toolkits, not replacements for physical tools.

On using the terms ratio, rate, and proportion. In these materials, a quantity is a measurement that is or can be specified by a number and a unit, e.g., 4 oranges, 4 centimeters, “my height in feet,” or “my height” (with the understanding that a unit of measurement will need to be chosen). The term ratio is used to mean an association between two or more quantities and the fractions $$\frac{a}{b}$$ and $$\frac{b}{a}$$ are never called ratios. The fractions $$\frac{a}{b}$$ and $$\frac{b}{a}$$ are identified as “unit rates” for the ratio $$a : b$$. The word “per” is used with students in interpreting a unit rate in context, as in “\\$3 per ounce,” and “at the same rate” is used to signify a situation characterized by equivalent ratios.

In grades 6–8, students write rates without abbreviated units, for example as “3 miles per hour” or “3 miles in every 1 hour.” Use of notation for derived units such as $$\frac{\text{mi}}{\text{hr}}$$ waits for high school—except for the special cases of area and volume. Students have worked with area since grade 3 and volume since grade 5. Before grade 6, they have learned the meanings of such things as sq cm and cu cm. After students learn exponent notation in grade 6, they also use $$\text{ cm}^2$$ and $$\text{ cm}^3$$.

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 $$\frac{a}{b}$$ as “ratios.”

A proportional relationship between two quantities represented by $$a$$ and $$b$$ is associated with two constants of proportionality: $$\frac{a}{b}$$ and $$\frac{b}{a}$$. Throughout the unit, the convention is if $$a$$ and $$b$$ are represented by columns in a table and the column for $$a$$ is to the left of the column for $$b$$, then $$\frac{b}{a}$$.is the constant of proportionality for the relationship represented by the table.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as representing, generalizing, and explaining. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Represent

• situations involving proportional relationships (Lesson 1)
• constants of proportionality in different ways (Lesson 3)
• slope using expressions (Lesson 7)
• linear relationships using graphs, tables, equations, and verbal descriptions (Lesson 8)
• situations using negative slopes and slopes of zero (Lesson 9)
• situations by graphing lines and writing equations (Lesson 12)
• situations involving linear relationships (Lesson 14)

Generalize

• categories for graphs (Lesson 2)
• about equations and linear relationships (Lesson 7)
• in order to make predictions about the slope of lines (Lesson 10)

Explain

• how to graph proportional relationships (Lesson 3)
• how to use a graph to determine information about a linear situation (Lessons 5 and 6)
• how to graph linear relationships (Lesson 10)
• how slope relates to changes in a situation (Lesson 11)

In addition, students are expected to describe observations about the equation of a translated line and describe features of an equation that could make one variable easier or harder to solve for than the other. Students will also have opportunities to use language to interpret situations involving proportional relationships, interpret graphs using different scales, interpret slopes and intercepts of linear graphs, justify reasoning about linear relationships, justify correspondences between different representations, and justify which equations correspond to graphs of horizontal and vertical lines.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow the one in which it was first introduced.

lesson new terminology
receptive productive
8.3.1 represent
scale
label
constant of proportionality
8.3.2 equation
8.3.3 rate of change equation
8.3.5 linear relationship
constant rate
slope
8.3.6 vertical intercept
$$y$$-intercept
8.3.7 initial (value or amount) constant rate
8.3.8 relate
8.3.9 horizontal intercept
$$x$$-intercept
8.3.10 intersection point rate of change
vertical intercept

$$y$$-intercept
8.3.11 constraint horizontal line
vertical line
8.3.12 solution to an equation with two variables
variable
combination
set of solutions

### Unit 4: Linear Equations and Linear Systems

In this unit, students build on their grades 6 and 7 work with equivalent expressions and equations with one occurrence of one variable, learning algebraic methods to solve linear equations with multiple occurrences of one variable. Students learn to use algebraic methods to solve systems of linear equations in two variables, building on their grades 7 and 8 work with graphs and equations of linear relationships. Understanding of linear relationships is, in turn, built on the understanding of proportional relationships developed in grade 7 that connected ratios and rates with lines and triangles.

The unit begins with a lesson on “number puzzles” in which students are shown a number line diagram that displays numerical changes (e.g., as in grade 7 work with signed numbers) and asked to write descriptions of situations and equations that the diagram could represent. Students are then given descriptions of situations in which an unknown quantity is linearly related to a combination of known quantities and asked to determine the unknown quantities in any way they can, e.g., using diagrams or writing equations.

In the second and third sections of the unit, students write and solve equations, abstracting from contexts (MP2) to represent a problem situation, stating the meanings of symbols that represent unknowns (MP6), identifying assumptions such as constant rate (MP4), selecting methods and representations to use in obtaining a solution (MP5), reasoning to obtain a solution (MP1), interpreting solutions in the contexts from which they arose (MP2) and writing them with appropriate units (MP6), communicating their reasoning to others (MP3), and identifying correspondences between verbal descriptions, tables, diagrams, equations, and graphs, and between different solution approaches (MP1).

The second section focuses on linear equations in one variable. Students analyze “hanger diagrams” that depict two collections of shapes that balance each other. Assuming that identical shapes have the same weight, they decide which actions of adding or removing weights preserve that balance. Given a hanger diagram that shows one type of shape with unknown weight, they use the diagram and their understanding of balance to find the unknown weight. Abstracting actions of adding or removing weights that preserve balance (MP7), students formulate the analogous actions for equations, using these along with their understanding of equivalent expressions to develop algebraic methods for solving linear equations in one variable. They analyze groups of linear equations in one unknown, noting that they fall into three categories: no solution, exactly one solution, and infinitely many solutions. They learn that any one such equation is false, true for one value of the variable, or (using properties of operations) true for all values of the variable. Given descriptions of real-world situations, students write and solve linear equations in one variable, interpreting solutions in the contexts from which the equations arose.

The third section focuses on systems of linear equations in two variables. It begins with activities intended to remind students that a point lies on the graph of a linear equation if and only if its coordinates make the equation true. Given descriptions of two linear relationships students interpret points on their graphs, including points on both graphs. Students categorize pairs of linear equations graphed on the same axes, noting that there are three categories: no intersection (lines distinct and parallel, no solution), exactly one intersection (lines not parallel, exactly one solution), and same line (infinitely many solutions).

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as critiquing, justifying, and generalizing. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Critique

• strategies for solving puzzles (Lesson 1)
• reasoning about maintaining balance in equations (Lesson 3)
• solutions of linear equations (Lessons 4 and 5)
• reasoning about structures of systems of equations (Lesson 14)
• explanations of solutions (Lesson 16)

Justify

• strategies for solving puzzles (Lessons 1 and 5)
• predictions about maintaining balance (Lesson 2)
• predictions about solutions of linear equations (Lesson 6)

Generalize

• about the structures of equations that have one, infinite, and no solutions (Lessons 7 and 8)
• about the structures of systems of equations (Lessons 14 and 15)

In addition, students are expected to use language to represent and interpret situations involving systems of linear equations, compare solutions of linear equations, and describe graphs of systems of linear equations.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow the one in which it was first introduced.

lesson new terminology
receptive productive
8.4.1 representation
8.4.2 expression
8.4.3 solution to an equation
distribute
8.4.4 substitute equation
8.4.5 term
like terms
distributive property
factor
8.4.6 term
like terms
distribute
common denominator
8.4.7 no solution
(only) one solution
8.4.8 constant term
coefficient

linear equation
infinitely many solutions
expression
variable
8.4.11 ordered pair
8.4.12 system of equations
solution to a system of equations
8.4.13 substitution substitute
no solution
(only) one solution
infinitely many solutions
8.4.14 algebraically
8.4.15 system of equations
substitution

### Unit 5: Functions and Volume

In this unit, students are introduced to the concept of a function as a relationship between “inputs” and “outputs” in which each allowable input determines exactly one output. In the first three sections of the unit, students work with relationships that are familiar from previous grades or units (perimeter formulas, proportional relationships, linear relationships), expressing them as functions. In the remaining three sections of the unit, students build on their knowledge of the formula for the volume of a right rectangular prism from grade 7, learning formulas for volumes of cylinders, cones, and spheres. Students express functional relationships described by these formulas as equations. They use these relationships to reason about how the volume of a figure changes as another of its measurements changes, transforming algebraic expressions to get the information they need (MP1).

The first section begins with examples of “input–output rules” such as “divide by 3” or “if even, then . . . ; if odd, then . . . ” In these examples, the inputs are (implicitly) numbers, but students note that some inputs are not allowable for some rules, e.g., $$\frac 12$$ is not even or odd. Next, students work with tables that list pairs of inputs and outputs for rules specified by “input–output diagrams,” noting that a finite list of pairs does not necessarily determine a unique input–output rule (MP6). Students are then introduced to the term “function” as describing a relationship that assigns exactly one output to each allowable input.

In the second section, students connect the terms “independent variable” and “dependent variable” (which they learned in grade 6) with the inputs and outputs of a function. They use equations to express a dependent variable as a function of an independent variable, viewing formulas from earlier grades (e.g., $$C = 2 \pi r$$), as determining functions. They work with tables, graphs, and equations for functions, learning the convention that the independent variable is generally shown on the horizontal axis. They work with verbal descriptions of a function arising from a real-world situation, identifying tables, equations, and graphs that represent the function (MP1), and interpreting information from these representations in terms of the real-world situation (MP2).

The third section of the unit focuses on linear and piecewise linear functions. Students use linear and piecewise linear functions to model relationships between quantities in real-world situations (MP4), interpreting information from graphs and other representations in terms of the situations (MP2). The lessons on linear functions provide an opportunity for students to coordinate and synthesize their understanding of new and old terms that describe aspects of linear and piecewise functions. In working with proportional relationships in grade 7, students learned the term “constant of proportionality,” and that any proportional relationship can be represented by an equation of the form $$y = kx$$ where $$k$$ is the constant of proportionality, that its graph lies on a line through the origin that passes through Quadrant I, and that the constant of proportionality indicates the steepness of the line. In an earlier grade 8 unit, students were introduced to “rate of change” as a way to describe the rate per 1 in a linear relationship and noted that its numerical value is the same as that of the slope of the line that represents the relationship. In this section, students connect their understanding of “increasing” and “decreasing” from the previous section with their understanding of linear functions, noting, for example, that if a linear function is increasing, then its graph has positive slope, and that its rate of change is positive. Similarly, they connect their understanding of $$y$$-intercept (learned in an earlier unit) with the new term “initial value,” noting, for example, when the numerical part of an initial value of a function is given by the $$y$$-intercept of its graph (MP1).

In the remaining three sections of the unit, students work with volume, using abilities developed in earlier work with geometry and geometric measurement.

Students’ work with geometry began in kindergarten, where they identified and described two- and three-dimensional shapes that included cones, cylinders, and spheres. They continued this work in grades 1 and 2, composing, decomposing, and identifying two- and three-dimensional shapes.

Students’ work with geometric measurement began with length and continued with area and volume. 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 connected rectangle area with multiplication, understanding why (for whole-number side lengths) multiplying the side lengths of a rectangle yields the number of unit squares that tile the rectangle. They used area models to represent instances of the distributive property. In grade 4, students used area and perimeter formulas for rectangles to solve real-world and mathematical problems. In grade 5, students extended the formula for the area of rectangles to rectangles with fractional side lengths. They found volumes of right rectangular prisms by viewing them as layers of arrays of cubes and used formulas to calculate these volumes as products of edge lengths or as products of base area and height. In grade 6, students extended the formula for the volume of a right rectangular prism to right rectangular prisms with fractional side lengths and used it to solve problems. They extended their reasoning about area to include shapes not composed of rectangles and 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 analyzed and described cross-sections of prisms (including prisms with polygonal but non-rectangular bases), pyramids, and polyhedra, and used the formula for the volume of a right rectangular prism (volume is area of base times height of prism) to solve problems involving area, surface area, and volume.

In this grade 8 unit, students extend their understanding of volume from right rectangular prisms to right cylinders, right cones, and spheres. They begin by investigating the volume of water in a graduated cylinder as a function of the height of the water, and vice versa. They examine depictions of of a cylinder, prism, sphere, and cone, in order to develop their abilities to identify radii, bases, and heights of these objects. They estimate volumes of prisms, cylinders, cones, and spheres, in order to reinforce the idea that a measurement of volume indicates the amount of space within an object. Students use their abilities to identify radii, bases, and heights, together with the geometric abilities developed in earlier grades, to perceive similar structure (MP7) in formulas for the volume of a rectangular prism and the volume of a cylinder—both are the product of base and height. After gaining familiarity with a formula for the volume of a cylinder by using it to solve problems, students perceive similar structure (MP7) in a formula for the volume of a cone.

The fifth section of the unit begins with an examination of functional relationships between two quantities that are illustrated by changes in scale for three-dimensional figures. For example, if the radius of a cylinder triples, its volume becomes nine times larger. This work combines grade 7 work on scale and proportional relationships. In grade 7, students studied scaled copies of two-dimensional figures, recognizing lengths are scaled by a scale factor and areas by the square of the scale factor, and applied their knowledge to scale drawings, e.g., maps and floor plans. In their study of proportional relationships, grade 7 students solved problems set in contexts commonly associated with proportional relationships such as constant speed, unit pricing, and measurement conversions, and learned that any proportional relationship can be represented by an equation of the form $$y = kx$$ where $$k$$ is the constant of proportionality. In this section, students use their knowledge of scale, proportional relationships, and volume to reason about how the volume of a prism, cone, or cylinder changes as another measurement changes.

In the last section of the unit, students reason about how the volume of a sphere changes as its radius changes. They consider a situation in which water flows into a cylinder, cone, and sphere at the same constant rate. Information about the height of the water in each container is shown in an equation, graph, or table, allowing students to use it strategically (MP5) to compare water heights and capacities for the containers.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as generalizing, justifying, and comparing. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Generalize

• about what happens to inputs for each rule (Lesson 1)
• about dimensions of cylinders (Lesson 14)
• about the relationship between the volumes of cylinders and cones (Lesson 15)
• about dimensions of cones (Lesson 16)
• about volumes of spheres, cones, and cylinders as functions of their radii (Lesson 21)

Justify

• claims about what can be determined from given information (Lesson 2)
• claims about volumes of cubes and spheres based on graphs (Lesson 7)
• claims about approximately linear relationships (Lesson 10)
• reasoning about the volumes of spheres and cones (Lesson 21)

Compare

• different representations of functions (Lesson 3)
• features of graphs, equations, and situations (Lesson 4)
• features of a situation with features of a graph (Lesson 6)
• temperatures shown on a graph with different temperatures given in a table (Lesson 7)
• the volumes of cones with the volumes of cylinders (Lesson 16)
• methods for finding and approximating the volume of a sphere as function of its radius (Lesson 20)

In addition, students are expected to interpret representations of volume functions of cylinders, cones, and spheres; describe quantities in a situation; describe volume measurements and features of three dimensional figures; describe the effects of varying dimensions of rectangular prisms and cones on their volumes; and describe and represent approximately linear relationships. Students are also expected to use language to represent relationships between volume and variable side length of a rectangular prism and relationships between volume and variable height of a cylinder; explain and represent how height and volume of cylinders are related; and explain reasoning about finding the volume of a cylinder and about the relationship between volumes of hemispheres and volumes of boxes, cylinders, and cones.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow the one in which it was first introduced.

lesson new terminology
receptive productive
8.5.1 input
output
8.5.2 function input
output
depends on
8.5.3 independent variable
dependent variable
8.5.5 prediction
8.5.7 volume
cube
8.5.8 functional relationship
linear function
function
8.5.9 mathematical model prediction
8.5.10 piecewise linear function linear function
constant rate
8.5.11 cylinder
three-dimensional
8.5.12 cone
sphere

dimension
cylinder
cube
cubic centimeter
rectangular prism
8.5.13 base (of a cylinder or cone)
approximation for $$\pi$$
base (of a cylinder or cone)
8.5.16 cone
8.5.19 hemisphere
8.5.20 sphere
8.5.21 spherical volume
8.5.22 approximate range

### Unit 6: Associations in Data

In this unit, students analyze bivariate data—using scatter plots and fitted lines to analyze numerical data, and using two-way tables, bar graphs, and segmented bar graphs to analyze categorical data.

The unit begins with an investigation of a table of data. Measurements of a leg and perimeter of an isosceles right triangle are shown in each row, but column entries are not in order, making it hard to discern a pattern. Students manipulate the data to look for patterns in the table (MP7), then examine a scatter plot of the same data. This motivates the need to use different representations of the same data to find and analyze any patterns.

The second section begins with investigation of two questions: “Are older students always taller?” and “Do taller students tend to have bigger hands?” Students collect data (measurements of each student’s arm span, hand span, and height) and record each student’s measurements together with the student’s age in months. They make a scatter plot for height vs. hand span and select their own methods to display the height data (MP5).

The second section focuses on using scatter plots and fitted lines to analyze numerical data. Students make and examine scatter plots, interpreting points in terms of the quantities represented (MP2) and identifying scatter plots that could represent verbal descriptions of associations between two numerical variables (MP1). They see examples of how a line can be used to model an association between measurements displayed in a scatter plot and they compare values predicted by a linear model with the actual values given in the scatter plot (MP4). They draw lines to fit data displayed in scatter plots and informally assess how well the line fits by judging the closeness of the data points to the lines (MP4). Students compare scatter plots that show different types of associations (MP7) and learn to identify these types, making connections between the overall shape of a cloud of points and trends in the data represented, e.g., a scatter plot of used car price vs. mileage shows a cloud of points that descends from left to right and prices of used cars decrease with increased mileage (MP2). They make connections between the overall shape of a cloud of points, the slope of a fitted line, and trends in the data, e.g., “a line fit to the data has a negative slope and the scatter plot shows a negative association between price of a used car and its mileage.” Outliers are informally identified based on their relative distance from other points in a scatter plot. Students examine scatter plots that show linear and non-linear associations as well as some sets of data that show clustering, describing their differences (MP7). They return to the data on height and arm span gathered at the beginning of the unit, describe the association between the two, and fit a line to the data (MP4).

The third section focuses on using two-way tables to analyze categorical data (MP4). Students use a two-way frequency table to create a relative frequency table to examine the percentages represented in each intersection of categories to look for any associations between the categories. Students also examine and create bar and segmented bar graphs to visualize any associations.

The unit ends with a lesson in which students collect and analyze numerical data using a scatter plot, then categorize the data based on a threshold and analyze the categories based on a two-way table (MP4).

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as explaining, representing, and interpreting. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Explain

• how to estimate using available data (Lesson 1)
• how to use tables and scatter plots to make estimates and predictions (Lesson 3)
• the meaning of slope for a situation (Lesson 6)
• how to use lines to show associations, identify outliers, and answer questions (Lesson 8)

Represent

• data in organized ways (Lesson 1)
• data using two-way tables, bar graphs, and segmented bar graphs (Lessons 9 and 10)
• data using scatter plots (Lesson 11)

Interpret

• situations and graphs involving bivariate data (Lesson 2)
• tables and scatter plots of bivariate data (Lesson 3)
• tables, scatter plots, equations, and situations involving bivariate data (Lesson 4)

In addition, students are expected to compare different representations of the same situation, describe and compare features of scatter plots, justify whether or not lines are good fits for a situation, and justify associations between bivariate data. Students also have opportunities to use language to generalize about what makes a line fit a data set well and generalize about categories for sorting scatter plots.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow the one in which it was first introduced.

lesson new terminology
receptive productive
8.6.1 scatter plot
8.6.2 data display
attribute
numerical data
categorical data
8.6.4 outlier
predict
overpredict
underpredict
linear model
8.6.5 positive association
negative association

linear association
8.6.6 nonlinear association
no association
fitted line
8.6.7 cluster
8.6.8 independent variable
dependent variable
positive association
negative association

linear association
8.6.9 segmented bar graph
relative frequency
two-way (frequency) table
8.6.11 scatter plot
outlier

cluster

### Unit 7: Exponents and Scientific Notation

Students were introduced to exponent notation in grade 6. They worked with expressions that included parentheses and positive whole-number exponents with whole-number, fraction, decimal, or variable bases, using properties of exponents strategically, but did not formulate rules for use of exponents.

In this unit, students build on their grade 6 work. The first section of the unit begins with a lesson that reviews exponential expressions, including work with exponential expressions with bases 2 and $$\frac 12$$. In the next two lessons, students examine powers of 10, formulating the rules $$10^n \boldcdot 10^m = 10^{n+m}$$, $$({10^n})^m = 10^{n \boldcdot m}$$, and, for $$n>m$$, $$\frac{10^n}{10^m} = 10^{n-m}$$ where $$n$$ and $$m$$ are positive integers. After working with these powers of 10, they consider what the value of $$10^0$$ should be and define $$10^0$$ to be 1. In the next lesson, students consider what happens when the exponent rules are used on exponential expressions with base 10 and negative integer exponents and define $$10^{-n}$$ to be $$\frac{1}{10^n}$$. In the next two lessons, they expand their work to numerical bases other than 10, using exponent rules with products of exponentials with the same base and contrasting it with products of exponentials with different bases. They note numerical instances of $$a^n \boldcdot b^n = (a \boldcdot b)^n$$.

The third section of the unit returns to powers of 10 as a prelude to the introduction of scientific notation. Students consider differences in magnitude of powers of 10 and use powers of 10 and multiples of powers of 10 to describe magnitudes of quantities, e.g., the distance from Earth to the Sun or the population of Russia. Initially, they work with large quantities, locating powers of 10 and positive-integer multiples of powers of 10 on the number line. Most of these multiples are products of single-digit numbers and powers of 10. The remainder are products of two- or three-digit numbers and powers of 10, allowing students to notice that these numbers may be expressed in different ways, e.g., $$75 \boldcdot 10^5$$ can be written $$7.5 \boldcdot 10^6$$, and that some forms may be more helpful in finding locations on the number line. In the next lesson, students do similar work with small quantities.

In the remaining five lessons, students write estimates of quantities in terms of integer or non-integer multiples of powers of 10 and use their knowledge of exponential expressions to solve problems, e.g., How many meter sticks does it take to equal the mass of the Moon? They are introduced to the term “scientific notation,” practice distinguishing scientific from other notation, and use scientific notation (with no more than three significant figures) in order to make additive and multiplicative comparisons of pairs of quantities. They compute sums, differences, products, and quotients of numbers written in scientific notation (some with as many as four significant figures), using such calculations to estimate quantities. They make measurement conversions that involve powers of ten, e.g., converting bytes to kilobytes or gigabytes, choose appropriate units for measurements and express them in scientific notation.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as critiquing, representing, and justifying. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Critique

• reasoning about powers of powers (Lesson 3)
• reasoning about zero exponents (Lesson 4)
• applications of exponent rules (Lesson 7)
• reasoning about scientific notation (Lesson 15)

Represent

• situations using exponents (Lesson 1)
• large and small numbers using number lines, exponents, and decimals (Lesson 9–11)
• situations comparing quantities expressed in scientific notation (Lesson 14)

Justify

• reasoning about multiplying powers of 10 (Lesson 2)
• reasoning about powers of powers (Lesson 3)
• reasoning about dividing powers of 10 (Lesson 4)
• whether or not expressions are equivalent to exponential expressions (Lesson 6)
• reasoning about situations comparing powers of 10 (Lesson 12)

In addition, students are expected to use language to generalize reasoning about repeated multiplication and generalize about patterns when multiplying different bases and exponents; describe how negative powers of 10 affect placement of decimals; and interpret situations comparing quantities expressed in scientific notation. Students also have opportunities to compare correspondences between exponential expressions and base-ten diagrams; compare expressions in scientific notation to other expressions; explain how to simplify expressions with negative powers of 10; and explain how to place and order large numbers on a number line.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students&rsqup; use of a new term in the lessons that follow the one in which it was first introduced.

lesson new terminology
receptive productive
8.7.1 exponent
power
factor
reciprocal
repeated multiplication
8.7.2 powers of 10
8.7.3 base (of an exponent)
power of powers
8.7.4 expanded
positive exponent
zero exponent
8.7.5 negative exponent positive exponent
8.7.6 exponent
base (of an exponent)
power
zero exponent
8.7.7 reciprocal
evaluate
factor
power of powers
negative exponent
8.7.8 square (of a number)
8.7.9 billion
trillion
multiple of
8.7.10 integer
8.7.12 multiple of
8.7.13 scientific notation integer
8.7.14 powers of 10
billion
trillion
8.7.15 scientific notation

### Unit 8: Pythagorean Theorem and Irrational Numbers

Work in this unit is designed to build on and connect students’ understanding of geometry and numerical expressions. The unit begins by foreshadowing algebraic and geometric aspects of the Pythagorean Theorem and strategies for proving it. Students are shown three squares and asked to compare the area of the largest square with the sum of the areas of the other two squares. The comparison can be done by counting grid squares and comparing the counts—when the three squares are on a grid with their sides on grid lines and vertices on intersections of grid lines—using the understanding of area measurement established in grade 3. The comparison can also be done by showing that there is a shape that can be decomposed and rearranged to form the largest square or the two smallest squares. Students are provided with opportunities to use and discuss both strategies.

In the second section, students work with figures shown on grids, using the grids to estimate lengths and areas in terms of grid units, e.g., estimating the side lengths of a square, squaring their estimates, and comparing them with estimates made by counting grid squares. The term “square root” is introduced as a way to describe the relationship between the side length and area of a square (measured in units and square units, respectively), along with the notation $$\sqrt{}$$. Students continue to work with side lengths and areas of squares. They learn and use definitions for “rational number” and “irrational number.” They plot rational numbers and square roots on the number line. They use the meaning of “square root,” understanding that if a given number $$p$$ is the square root of $$n$$, then $$x^2 = n$$. Students learn (without proof) that $$\sqrt 2$$ is irrational. They understand two proofs of the Pythagorean Theorem—an algebraic proof that involves manipulation of two expressions for the same area and a geometric proof that involves decomposing and rearranging two squares. They use the Pythagorean Theorem in two and three dimensions, e.g., to determine lengths of diagonals of rectangles and right rectangular prisms and to estimate distances between points in the coordinate plane.

In the third section, students work with edge lengths and volumes of cubes and other rectangular prisms. (In this grade, all prisms are right prisms.) They are introduced to the term “cube root” and the notation $$\sqrt{}$$. They plot square and cube roots on the number line, using the meanings of “square root” and “cube root,” e.g., understanding that if a given number $$x$$ is the square root of $$n$$ and $$n$$ is between $$m$$ and $$p$$, then $$x^2$$ is between $$m$$ and $$p$$ and that $$x$$ is between $$\sqrt{m}$$ and $$\sqrt{p}$$.

In the fourth section, students work with decimal representations of rational numbers and decimal approximations of irrational numbers. In grade 7, they used long division in order to write fractions as decimals and learned that such decimals either repeat or terminate. They build on their understanding of decimals to make successive decimal approximations of $$\sqrt 2$$ and $$\pi$$ which they plot on number lines.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as explaining, justifying, and comparing. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Explain

• strategies for finding area (Lesson 1)
• strategies for approximating and finding square roots (Lesson 4)
• strategies for finding triangle side lengths (Lesson 6)
• predictions about situations involving right triangles and strategies to verify (Lesson 10)
• strategies for finding distances between points on a coordinate plane (Lesson 11)
• strategies for approximating the value of cube roots (Lesson 13)

Justify

• which squares have side lengths in a given range (Lesson 1)
• ordering of irrational numbers (Lesson 5)
• ordering of hypotenuse lengths (Lesson 9)

Compare

• rational and irrational numbers (Lesson 3)
• lengths of diagonals in rectangular prisms (Lesson 10)
• strategies for approximating irrational numbers (Lesson 15)

In addition, students are expected to use language to generalize about area of squares, square roots, and approximations of side lengths and generalize about the distance between any two coordinate pairs; critique reasoning about square root approximations and critique a strategy to represent repeating decimal expansions as fractions; describe observations about the relationships between triangle side lengths and describe hypotenuses and side lengths for given triangles; interpret diagrams involving squares and right triangles; interpret equations and approximations for the value of square and cube roots; and represent relationships between side lengths and areas.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow the one in which it was first introduced.

lesson new terminology
receptive productive
8.8.2 square root square (of a number)
8.8.3 irrational number
square root symbol
rational number
8.8.4 diagonal
decimal approximation
8.8.5 square root
square root symbol
8.8.6 Pythagorean Theorem
hypotenuse
legs
right triangle
8.8.9 converse of the Pythagorean Theorem Pythagorean Theorem
8.8.10 edge length hypotenuse
legs
8.8.12 cube root
8.8.13 cube root
edge length
8.8.14 repeating decimal
decimal representation
finite decimal expansion
8.8.15 infinite decimal expansion irrational number
repeating decimal

### Unit 9: Putting It All Together

In these optional lessons, students solve complex problems. In the first several lessons, they consider tessellations of the plane, understanding and using the terms “tessellation” and “regular tessellation” in their work, and using properties of shapes (for example, the sum of the interior angles of a quadrilateral is 360 degrees) to make inferences about regular tessellations. These lessons need to come after unit 8.1 has been done. In the later lessons, they investigate relationships of temperature and latitude, climate, season, cloud cover, or time of day. In particular, they use scatter plots and lines of best fit to investigate the question of modeling temperature as a function of latitude. These lessons need to come after units 8.5 and 8.6 have been done.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes such as describing, representing, and justifying. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Describe

• tessellations (Lesson 1)
• associations in bivariate data (Lesson 5)

Represent

• the relationship between latitude and weather (Lesson 5)

Justify

• claims about shapes that can and cannot be used to produce regular tessellations (Lesson 2)

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow the one in which it was first introduced.

lesson new terminology
receptive productive
8.9.1   tessellation
pattern
8.9.2 regular tessellation regular polygon
8.9.6   mathematical model