In this unit students learn what makes figures similar and justify claims of similarity. They are introduced to the slope of a line and use properties of similar triangles to write equations that can describe all points \((x,y)\) on a given line.

In prior grades, students learned about the relationship between scale factors and scaled copies. Students expand on this in the first section where they learn about dilations as a new transformation that creates scaled copies. 

In the next section, students connect dilations to earlier work with rigid transformations as they explain why two figures are similar by describing a sequence of translations, reflections, rotations, and dilations that take one figure to the other. They discover that angle measures in similar figures are preserved, which can be used to justify that two triangles are similar if they share two (or three) angle measures. Students also find that the quotients of corresponding side lengths in similar figures are equal. This along with the fact that side lengths in similar figures are all multiplied by the same scale factor allows students to calculate unknown lengths in similar figures.

In the following section, students use the similarity of slope triangles to understand why any two distinct points on a line determine the same slope. Using these same properties of similar triangles, students practice writing equations for a given line, though students are not expected at this time to write equations in the form \(y=mx+b\).

The lessons in this unit ask students to work on geometric figures that are not set in a real-world context, as those tasks are sometimes contrived and hinder rather than help understanding. Students do have opportunities to tackle real-world applications in the culminating activity of the unit where students examine shadows cast by objects. 

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 in this course, 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 that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

This unit introduces students to nonproportional linear relationships by building on earlier work with rates and proportional relationships from grade 7, and on earlier grade 8 work around similarity and slope.

The unit begins by revisiting different representations of proportional relationships. Students create graphs, tables, and equations in order to interpret the constant of proportionality in a context. They see the constant of proportionality between two variables as the rate of change of one variable with respect to the other.

Next, students analyze a relationship that is linear but not proportional. In this context, students see that the rate of change has a numerical value that is the same as the slope of the line that represents the relationship. Students also view the graph of a line in the coordinate plane as the vertical translation of a proportional relationship.

In the following section, students are introduced to lines with non-positive slopes and vertical intercepts. They consider situations represented by linear relationships with negative rates of change and establish a way to compute the slope of a line from any two distinct points on the line. Students also write equations of horizontal and vertical lines.

In the last section, students consider what it means for a pair of values to be a solution to an equation and the correspondence between coordinates of points on a graph and solutions of an equation.

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 10).
• Linear relationships using graphs, tables, equations, and verbal descriptions (Lesson 5).
• Situations using negative slopes and slopes of zero (Lesson 9).
• Situations by graphing lines and writing equations (Lesson 13).
• Situations involving linear relationships (Lesson 15).

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 and 11).
• 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. 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 in this course, 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 that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

In this unit, students work with writing equivalent equations and use reasoning to solve equations for a variable. Then students solve systems of linear equations using graphic and algebraic methods.

The unit begins with a focus on moves that can be done to write equivalent equations. At first, students use hanger diagrams as an intuitive representation of equality and represent their reasoning by labeling arrows that connect equivalent representations. With the reintroduction of negative values, students move away from hanger diagrams to algebraic equations and writing equivalent equations with the intention of solving for a variable.

Next, students examine the conditions under which equations could have 0, 1, or infinite solutions as a transition to thinking about similar situations involving systems of equations. Students finish the unit by examining systems of equations graphically and then finding solutions algebraically. They build on their understanding that the line representing an equation with 2 variables is made up of coordinate pairs that make the equation true. They find that the intersection of 2 lines is the point that makes both equations for the system true. Students also recognize when systems have no solution or infinite solutions based on the graphs and the slope and intercept.

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 writing equivalent equations (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 writing equivalent equations (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 in this course, 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 that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

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. They study the different ways functions can be represented, making connections between the representations and interpreting what they mean in context. Linear functions are a focus of the third section, and students will continue to work with linear functions in a later unit to model data. The use of function notation is left for a future course.

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, focusing on situations involving proportional relationships. They use these relationships to reason about how the volume of a figure changes as one of its dimensions changes, transforming algebraic expressions to get the information they need. In future courses, students will continue this thinking as they study nonlinear relationships and question how, for example, the volume of a sphere changes as the radius increases.

\(r\)	\(V\)
0	0
2	\(\frac{32}{3} \pi\)
6	\(288 \pi\)
\(r\)	\(\frac{4}{3} \pi r^3\)

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 a function of its radius (Lesson 20).

In addition, students are expected to interpret representations of volume functions of cylinders, cones, and spheres and expected to describe the following: quantities in a situation, volume measurements and features of three-dimensional figures, the effects of varying dimensions of rectangular prisms and cones on their volumes, 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 in this course, 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 that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

In this unit, students analyze bivariate data. They will use scatter plots and fitted lines to analyze numerical data, and two-way tables, bar graphs, and segmented bar graphs to analyze categorical data. Students advance their understanding of lines by examining slopes in the context of data. They will revisit these data analysis topics in a later course in more depth. At this level, students should be able to construct and interpret points on a scatter plot, informally fit linear models to data, interpret a given linear model in the context of data, and generally recognize patterns of association using relative frequencies in a two-way table.

In prior grades, students analyzed data collected about one variable using dot plots, histograms, and box plots. This unit expands on that by considering the possible influence of a second variable on measurements about individuals. 
In the first section, students are introduced to scatter plots and are reminded how to interpret points on a graph using a context. They also begin to recognize general trends in data.

In the second section, students look more closely at associations in data by informally drawing lines that model the general trend of the data. They also classify associations as positive, negative, linear, and non-linear by looking at the shape of the data in a scatter plot.

In the third section, students look at categorical data using two-way tables and relative frequencies. They then informally look at the relative frequencies to notice whether the variables are associated or not.

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.

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 about categories for sorting scatter plots.

The table shows lessons where new terminology is first introduced in this course, 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 that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

In this unit, students deepen their understanding of exponents, powers of 10, and place value before being introduced to scientific notation. They build on work done in a previous course where students focused on whole-number exponents with whole-number, fraction, decimal, or variable bases, but did not formulate rules regarding the use of exponents.

Students begin this unit by identifying patterns that emerge when multiplying and dividing powers of 10, and when raising powers of 10 to another power. Students generalize these patterns to develop exponent rules. They extend these rules to see why \(10^0\) must be equal to 1 and to understand what negative exponents mean.

Next, students determine that the rules developed for powers of 10 also work with other bases, as long as the bases in both expressions are the same. They observe a new rule that applies when multiplying bases that are different if the exponents are the same.

In the next section, students return to working with powers of 10 as they use multiples of powers of 10 to describe magnitudes of very large and very small quantities, such as the distance from Earth to the sun in kilometers or the mass of a proton in grams. Students plot these large and small values on number lines labeled using exponents and see how these numbers can be expressed in different ways — for example as \(75\boldcdot10^5\) or \(7.5\boldcdot10^6\).

After building a foundation connecting powers of 10 with place value, students are finally introduced to scientific notation as a specific and useful way of writing numbers as a power of 10. They compute sums, differences, products, and quotients of numbers written in scientific notation to make additive and multiplicative comparisons, estimate quantities, and make measurement conversions.

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, 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 in this course, 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 that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

This unit introduces students to irrational numbers with a focus on connecting geometric and algebraic representations of square roots, cube roots, and the Pythagorean Theorem. 

In the first section, students extend work from grade 6, composing and decomposing shapes to find the areas of tilted squares. They see “square root of \(n\)” and \(\sqrt{n}\) to mean the side length of a square with area \(n\) square units, and understand that finding the solution to equations of the form \(x^2=n\) means determining which values of \(x\) make the equation true. Students learn and use definitions for “rational number” and “irrational number,” learn (without proof) that \(\sqrt{2}\) is irrational, and plot square roots on the number line.

Three right triangles are indicated. A square is drawn using each side of the triangles. The triangle on the left has the square labels “a squared equals 16” and “b squared equals 9” attached to each of the legs. The square labeled “c squared equals 25” is attached to the hypotenuse. The triangle in the middle has the square labels “a squared equals 16” and “b squared equals 1” attached to each of the legs. The square labeled “c squared equals 17” is attached to the hypotenuse. The triangle on the right has the square labels “a squared equals 9” and “b squared equals 9” attached to each of the legs. The square labeled “c squared equals 18” is attached to the hypotenuse.

In the second section, students continue using tilted squares as they investigate relationships between side lengths of right and non-right triangles. Students are encouraged to notice patterns among the triangles before being shown geometric and algebraic proofs of the Pythagorean Theorem. They use the Pythagorean Theorem and its converse to solve problems in two and three dimensions, for example, 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 see that “cube root of \(n\)"  and \(\sqrt[3]{n}\) mean the side length of a cube with volume \(n\) cubic units. They also represent a cube root as a decimal approximation and as a point on the number line.

In the fourth section, students consider the decimal expansions of rational and irrational numbers. They learn how to rewrite fractions as a repeating decimal, how to rewrite a repeating decimal as a fraction, and reinforce their understanding that irrational numbers have a place on the number line even if they cannot be written as a fraction of integers.

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 5).
• Strategies for finding triangle side lengths (Lesson 7).
• Predictions about situations involving right triangles and strategies to verify (Lesson 11).
• Strategies for finding distances between points on a coordinate plane (Lesson 13).
• Strategies for approximating the value of cube roots (Lesson 15).

Justify

• Which squares have side lengths in a given range (Lesson 2).
• Ordering of irrational numbers (Lesson 6).
• Ordering of hypotenuse lengths (Lesson 10).

Compare

• Rational and irrational numbers (Lesson 4).
• Lengths of diagonals in rectangular prisms (Lesson 11).
• Strategies for approximating irrational numbers (Lesson 17).

In addition, students will do the following: Use language to generalize about area of squares, square roots, and approximations of side lengths and 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 in this course, 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 that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

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, as well as using properties of shapes (for example, the sum of the interior angles of a quadrilateral is 360 degrees) and transformations to make inferences about regular tessellations. These lessons need to come after unit 8.1 has been done. In the later lessons, they investigate factors that impact predicting the temperature. In particular, they use scatter plots and lines of best fit to model the association between temperature and latitude. These lessons need to come after units 8.5 and 8.6 have been done.

All related standards in this unit have been addressed in prior units. These sections provide an optional opportunity for students to go more deeply and make connections between domains.

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 in this course, 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 that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow the one in which it was first introduced.