This unit begins with constructions, continues to rigid transformations, and concludes with an introduction to proof writing. In grade 8, students determined the angle-preserving and length-preserving properties of rigid transformations experimentally, mostly with the help of a coordinate grid. Students have also previously studied the angle properties that they will prove in this unit.

Constructions play a significant role in the logical foundation of geometry. A focus of this unit is for students to explore properties of shapes in the plane without the aid of given measurements. At this point, students have worked so much with numbers, equations, variables, coordinate grids, and other quantifiable structures that it may come as a surprise just how far they can push concepts in geometry without measuring distances or angles. 

At the beginning of the unit, students have the opportunity to move from informal explorations of lines and arcs to generating conjectures and writing justifications based on their constructions. The definition of a circle is an important foundation for concepts in this unit and throughout the course.

Next, students recall transformations from previous grades and learn to create rigid motions using construction tools instead of a grid. This leads to rigorous definitions of rotations, reflections, and translations without reference to a coordinate grid. 
 

Finally, students begin to use the definitions they have learned to prove theorems. They begin to express their reasoning more formally, moving from vague statements (“It looks like . . .”) toward the more formal point-by-point transformation proofs used by mathematicians.

Starting in the second section, a blank reference chart is provided for students, and a completed reference chart is provided for teachers. The reference chart is a resource for students to refer to as they make formal arguments. Students will continue adding to it throughout the course. Refer to the Course Guide for more information.

These materials use words rather than symbolic notation to allow students to focus on the content. By using words, students do not need to translate the meaning of the symbol while reading. To increase exposure to different notations, images with given information marked using ticks, right angle marks, or arrows also have a caption with the symbolic notation (\(\overline{AB} \cong \overline{AC}, \overline{AB} \perp \overline{AC} \text{, or }\overline{AB} \parallel \overline{AC}\)). Feel free to use the symbolic notation when recording student responses, as that is an appropriate use of shorthand.

Students have the opportunity to choose appropriate tools (MP5) in nearly every lesson as they select among the options in their geometry toolkit as well as dynamic geometry software. For this reason, this math practice is only highlighted in lessons where it's particularly salient.

In this unit, students prove a variety of figures congruent. They start with segments (2 vertices), then triangles (3 vertices), and finally quadrilaterals (4 vertices). Before starting this unit, students are familiar with rigid transformations and congruence from a previous unit. The triangle congruence theorems in this unit lay the foundation for triangle similarity in a subsequent unit.

The first section focuses on establishing congruence. Students recall from middle school that if two figures are congruent, then each pair of corresponding parts of the figures are also congruent. Next, students use rigid transformations to justify the triangle congruence theorems of Euclidean geometry: Side-Side-Side Triangle Congruence, Side-Angle-Side Triangle Congruence, and Angle-Side-Angle Triangle Congruence. Students justify that for each set of criteria, a sequence of rigid motions exists that will take one triangle onto the other. 

In a previous unit, students began justifying their responses. In this unit, students write more rigorous proofs. At first, students may use imprecise language to convey their ideas. Throughout the unit they will read examples, practice explaining ideas to a partner, and build a reference of precise statements to use in future proofs. Writing proofs doesn’t only mean providing the reasons for someone else’s claim, constructing a viable argument includes writing conjectures and detailing given information.

Students write congruence proofs using transformations so the resulting theorems can be used in future work without repeating the argument. Many of the applications students explore involve quadrilaterals. Students will use the theorems they prove about quadrilaterals later in coordinate geometry as they use algebraic methods to prove additional results about quadrilaterals.

Note on materials: For most activities in this unit, students have access to a geometry toolkit that includes many tools that students can choose from strategically: compass and straightedge, tracing paper, colored pencils, and scissors. In some lessons, students will also need access to a ruler and protractor. When students work with quadrilaterals, instructions for making 1-inch strips cut from cardstock with evenly spaced holes are included. These strips allow students to explore dynamic relationships among sides and diagonals of quadrilaterals. Finally, there are some activities that are best done using dynamic geometry software, and these lessons indicate that digital materials are preferred. Students will continue to use and add to their reference charts. The completed reference chart for this unit is provided for teacher reference.

In this unit, students explore similar figures as pairs of figures for which one can be taken onto the other through a sequence of rigid transformations and a dilation. The work of this unit expands upon students’ understanding of similarity, previously encountered in grade 8, and their work in an earlier unit on congruence.

The unit begins with an exploration of dilation properties, including the idea that angles are preserved through dilation and lines are taken to themselves or parallel lines depending on the center of dilation. Then students practice using a dilation along with rigid transformations to show that a pair of figures are similar and look for other ways to know that a pair of figures will be similar. The unit then focuses more closely on similar triangles, introducing theorems such as the Angle-Angle Similarity Theorem and connections to the Pythagorean Theorem.

Students focus on writing conjectures and proving them throughout the unit. Students should become more familiar with the process of noticing a pattern, making a conjecture, then looking to find counterexamples or justifying the conjecture with a proof. 

Note on materials: For most activities in this unit, students have access to a geometry toolkit that includes tools that students can choose from strategically: compass and straightedge, tracing paper, colored pencils, and scissors. In some lessons, students will also need access to a ruler and a protractor. Students are given access to measuring tools in certain activities, to ensure that their focus during most activities is on logic and reasoning. Using a straightedge without markings on it forces students to attend to attributes of diagrams other than the specific length. In the final section, “Let's Put It to Work,” there are optional activities involving going outside to indirectly measure the heights of tall objects. Students will need measuring tools and may also choose to use speciality materials such as straws or small mirrors. Finally, there are some activities that are best done using dynamic geometry software, and these lessons encourage teachers to prepare to give students access to the digital version of the student materials. Students will continue to use and add to their reference charts. The completed reference chart for this unit is provided for teacher reference.

In this unit students build an understanding of ratios in right triangles, which leads to naming cosine, sine, and tangent as trigonometric ratios.

Prior to beginning this unit, students will have considerable familiarity with right triangles and similarity. They learned to identify right triangles in grade 4. Students studied the Pythagorean Theorem in grade 8, and used similar right triangles to build the idea of slope. This unit builds on this extensive experience and grounds trigonometric ratios in familiar contexts.

Several concepts build throughout the unit. Students begin by using similar triangles to create a table of ratios of the side lengths in right triangles. At first, their table includes only the bottom rows of the table shown here. Taking the time to both build and use the table helps students construct a solid foundation before they learn the names of trigonometric ratios.

Students notice patterns between the columns for cosine and sine before they first hear the terms “cosine” and “sine.” In a subsequent lesson they investigate that relationship, proving the two ratios are equal for complementary angles. Finding the measures of acute angles in a right triangle follows a similar arc, where students first use the table to estimate and then in a subsequent lesson learn how to calculate an angle measure given the side measures by using arcsine, arccosine, and arctangent.

As students measure side lengths and compute ratios, there is an opportunity to discuss precision. In this unit, students will round side lengths to the nearest tenth and angle measures to the nearest degree in most cases. When students solve problems in context they grapple with whether or not their answer is reasonable, as well as the appropriate degree of precision to report.

Students will continue to use and add to their reference charts. The completed reference chart for this unit is provided for teacher reference.

In this unit, students practice spatial visualization in three dimensions. They sketch cross-sections, study dilations, derive volume formulas, and apply their understandings to solve problems.

In previous grades, students studied multiple aspects of solids. In grade 6, they started calculating surface areas and volumes of right rectangular prisms. They extended this understanding to the volume and surface area of right prisms in grade 7, and to that of spheres, cones, and cylinders in grade 8. Students also described cross-sections of three-dimensional figures in grade 7. 

In future courses, the visualization skills developed in this unit will be applicable when using calculus to compute volumes or when using linear algebra in three or more dimensions.

Students begin the unit by examining solids of rotation and cross-sections of a variety of solids. They make a connection between cross-sections and dilation to see that cross-sections of a pyramid may be viewed as dilations of the base for scale factors between 0 and 1. Later in the unit they learn Cavalieri’s Principle: Suppose two solids have equal heights. If at all distances from the base, the cross-sections of the two solids have equal areas, then the solids have equal volumes. Combining the concepts of dilations, cross-sections, and Cavalieri’s Principle with dissection allows students to derive the volume formulas for a pyramid or cone. 

When students establish that dilating by a scale factor of \(k\) multiplies areas by \(k^2\) and volumes by \(k^3\), it provides an opportunity to use square root and cube root graphs to illustrate the relationship between scaled area or volume and scale factors. While this unit is the primary opportunity to study root functions, students will continue to graph functions and interpret the meaning of points, coefficients, and constants in future courses. 

In this unit, students may assume that cylinders or prisms that appear to be oblique are indeed oblique, and those that appear to be right are right. For right cylinders and prisms, right angles will not be marked to indicate that the bases are at right angles to the lateral surfaces. Unless otherwise stated, all responses given in decimal form are rounded to the nearest tenth.

Students will continue to use and add to their reference charts. The completed reference chart for this unit is provided for teacher reference.

Prior to beginning this unit, students will have spent most of the course studying geometric figures not described by coordinates. However, students have seen figures on the grid (notably transformations in grade 8) as well as lines and curves on the coordinate plane (in previous courses). This unit brings together students’ experience from previous years with their new understanding from this course for an in-depth study of coordinate geometry.

The first few lessons examine transformations in the plane. Students encounter a new coordinate transformation notation that connects transformations to functions. Students transform figures using rules such as \((x,y) \rightarrow (x+3, y+1)\) and connect the geometric definitions of reflections and dilations to coordinate rules that produce them. They prove that objects are similar or congruent, using reasoning including distance (via the Pythagorean Theorem), angles (calculated using trigonometry), and definitions of transformations.

The next set of lessons focuses on building equations from definitions. Students examine circles and parabolas through the lens of distance. A circle is the set of points the same distance from a given center, and a parabola is the set of points equidistant from a given point (the focus) and line (the directrix). Based on these definitions, students develop a general equation for a circle, and they write equations that represent specific parabolas.

The unit progresses next to coordinate proof. Students build the point-slope form of the equation of a line. They then write and prove conjectures about slopes of parallel and perpendicular lines, applying concepts of transformations in the proofs. They apply these ideas to other proofs, such as classifying quadrilaterals, and they use graphs to solve simple systems of equations that include a linear equation and a quadratic equation.

At the end of the unit, students use weighted averages to partition segments, scale figures, and locate the intersection points of the medians of a triangle. Students locate the intersection points of the altitudes of a triangle. Then there are several optional activities that offer diverging paths toward the Euler line or toward practicing equations of lines through constructing and describing tessellations.

In the final lesson, students apply their understanding of slope and distances in a plane, as they explore the Nazca lines in a real-world situation. 

In the images in this unit, students may assume that a point that appears to be the center of a circle is indeed the true center. Students will continue to use and add to their reference charts. The completed reference chart for this unit is provided for teacher reference.

In this unit, students investigate the geometry of circles more closely. In grade 7, students used formulas for the area and circumference of a circle to solve problems. Earlier in this course, students made formal geometric constructions, studied similarity and proportional reasoning, and proved theorems about lines and angles. This unit builds on these skills and concepts. In Algebra 2, the concepts learned in this unit will be helpful as students connect the unit circle to trigonometric functions.

Students define the terms “chord,” “arc,” and “central angle” before observing that inscribed angles are half the measure of their associated central angles, and writing related proofs about congruent chords and similar triangles. Throughout this unit, students also construct lines tangent to circles and use their proofs that a tangent line is perpendicular to the radius drawn to the point of tangency.

Next, students prove properties of cyclic quadrilaterals, and they use their understanding of perpendicular bisectors from a previous unit to construct triangles with circumscribed circles and define “circumcenter.” Students then use angle bisectors to construct incenters of triangles and circles inscribed in triangles. 

In the next section, students develop methods for calculating sector areas and arc lengths, and then students define “radian measure of a central angle” as the quotient of the length of the arc defined by the angle and the radius of the circle. They develop fluency with radian measures by shading portions of circles and working with a double number line.

In the final lesson, students apply what they have learned about circles to solve problems in context.

In this unit, students will do several constructions. A particular choice of construction tools is not necessary. Paper folding and straightedge and compass moves are both acceptable methods.

Students will continue to use and add to their reference charts. The completed reference chart for this unit is provided for teacher reference.

In this unit, students extend their understanding of probability, sample spaces, and events from their introduction in grade 7. The chance experiments under consideration have multiple parts, such as rolling a number cube and then flipping a coin—allowing events within the sample space to be considered in new ways.

The unit begins with students creating different models for understanding sample spaces and probability. The models include tables, trees, lists, and Venn diagrams. Venn diagrams allow students to visualize various subsets of the sample space, such as “A and B,” “A or B,” or “not A.” Tables help students determine the probability of those subsets occurring, and support students’ understanding of the Addition Rule, \(P(\text{A or B}) = P(\text{A}) + P(\text{B}) - P(\text{A and B})\).

Conditional probability is discussed and applied using several games and connections to everyday situations. In particular, the Multiplication Rule \(P(\text{A and B}) = P(\text{A | B}) \boldcdot P(\text{B})\) is used to determine conditional probabilities. Conditional probability leads to a study of independence of events. Students describe independence using everyday language and use the equations \(P(\text{A | B})=P(\text{A})\) and \(P(\text{A and B}) = P(\text{A}) \boldcdot P(\text{B})\) when events A and B are independent.

The unit closes with students making conjectures about the independence of events, when playing games with one another, and then testing those conjectures by collecting data and analyzing the results.