## Narrative

For the first several units, students practice generating conjectures and observations. This begins with work on compass and straightedge constructions. They gradually build up to formal proof, engaging in a cycle of conjecture, rough draft, peer feedback, and final draft narratives. To support their proof writing, students record definitions and theorems in a reference chart, which will be used and expanded throughout the course.

Students build on their middle school study of transformations of figures. Students use transformation-based definitions of congruence and similarity, allowing them to rigorously prove the triangle congruence and similarity theorems. They apply these theorems to prove results about quadrilaterals, isosceles triangles, and other figures. Students extend their understanding of similarity when they study right triangle trigonometry, which in future courses will be expanded into a study of periodic functions.

Next, students derive volume formulas and study the effect of dilation on both area and volume. They connect ideas from algebra and geometry through coordinate geometry, reviewing theorems and skills from prior units using the structure of the coordinate plane. They use transformations and the Pythagorean Theorem to build equations of circles, parabolas, parallel lines, and perpendicular lines from definitions, and they link transformations to the concept of functions.

Students analyze relationships between segments and angles in circles and develop the concept of radian measure for angles, which will be built upon in subsequent courses. They close the year by extending what they learned about probability in grade 7 to consider probabilities of combined events, including identifying when events are independent.

Within the classroom activities, students have opportunities to engage in aspects of mathematical modeling. Additionally, modeling prompts are provided for use throughout the course. Modeling prompts offer opportunities for students to engage in the full modeling cycle. These can be implemented in a variety of ways. Please see the course guide for a more detailed explanation of modeling prompts.

## Geometry Reference Chart

In order to write convincing arguments, students need to support their statements with facts. The reference chart is a way to keep track of those facts for future reference when they are trying to prove new facts. At the beginning of the course, the chart is blank. Students continue adding entries and referring to them through unit seven.

Charts can be printed double-sided to preserve paper. Students will need to keep track of these so there should be a system (examples: hole punch and keep in a binder, staple and tuck in the front of a notebook or the back of the workbook).

Each entry includes a statement, a diagram, a type and the date. The types are assertions, definitions, and theorems. An assertion is an observation that seems to be true, but is not proven. Sometimes assertions are not proven because they are axioms, other times because the proof is beyond the scope of this course. The chart includes the most essential definitions, but if there are additional definitions from previous courses students would benefit from, feel free to add them. For example, it is assumed that students recall the definition of isosceles. If this is not the case, that would be a useful definition to record.

date, type statement diagram
9/13/18
assertion
A rigid transformation is a translation, reflection, rotation, or any sequence of the three.
Rigid transformations take lines to lines, angles to angles of the same measure, and segments to segments of the same length.
9/13/18
definition
One figure is congruent to another if there is a sequence of translations, rotations, and reflections that takes the first figure exactly onto the second figure.
The second figure is called the image of the rigid transformation.
9/25/18
theorem
Translations take lines to parallel lines or to themselves.

Students are not expected to record all of their observations in the chart. Sometimes their conjectures will be proven in a subsequent lesson and added later as theorems rather than assertions. Other times they prove something that they will not need to use again. Students are welcome to use any proven statement in a later proof, but the reference chart is designed to be as concise as possible so it is a more useful reference than students’ entire notebooks.

The intention is for students to be able to use their reference charts at any time, including during assessments. The goal is to learn to apply statements precisely, not to memorize. Some teachers ask students to make a tally mark each time they use a statement in the chart to justify a response. This allows students to see which are the most powerful statements and teachers to see how students are using their charts. Including the date will help students to know if they missed a row when they were absent or to help students locate a statement if they remember approximately how long ago they added it.

In addition to the blank reference chart, there is also a scaffolded version of the reference chart. This is intended to provide access for students with disabilities (language based, low vision, motor challenges) and English learners. In this version, students are provided with sentence frames for the “statement” column. The diagrams are also partially provided so students can focus on annotating key information. There is a teacher version of the chart where the words to fill in the blanks and the missing annotations are highlighted.

## Notation

Within student-facing text, these materials use words rather than symbols to allow students to focus on content without needing to translate the meanings of symbols while reading. To increase exposure to different notation, images with given information marked using ticks or arrows include a caption with the symbolic notation (like $$\overline{AB} \cong \overline{CD}$$). Teachers are encouraged to use the symbolic notation when recording student responses, since that is an appropriate use of shorthand.

### Unit 1: Constructions and Rigid Transformations

In grade 8, students determine the angle-preserving and length-preserving properties of rigid transformations experimentally, mostly with the help of a coordinate grid. Students have previously studied angle properties, including the Triangle Angle Sum Theorem, but no formal proofs have been required. In this unit, students create rigid motions using construction tools with no coordinate grid. This leads to more rigorous definitions of rotations, reflections, and translations. Students begin to explain and prove angle relationships like the Triangle Angle Sum Theorem using these rigorous definitions and a few assertions.

In previous courses, students developed their understanding of the concept of functions. In this unit, the concept of a transformation is made somewhat more formal using the language of functions. While students do not use function notation, they do move away from describing transformations as “moves” that act on figures and towards describing them as taking points in the plane as inputs and producing points in the plane as outputs.

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. Constructions are used throughout several lessons to introduce students to reasoning about distances, generating conjectures, and attending to the level of precision required to define rigid motions later in the unit. The definition of a circle is an important foundation for concepts in this unit and throughout the course.

Then, students learn rigorous definitions of rigid motions without reference to a coordinate grid. In subsequent units, they use those definitions to prove theorems. To prepare students for future congruence proofs, students start to come up with a systematic, point-by-point sequence of transformations that will work to take any pair of congruent polygons onto one another. This point-by-point perspective also illustrates the transition from thinking about transformations as “moves” on the grid to thinking about transformations as functions that take points as inputs and produce points as outputs. Students also examine the rigid transformations that take some shapes to themselves, otherwise known as symmetries. The concept of transformations as functions is developed further in a later unit that explores coordinate geometry.

In the final lessons of the unit, students learn ways to express their reasoning more formally. Students create conjectures about angle relationships and prove them using what they know about rigid transformations. As a tool for communicating more precisely, students begin to label and mark figures to indicate congruence. In the culminating lesson of the study of constructions, students build on their experiences with perpendicular bisectors to answer questions about allocating resources in a real-world situation.

A blank reference chart is provided for students, and a completed reference chart for teachers. The purpose of the reference chart is to be a resource for students to reference as they make formal arguments. Students will continue adding to it throughout the course. Refer to About These Materials in the Geometry course for more information.

Students have the opportunity to choose appropriate tools (MP5) in nearly ever 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.

### Unit 2: Congruence

Before starting this unit, students are familiar with rigid transformations and congruence from grade 8. They have experimentally confirmed properties of rigid transformations, and informally justified that figures are congruent by finding a sequence of rigid motions that takes one figure onto the other. In this unit, rigid transformations are used 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 exist that will take one triangle onto the other. In middle school, they focused on specific examples and finding specific sequences of rigid motions (for example, students might justify that two triangles on the coordinate plane are congruent because they can find a reflection across the $$x$$-axis and a horizontal translation of two units that takes one triangle onto the other). In this unit, students learn to explain how two triangles with all three pairs of corresponding side lengths congruent can be taken onto one another using a more general sequence of rigid motions.

Students will justify how they know that a given sequence of transformations will result in the vertices coinciding. They practice making statements such as, “Since the vertices are the same distance along the same ray, they have to be in the same place. I know the points are the same distance from the endpoint because rigid transformations preserve distance.” At first, students may use imprecise language to convey this and other 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.

Students also get the opportunity to immediately apply theorems they have proven to new contexts in which those theorems help them prove new results. Many of the applications students explore involve quadrilaterals. Students learn to decompose quadrilaterals into congruent triangles, and prove many relationships within the quadrilateral hierarchy, such as that any parallelogram with at least one right angle must be a rectangle, or any quadrilateral with perpendicular diagonals that bisect each other must be a rhombus. Students will use these theorems 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.

### Unit 3: Similarity

Before starting this unit, students are familiar with dilations and similarity from work in grade 8. They have experimentally confirmed properties of dilations, and informally justified that figures are similar by finding a sequence of rigid motions and dilations that takes one figure onto the other. Students have primarily explored dilations on grids, where they have additional structure to help them precisely determine if two figures are dilations, or to draw a dilation of a figure precisely.

In a previous unit, students used rigid transformations to justify the triangle congruence theorems of Euclidean geometry: Side-Side-Side Triangle Congruence Theorem, Side-Angle-Side Triangle Congruence Theorem, and Angle-Side-Angle Triangle Congruence Theorem.

In this unit, students use dilations and rigid transformations to justify triangles are similar. They prove, if triangles have three pairs of congruent corresponding angles and three pairs of corresponding sides in a proportional relationship, the triangles are similar. Students are then able to prove the Angle-Angle Triangle Similarity Theorem. They also draw conclusions about figures they’ve proven to be similar: in similar figures, corresponding angles are congruent and corresponding sides are in a proportional relationship

The unit balances a focus on proof with a focus on using similar triangles to find unknown side lengths and angle measurements. Early in the unit, students prove theorems using rigid transformations and dilations. Later in the unit, students use similarity shortcuts, especially the Angle-Angle Triangle Similarity Theorem, to justify that triangles must be similar and to find unknown side lengths using the fact that side lengths in similar figures are in the same proportion. Since there is more calculation in this unit students may benefit from access to calculators to de-emphasize computation and allow them to focus on reasoning about the context. Alternately students may leave answers in forms that don’t require computation.

This unit previews many of the important concepts that students rely on to make sense of trigonometry in later units. The latter part of the unit focuses on similar right triangles. In addition, students are introduced to some of the applications of right triangles that they will explore in more depth in the trigonometry unit, such as finding the heights of objects through indirect measurement.

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 protractor. In the final section, Putting It All Together, there are optional activities involving going outside to 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.

### Unit 4: Right Triangle Trigonometry

Prior to beginning this unit students will have considerable familiarity with right triangles. 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.

The first few lessons of this unit examine some special cases of similar right triangles to solidify the idea that any right triangles with a single congruent acute angle are similar. Two of these three lessons are optional. While the standards do not specifically call for special right triangles they are an opportunity to practice, build on important ideas, and are frequently included on college entrance exams. From there students generate data for the side length ratios of many sets of right triangles. This data is organized into a table which students apply to problems. Taking the time to both build and use the table helps students build a solid foundation before they learn the names of trigonometric ratios. Once students have practiced estimating both side lengths and angle measures using the table, they learn the names cosine, sine, and tangent.

Students practice looking up the cosine, sine, or tangent of a given angle in a calculator with simple triangles, then they apply trigonometry to several contexts. 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. At the end of the unit students study how to approximate the value of $$\pi$$ using inscribed and circumscribed polygons and learn about how precisely mathematicians have defined $$\pi$$ throughout history.

Several concepts build throughout the unit. Students notice patterns between the columns for cosine and sine before they even learn the names 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.

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

### Unit 5: Solid Geometry

In previous grades, students solved problems involving area, surface area, and volume for various solids. In grade 6, students worked with areas of triangles and quadrilaterals, as well as surface areas and volumes of right rectangular prisms including those with fractional edge lengths. In grade 7, students found areas of circles, solved problems involving the volume and surface area of right prisms, and described plane sections of three-dimensional figures. In grade 8, students solved problems involving volumes of spheres, cones, and cylinders using given volume formulas.

In this unit, students practice spatial visualization in three dimensions, study the effect of dilation on area and volume, derive volume formulas using dissection arguments and Cavalieri’s Principle, and apply volume formulas to solve problems involving surface area to volume ratios, density, cube roots, and square roots.

Students first practice spatial visualization by examining solids of rotation, envisioning these solids by rotating paper figures using a pencil as an axis of rotation. Then they investigate cross sections of a variety of solids. They create physical representations to show that cross sections of a pyramid may be viewed as dilations of the base for scale factors between 0 and 1. Students study the effect of dilation on cross sections and other two-dimensional figures, establishing that dilating by a scale factor of $$k$$ multiplies areas by $$k^2$$. They’re introduced to the equation $$y=\sqrt{x}$$ in this geometric context. They create a graph representing $$y=\sqrt{x}$$ and use it to illustrate the relationship between scaled area and scale factors.

Then, students extend their study of scaling to solids. They conclude that dilating a solid by a scale factor of $$k$$ multiplies all lengths by $$k$$, surface areas by $$k^2$$, and volumes by $$k^3$$. They work backwards from a scaled volume or surface area to find the scale factor involved, requiring the introduction of cube roots. Students create a graph representing $$y=\sqrt[3]{x}$$ and use it to answer questions about how changes in volume affect changes in the corresponding scale factor.

The unit then builds on students’ prior knowledge about volumes of prisms to introduce 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 area, then the solids have equal volumes. This leads to the idea that the volume of a prism or cylinder whose base has an area of $$B$$ square units and height $$h$$ units is $$Bh$$ cubic units, regardless of the shape of the base and regardless of whether the solid is oblique.

Students now combine concepts of dilations, cross sections, and Cavalieri’s Principle with dissection to derive the formula for the volume of a pyramid or cone. First, they establish that any triangular pyramid whose base has area $$B$$ square units and whose height is $$h$$ units can be combined with two other triangular pyramids of equal volume to form a prism with the same base and height as the original pyramid. Each pyramid has $$\frac13$$ the volume of the prism. Therefore, the volume of the original pyramid is $$\frac13 Bh$$.

Next, students consider a general pyramid, and compare it to a triangular pyramid with equal height and a base of equal area. Earlier in the unit, students found that cross sections of a pyramid may be viewed as dilations of the base by some scale factor $$k$$ using the apex as a center. Now, they use this fact to determine that if two pyramids each have a base with area $$B$$ square units and height $$h$$ units, cross sections at equal heights of the pyramids each have area $$Bk^2$$ where $$k$$ is the scale factor of dilation that yields the particular cross section. Because all corresponding cross sections have equal area, Cavalieri’s Principle applies, and the two pyramids have equal volume. This extends the volume formula to all pyramids and cones, regardless of the particular shape of the base or whether the solid is oblique.

In the final lessons, students apply what they know about volume to solve problems. They calculate densities, analyze surface area to volume ratios, and use graphs that represent equations involving square roots and cube roots to answer questions about situations.

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.

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

### Unit 6: Coordinate Geometry

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 which 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 objects similar or congruent using reasoning, including distance (via the Pythagorean Theorem), angle (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, 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. In the final lesson, students locate the intersection points of the altitudes of a triangle. Then there are several optional activities which offer diverging paths toward the Euler line or toward practicing equations of lines through constructing and describing tessellations.

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.

### Unit 7: Circles

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 to investigate the geometry of circles more closely.

Students begin by defining the terms chord, arc, and central angle, and they use their new vocabulary to write a proof about congruent chords in circles. They observe that inscribed angles are half the measure of their associated central angles, and they use this idea to prove that two intersecting chords define similar triangles. Students construct lines tangent to circles, then prove that a tangent line is perpendicular to the radius drawn to the point of tangency. They apply this result to an analysis of the relationship between central and circumscribed angles.

Next, students connect their findings about inscribed angles to cyclic quadrilaterals, or quadrilaterals which can be circumscribed by a circle. This leads to an exploration of circumscribed circles of triangles. Students use a property of perpendicular bisectors from a previous unit to prove that the perpendicular bisectors of a triangle’s sides intersect at a single point, which allows for the construction of circumscribed circles for triangles. Then, angle bisectors are shown to be the set of points equidistant from the rays that form the angle, and students use this fact to construct incenters and inscribed circles for triangles. They observe that a triangle’s sides are tangent to its inscribed circle.

The next section begins with students developing methods for calculating sector areas and arc lengths. Students explore the relationship between arc length and the radius of a circle, noting that because all circles are similar, the ratio of arc length to radius is invariant for a given central angle. This leads to the definition of the 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. Students develop fluency with radian measures by shading portions of circles and working with a double number line. This is important for the transition towards Algebra 2. In that course, students will explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

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.

### Unit 8: Conditional Probability

In grade 7, students learned about probability by conducting chance experiments. Along with looking at experimental data, students created and analyzed sample spaces for situations. In this unit, students extend that knowledge by considering situations with two events, for example: roll a number cube and flip a coin. Students find probabilities when events are combined in various ways including both occurring, at least one occurring, and one event happening under the condition that the other happens as well.

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 the definition of independence of events. Students describe independence using everyday language and use the equation $$P(\text{A | B}) = P(\text{A})$$ 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, then testing those conjectures by collecting data and analyzing the results.