During this unit, students will engage in each phase of the proof-writing cycle as they study and review properties of triangles and quadrilaterals. Writing proofs doesn’t mean only providing the reasons for someone else’s claim. Constructing a viable argument includes generating conjectures, detailing specific statements to be proved, writing a proof, and critiquing a proof. Students write proofs, starting with informal justifications and ending with formal proofs using definitions, assertions, and theorems developed in Math 1.

This unit begins by asking students to make observations about triangles. Students use experimental information to develop conjectures about triangles and justify those conjectures informally, preparing students for proof writing. 

In the next section, students write rigorous proofs. Students use many of the properties of angles and triangles reviewed in the first section and the rigid transformations reviewed in this section. Though students may use imprecise language to convey their ideas at first, throughout the unit they will read examples, practice explaining ideas to a partner, and build a reference of precise statements to use in formal proofs.

Note on materials: For many 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 have the opportunity to choose appropriate tools (MP5) in nearly every lesson as they select among these options. For this reason, this math practice is highlighted only in lessons where it’s particularly noteworthy.  

Starting in the first section, a blank reference chart and reference material from a previous course are 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. To model clear communication with mathematical language, point students toward particular entries by describing their content instead of by number. For example, say “definition of ‘congruent’” instead of “definition 2.” Students will continue adding to the reference chart 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 a 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. Feel free to use the symbolic notation when recording student responses, as that is an appropriate use of shorthand.

This unit intentionally allows extra time for students to learn new routines and establish norms for the year.

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.

Prior to this unit, students have studied what it means for a relationship to be a function, used function notation, and investigated linear and exponential functions. In this unit, they look at some patterns that grow quadratically and contrast this growth with linear and exponential growth. They further observe that eventually these quadratic patterns grow more quickly than do linear patterns but more slowly than exponential patterns grow. 

Students examine the important example of free-falling objects whose height over time can be modeled with quadratic functions. They use tables, graphs, and equations to describe the movement of these objects, eventually looking at the situation in which a projectile is launched upward. They interpret the meaning of each term in this context and work toward understanding how the coefficients influence the shape of the graph. Additional situations, such as revenue and area, are also introduced.

Next, students examine standard, factored, and vertex forms of quadratic functions. They recognize what information about the graph is easily obtained from each form and how the different values in each form influence the graph. In particular, they begin to generalize ideas of how horizontal and vertical translation, as well as vertical and horizontal stretching of graphs, relate to modifying the equation of a function.

Then students use the idea that a parabola can be defined as all the points equally distant from a point called the “focus” and a line called the “directrix.” Students use the Pythagorean Theorem to write functions that describe parabolas with a given focus and directrix, then recognize that the functions must be quadratic.

Note on materials: Access to graphing technology is necessary for many activities. Examples of graphing technology are: a handheld graphing calculator, a computer with a graphing calculator application installed, and an internet-enabled device with access to a site like desmos.com/calculator or geogebra.org/graphing. For students using the digital version of these materials, a separate graphing calculator tool isn’t necessary. Interactive applets are embedded throughout, and a graphing calculator tool is accessible in the student math tools.

Graph of the quadratic function \(h(t) = 5 + 60t - 16t^2\) on a coordinate plane, origin \(O\). Horizontal axis scale 0 to 4 by 1’s, labeled “time (seconds)”. Vertical axis scale 0 to 80 by  20’s, labeled “distance above ground (feet)”. Some of the points of this function are (0 comma 5), (1 comma 49), to a maximum near (1 point 9 comma 61 point 2 5) then decreasing through (2 comma 61), (3 comma 41) and (3.8 comma 0).

In this unit, students interpret, write, and solve equations algebraically.

Previously, students represented quadratic functions using expressions, tables, and descriptions. They connected important features of graphs to standard, factored, and vertex forms and expanded expressions into standard form.

Students begin with solving quadratic equations through reasoning without much algebraic manipulation. Then, they examine solving using the zero product property for quadratic equations that can be written in factored form. They notice patterns that help them rewrite quadratic expressions in factored form and recognize that not all of them are easily factorable.

This motivates finding another process of solving quadratic equations. Students recognize the benefits of equations of the form \((x-n)^2=q\) and develop the method of completing the square. Then, they generalize this process to the quadratic formula.

Throughout the unit, students analyze quadratic equations to recognize that there can be 0, 1, or 2 solutions. Solutions can be rational, irrational, or combinations of these. Students interpret the solutions that arise in different contexts.

The unit concludes with rewriting quadratic expressions from standard form into vertex form to find maximum and minimum values, then apply all of their understanding to applied problems.

In this unit, students use what they know about exponents and radicals to extend exponent rules to include rational exponents, use square roots to develop an understanding of complex numbers, and solve quadratic equations that include complex roots.

The unit opens with an optional review of exponent rules before using those rules to justify why \(x^{\frac{a}{b}} = \sqrt[b]{x^a}\) when \(a\) and \(b\) are integers and \(x\) is positive. The new rule leads to an exploration of square and cube roots as solutions to equations of the form \(x^2 = c\) or \(x^3 = c\). In particular, students learn that positive numbers have two square roots and that \(\sqrt{c}\) represents only the positive root.

The next section introduces imaginary and complex numbers by proposing \(i\) as a solution to the equation \(x^2 = \text{-}1\). Students explore the implications of this new type of number by representing it on an imaginary axis off of the real number line. This exploration includes adding imaginary and real numbers together to get complex numbers, and then adding and multiplying complex numbers.

Next, students revisit the methods of completing the square and using the quadratic formula to solve any quadratic equation, including those with complex solutions.

\(\displaystyle \begin{align} x^2 - 12x + 36 + 5 &= 0 \\ (x - 6)^2 + 5 &= 0 \\ (x - 6)^2 &= \text- 5 \\ x-6 &= \pm i \sqrt{5} \\ x &= 6 \pm i \sqrt{5} \end{align}\)

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. In a previous course, students made formal geometric constructions. Earlier in this course, students studied similarity and proportional reasoning, and proved theorems about lines and angles. This unit builds on these skills and concepts. In later courses, the concepts learned in this unit will be helpful as students connect the unit circle to trigonometric functions.

First, students make connections between prior work with distance in the coordinate plane and the definition of a circle to create a generalized equation for a circle in the coordinate plane: \((x-h)^2+(y-k)^2=r^2\).

Then, 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.