This unit begins with constructions, then continues to rigid transformations. In grade 8, students determined the angle-preserving and length-preserving properties of rigid transformations experimentally, mostly with the help of a coordinate grid.

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.

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. 
 

Starting in the second section, a blank reference chart is provided for students, and a completed reference chart is provided for teachers. The purpose of the reference chart is to be 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 move on to triangles (3 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 later course.

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.

The proofs in these materials are all written in narrative form. The narrative format matches the discussion students might have to use to convince their partner, and it also matches the way mathematicians write proofs. While students may use other formats to support their organization, it is important that students can see the flow of reasoning that exists in a well-written proof. A two-column proof can be thought of like an outline for an essay. Outlines help organize thoughts, but an outline is less persuasive than a well-written essay. Students should learn to write a well-written justification in the form of a narrative proof.

Students are learning ways to express their reasoning more formally. Expect students to put together explanations with various levels of formality. Mastery of proof writing is not expected by the end of this unit. Students will have more practice writing proofs in a later course.

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. 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 collect, display, and analyze data using statistics such as mean, median, interquartile range, and standard deviation.

In grades 6–8, students used histograms, dot plots, and box plots as a way to summarize data and worked with basic measures of center (mean and median) as well as measures of variability (mean absolute deviation and interquartile range). These concepts are revisited in the first two sections of this unit, but with a focus on interpretation and what they reveal about the data in addition to the mechanics of constructing the data displays.

The optional third section is available to familiarize students with spreadsheets and technology that will be used to calculate statistics such as mean, median, quartiles, and standard deviation as well as create data displays.

The fourth section introduces additional ways to interpret data using standard deviation and outliers. They finish the unit by using these tools to compare related data sets using measures of center and measures of variability.

The last lesson gives students a chance to practice their skills by posing a statistical question, designing an experiment, collecting data, and analyzing their data.

Because the first half of the unit mostly revisits material from middle school, a Mid-unit Assessment is not included in this unit. The Cool-downs and Checkpoints can be used to monitor student understanding.

In this unit, only the population standard deviation is used. Sample standard deviation is introduced in a later course.

Geogebra’s spreadsheets are chosen for their versatility for the on-level mathematics in this course. While other spreadsheet programs have additional functionality and uses, they are limited in other ways. That said, please adapt the materials to the needs of your students.

In this unit, students examine solving and graphing linear equations and systems of linear equations.

The unit builds on learning from middle school when students used variables to write equations, manipulated equations using valid moves such as the distributive property, and solved basic systems of linear equations using graphs and substitution.

In the first section, students recall writing equations to represent situations. In the second section, they use valid moves to write equivalent equations that can be used to solve for unknown values or to isolate variables. The third section examines solving systems of equations using graphs, substitution for variables, and elimination of variables. Students use their understanding of writing equivalent equations to understand why each of the methods works for finding the solution.

Graph of 2 intersecting lines, origin O. Horizontal axis 0 to 280, by 20’s, labeled, yards of silver thread. Vertical from 0 to 280, by 20’s, labeled yards of gold thread. Line 1 starts at 240 comma 0, passes through 60 comma 180, ends at 240 comma 0. Line 2 starts at 0 comma 214 point 2 9, passes through 60 comma 180, ends at 280 comma 54 point 2 9.  

Prior to beginning this unit, students have studied geometric figures not described by coordinates. Students have seen figures on a grid (notably transformations in grade 8) and lines on a coordinate plane (in previous units). This unit brings together students’ experience from previous years and their new understanding from this course, for an in-depth study of the coordinate geometry of lines and polygons.

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. Students prove that objects are similar or congruent, using reasoning including distance (via the Pythagorean Theorem) and definitions of transformations.

The next section focuses on relationships between lines. 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 also create equations that are parallel or perpendicular to another line, through given points. 

Students next look deeply at the concept of distance between points in a coordinate plane. They move from finding the distance between specific points to generalizing an equation for the distance between any two points in a plane. They use this equation to find the lengths of line segments and the side lengths of figures in a plane. Finally, students apply these ideas to classifying quadrilaterals as parallelograms, rectangles, rhombuses, and squares.

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. 

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 use statistical methods to look for associations in bivariate data. The unit begins with students analyzing categorical data arranged in two-way tables. Students use the relative frequencies of the combinations of those categorical variables to check for evidence of any associations in the data.

The unit then transitions to bivariate numerical data, which are visualized using scatter plots and lines of best fit. Students use technology to compute the lines of best fit and observe how well the linear models match the data. Residuals and correlation coefficients are used to quantify the goodness of fit for linear models.

The unit closes with an exploration of the difference between correlation and causal relationships, and it is also an opportunity to apply this learning to areas of interest, like anthropology and sports.

In grade 8, students informally constructed scatter plots and lines of fit, noticed linear patterns, and observed associations in categorical data using two-way tables. In this unit, students build on this previous knowledge by assessing how well a linear model matches the data by using residuals as well as the correlation coefficient for best-fit lines (found using technology).

There are opportunities to practice concepts from a previous unit by interpreting the slope and intercept of a linear model in context as well as using the models to predict one variable given information about the other.

A scatterplot. Horizontal, from 0 to 3, by 0 point 5's, labeled weight in pounds. Vertical, 0 to 2 point 5, by 0 point 25s, labeled price in dollars. 12 dots trending upward and to the right. A line of best fit passes through the y axis at 0 comma 0 point 92, and trends upwards and to the right, passing through three dots.  

In this unit, students examine solving and graphing linear inequalities and systems of linear inequalities.

The unit builds on concepts from middle school when students write and solve inequalities by reasoning about quantities. It further builds on concepts from an earlier unit in which students solve linear equations and systems of equations by writing equivalent equations.

To start, students solve linear inequalities in one variable and graph the solutions on a number line by writing equivalent inequalities. In the second section, they solve linear inequalities with two variables by looking at the related equation, graphing it on a coordinate plane, and testing points on either side of the line to determine the solution region. The third section is about solving systems of linear inequalities considering multiple linear inequalities as conditions for situations and finding a solution region that satisfies all of the inequalities.

A graph of two intersecting inequalities on a coordinate plane, origin O. Each axis from negative 10 to 5, by 5’s. The first dashed line starts below x axis and right of y axis, goes through negative 2 point 5 comma negative 5, 0 comma 0, and 2 point 5 comma 5. The region above the dashed line is shaded. Second line at on the y axis at 10, goes through 5 comma 5, end on x axis at 10. The region below the dashed line is shaded.

In this unit, students expand and deepen their understanding of functions. They begin with a reminder of the definition of a function (a rule that assigns exactly one output to each input) that they previously saw in grade 8, then get familiar with function notation and use it to compare and analyze functions, write rules for functions, and solve for inputs or outputs.

Then, students explore graphs of functions to describe features such as “maximum,” “minimum,” “intercepts,” “increasing,” “decreasing,” and “average rate of change” and make connections between the graphs and real-life situations. They use situations to discuss the domain and range of a function and make sense of piecewise-defined functions. In particular, students examine the absolute value function and some basic transformations of it. Later, students explore inverses of linear functions as a way to find corresponding input values when output values are known. Throughout the unit, students have chances to mathematically model real-world situations.

In this unit, students build on their understanding of linear functions, properties of exponents, and percent change to explore exponential relationships. Students learn that exponential relationships are characterized by a constant quotient over equal intervals, and compare it to linear relationships which are characterized by a constant difference over equal intervals. They encounter contexts that change exponentially. These contexts are presented verbally and with tables and graphs. They construct equations and use them to model situations and solve problems. At first, students investigate these exponential relationships without using function notation and language so that they can focus on gaining an appreciation for critical properties and characteristics of exponential relationships.

Later, students view these relationships as functions and employ the notation and terminology of functions. They study graphs of exponential functions both in terms of contexts they represent and abstract functions that don’t represent a particular context, observing the effect of different values of \(a\) and \(b\) on the graph of the function \(f\) represented by \(f(x)=ab^x\). 

The contexts used early in this unit lead to functions where the domain is the integers, then students explore fractional exponents and their connection to roots.

Note on materials: Students should have access to a calculator with an exponent button throughout the unit. Access to graphing technology is necessary for some activities and encouraged throughout the unit. Examples of graphing technology include a handheld graphing calculator, a computer with a graphing calculator application installed, or an internet-enabled device with access to a site like desmos.com/calculator or geogebra.org/graphing. Interactive applets are embedded throughout, and a graphing calculator tool is accessible in the Math Tools in the digital version.

Graph of 3 functions labeled, p, q, and r on grid, origin O. Horizontal axis, time in hours, from 0 to 10, by 2's. Vertical axis, number of bacteria, from 0 to 40,000 by 20,000's. All three functions start at 0 comma 5,000 and trend upward and to the right. Function p passes through 2 comma 20,000 and trends rapidly upward and right. Function q passes through 4 comma11,250 and trends steadily upward and right. Function r passes through 6 comma 7,200 and trends more slowly upward and right.