The course begins with students using a compass and a straightedge to improve their logical-reasoning skills in a geometric setting. Students gradually build a toolkit of constructions that lead to rigid transformations and showing congruence of figures. In particular, they examine conditions needed to guarantee triangle congruence.

Students describe the shape of data distributions, using measures of center and variability. This leads them to model how multiple variables are related, using linear equations and systems of linear equations. Students write, evaluate, graph, and solve equations, explaining and validating their reasoning with increased precision.

By examining how transformations affect graphs, students connect their geometric understanding of rigid transformations to their understanding of linear equations on a coordinate plane. These insights lead into a unit on two-variable statistics in which students examine relationships between variables, using two-way tables, scatter plots, and linear models. Students continue their exploration of graphs by solving linear inequalities and systems of linear inequalities to represent constraints in situations.

Shifting focus, students deepen their understanding of functions by representing, interpreting, and communicating about them, using function notation, domain and range, average rate of change, and features of graphs. They also see categories of functions, starting with linear functions (including their inverses) and piecewise-defined functions (including absolute-value functions), followed by exponential functions. For each function type, students investigate real-world contexts, look closely at the structural attributes of the function, and analyze how these attributes are expressed in different representations.

Within the classroom activities, students have opportunities to engage in aspects of mathematical modeling. Additionally, modeling prompts are provided for use throughout the course, offering opportunities for students to engage in the full modeling cycle. Implement these in a variety of ways. Please see the Mathematics Modeling Prompts section of this Course Guide for a more detailed explanation.

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 students are trying to prove new facts. At the beginning of the course, the chart is blank. Students continue adding entries and referring to them in the geometry units of this course.

Print charts double sided to save paper. There should be a system for students to keep track of their charts (for example, hole punch and keep in a binder, or 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. A statement can be one of these three types: assertion, definition, or theorem. An assertion is an observation that seems to be true but is not proven. Sometimes assertions are not proven, because they are axioms or because the proof is beyond the scope of this course. The chart includes the most essential definitions. If there are additional definitions from previous courses that 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. Here are some entries to show the chart’s structure: 

date, type	statement	diagram
[date] 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.
[date] definition	Two figures are congruent if there is a sequence of translations, rotations, and reflections that takes one figure exactly onto the other.  The second figure is called the image of the rigid transformation.
[date] theorem	Translations take lines to parallel lines or to themselves.

Students are not expected to record all of their observations in the chart. Sometimes students’ conjectures will be proven in a subsequent lesson and added later as theorems rather than assertions. Other times students 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 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. The scaffolded version 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 in which the words needed 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 instead of translating the meanings of symbols while reading. To increase exposure to different notation, images with information that is given by tick marks 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.