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.