Geometry Supports
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.
-Taken from Illustrative Math Geometry Narrative
Students use dilations and rigid transformations to justify the the triangle similarity theorems of Euclidean geometry building to the shortcut specific to similarity: Angle-Angle Triangle Similarity. Students explicitly build on their work with congruence and rigid motions, establishing that triangles are similar by dilating them strategically and showing that after dilation, the congruence criteria they already established are met, and therefore a sequence of rigid transformations takes one triangle onto the dilation of the other, and the original triangles are related by a sequence of rigid motions and a dilation. By the definition of similarity in terms of transformations, the triangles are similar. The unit balances a focus on proof with a focus on using similar triangles to find unknown side lengths and angle measurements.
-taken from Illustrative Math, Geometry Unit 3 Overview.
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.