In this section, students use rigid compasses and straightedges to develop a catalog of construction techniques that they can use to construct a variety of figures.

Students begin by following instructions and making observations about figures. Then they write their own instructions and identify ways to attend to precision in their communication. Multiple lessons include perpendicular lines. This key building block allows students to construct parallel lines and squares in this section. The perpendicular bisector is used for the definition of the term “reflection” later in this unit and in the proof of the Side-Side-Side Triangle Congruence Theorem in a subsequent unit. Each construction technique supports students in refining their definitions of geometric objects and their ability to explain their thinking. In the culminating lesson, students build on their experiences with perpendicular bisectors to answer questions about allocating resources in a real-world situation.

Two circles, each passing through the center of the other.  A horizontal line is drawn through each of the circles's centers.  A vertical line is drawn through the two circle's intersection points. 

All constructions in this section are accessible using physical straightedges and rigid compasses. If students have ready access to digital materials in class, they can choose to perform any or all construction activities with the GeoGebra Constructions Tool accessible in the Math Tools or available at https://im612.org/construction-tool. If students do not have ready access to this digital tool in class, consider using the GeoGebra Constructions Tool to demonstrate constructions during the corresponding Activity Synthesis or Lesson Synthesis. There is also an optional lesson which allows students who have infrequent access to technology to experience this digital tool.

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

In this section, students examine the rigid transformations that take some shapes to themselves, otherwise known as symmetries. Then students continue describing sequences of rigid transformations that take one figure to another, congruent, figure.

To prepare students for future congruence proofs, students start to come up with a systematic, point-by-point sequence of transformations that will work to take any pair of congruent polygons onto one another. There is an optional lesson if students need additional practice with this strategy. The point-by-point perspective also illustrates the transition from thinking about transformations as “moves” on the grid to thinking about transformations as functions that take points as inputs and produce points as outputs. The concept of transformations as functions is developed further in a later unit that explores coordinate geometry.

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

In this section, students study transformations both on and off of grids. Providing the structure of a grid allows students to make connections with their learning in previous courses. Then students learn precise definitions for the rigid motions, translations, reflections, and rotations that apply off the grid. 

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Students recall that the figures are called congruent if there is a rigid transformation that takes one figure to the other. They examine the idea that rigid motions preserve distances and angles in a variety of contexts. In subsequent sections and units, students use the precise definitions generated here to prove theorems. 

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

In this final section, students have the opportunity to apply their thinking from throughout the unit. As this is a short section followed by an End-of-Unit Assessment, there are no section goals or checkpoint questions.

In this section students create conjectures about angle relationships and prove them using what they know about rigid transformations.

The primary work is on proof-writing skills. In order to focus on justification, many proofs are for ideas students were first introduced to in previous grades, such as supplementary, complementary, vertical, and adjacent angles. Students also prove the Triangle Angle Sum Theorem using the rigorous definitions from the previous section and a few assertions.

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 section. A more reasonable goal is for students to consistently label and mark figures to indicate congruence which helps them communicate more precisely.

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.

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