Students begin connecting their geometric transformation understanding of parallel lines with slope by calculating the slope of translated lines. Identify students with different starting slopes so the class can see that the slope remains constant for positive, negative, steep, and shallow lines.

The purpose of the discussion is to elicit the conjecture that slopes of parallel lines are equal. Select students with positive, negative, steep, and shallow slopes to share. Here are some questions for discussion:

• “What do you notice about the slopes of the translated lines?” (The slopes stayed the same when the lines were translated.)
• “What do you know about the 3 lines you drew?” (They are parallel because translations take lines to parallel lines or the same line.)
• “Make a conjecture about slopes of parallel lines.” (They appear to be equal.)

Tell students that they will prove this conjecture in the next activity.

In this activity, students provide reasons for a proof that parallel lines must have equal slopes, as they prepare to construct an argument (MP3). To prepare them for this reasoning, students begin by noticing and wondering about an image of parallel lines on a coordinate plane. 

Monitor for students who approach the proof using these methods, from more general to more precise:

• Describing properties of the diagram and transformations.
• Annotating the diagram with congruent angles and segments.
• Labeling points and specifying congruent angles and segments using the labels.

Note that this reasoning works only for unique, non-vertical parallel lines that have a nonzero slope. It is not necessary that students make this observation at this time, but it may be worth pointing out if students ask.

In this activity, students use the result of a previous proof that parallel lines have the same slope to show that the opposite sides of a quadrilateral have the same slope and are therefore parallel. This leads to the conclusion that the quadrilateral is a parallelogram. This gives an opportunity for students to apply the proof and prepares them for proofs about parallelograms in later courses. Students do not need to formally prove that a figure on a grid is a parallelogram at this time, but students may recognize that two pairs of intersecting parallel lines form a parallelogram. Students make use of the structure of translation by a directed line segment to make observations about slope and parallel lines (MP7).

Monitor for students who use different equivalent equations to represent the parallel lines and ask them to share in the whole-class discussion.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).