The goal of this activity is to identify the coordinates of points on a line by looking at dilations of a single slope triangle. Since all dilations of the triangle are similar, their longest sides all lie on the same line, and the coordinates of the points on that line have a structure linked with the dilations used to produce them. 

For the question where is dilated by scale factor , monitor for students who:

• Look for and express a pattern for the coordinates of the points from earlier questions (MP8). For example, a scale factor of 1 gives , a scale factor of 2 gives , and a scale factor of 2.5 gives . So the -coordinate appears to be twice the scale factor while the -coordinate appears to be 1 more than the scale factor.

• Use the structure of triangle and the definition of dilations (MP7). For example, since the horizontal length of triangle is 2 and the vertical length is 1, a dilation with scale factor would make the horizontal length and the vertical length . Since the vertical side of triangle is 1 unit above the -axis, the dilation would result in a -coordinate that is 1 greater than the dilated vertical length. No adjustment is needed for the dilated -coordinate because the horizontal side of triangle  starts on the -axis.

The goal of this discussion is for students to see that the structure of the coordinates of points on a line can be derived from properties of dilations. Begin by inviting previously selected students, as described in the Activity Narrative, to share their reasoning for the last question. Here are some questions for discussion:

• “What do the different strategies have in common? How are they different?” (Both use the structure of the grid. One strategy finds a pattern, and the other strategy uses the definition of dilations.)

• “Why do the different approaches lead to the same outcome?” (Finding a pattern helps to determine the coordinates for point while thinking about dilations explains why the   coordinate doubles and the coordinate is 1 more than the scale factor. The outcome applies to all scale factors in this situation, so a correct strategy will result in the same outcome.)

This activity has students write an equation satisfied by the points on a line and then use it to check whether or not specific points lie on that line.  

Note that the -intercept was intentionally left off of this activity's diagram so that students are encouraged to engage with thinking about similar triangles.

There are many slope triangles that students can draw, but the one joining and is the most natural for calculating the slope. Then and either or can be used to find an equation. Monitor for students who use each of these points in their equation and invite them to present during the discussion.

This is the first time Math Language Routine 3: Critique, Correct, Clarify is suggested in this course. In this routine, students are given a “first draft” statement or response to a question that is intentionally unclear, incorrect, or incomplete. Students analyze and improve the written work by first identifying what parts of the writing need clarification, correction, or details, and then writing a second draft (individually or with a partner) (MP3). Finally, the teacher scribes as a selected second draft is read aloud by its author(s), and the whole class is invited to help edit this “third draft” by clarifying meaning and adding details to make the writing as convincing as possible to everyone in the room. Typical prompts are “Is anything unclear?” and “Are there any reasoning errors?” The purpose of this routine is to engage students in analyzing mathematical writing and reasoning that is not their own, and to solidify their knowledge and use of language.

The purpose of this discussion is for students to examine multiple ways of finding an equation for a line and to understand how to use that equation to determine if a point lies on the line. Begin by inviting previously selected students to show how they arrived at their equation, making sure to have different equations shown as described in the Activity Narrative. Here are some questions for discussion:

• “How are each of these equations similar?” (All of the equations have the slope somewhere in the equation. All of the equations have an and a .)

• “How are each of these equations different?” (Some equations used the point and some used the point for the slope triangle.)

Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response for the last question about point by correcting errors, clarifying meaning, and adding details. 

• Display this first draft:

“When I put the numbers 100 and 193 into the equation , it doesn’t work.”

• Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement. If not mentioned by students, display and review these ideas: 

  • The meaning of slope and how it is seen in the equation 

  • How the numbers 100 and 193 are used in the equation

  • How to use more precise language about what “doesn’t work”

• Give students 2–4 minutes to work with a partner to revise the first draft. 

• Select 1–2 individuals or groups to read their revised draft aloud slowly enough to record for all to see. Scribe as each student shares, then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.

Highlight that using and for a slope triangle gives an equation while using and gives the equation . These equations look different, but they will both work to check whether or not a point is on the line. 

Using algebra to show that these two equations are equivalent is not necessary (or appropriate in grade 8), but students can see from the graphed line that either equation can be used to test whether or not a point is on the line. If students have not done so already when they share their solutions, draw and label the two slope triangles that correspond to these two equations.