In this activity, students check whether several points lie on a given line and determine a rule to test whether any given point is also on the line (MP8). 

Monitor for students who use these methods for writing a rule in the last question. These approaches are ordered from specific to general, starting with testing the quotient of a single point, to testing a ratio, to testing an equation that can be satisfied by any point :

• With words and arithmetic: For example, divide the -coordinate by the -coordinate and see if it is equivalent to 0.75.

• With a table and proportional relationships: For example, since and are in a proportional relationship and  when , see if the constant of proportionality applies to the coordinates of the given point.

• With an equation involving quotients of vertical and horizontal side lengths: For example, see if the point satisfies the equation .

If a student writes , this can be presented last but it is not essential that students see this now.

After reviewing how students answered the first four questions, invite previously selected students to share their response for the last question. Sequence the discussion of the responses in the order listed in the Activity Narrative. If possible, record and display their work for all to see.

Connect the different methods by asking questions such as:

• “How do you see the value in all of the methods?” (In the first method, the quotient had to be equivalent to . In the second method, the constant of proportionality was . In the third method, one side of the equation was .) 

• “What is the slope of this line?” ()

• “Which of these methods would describe the line most clearly without drawing it?” (Answers vary.) 

• “These methods all look different, but how do they show the same things?” (These methods all describe a proportional relationship with a constant of proportionality of 0.75. They all show how the constant of proportionality can be seen as the relationship between the - and -coordinates of a point on the line and also as the slope of the line.)

• Would these methods work if the line did not go through ? Explain your thinking. (No, these methods would not work because the line would not represent a proportional relationship.)

Display this image for all to see, or draw a similar diagram.

Ask students how the equation relates to this situation. (The point is on the line when .) The values and are equal because they both represent the quotient of the vertical and horizontal sides of slope triangles for the same line.

The structure of coordinates for points on a line will be examined in greater detail in upcoming lessons. The goal of this activity is to connect student understanding of slope triangles with what it means to be on a line. It is not necessary for students to be able to write equations of a line in the form at this time.

The goal of this activity is for students to write an equation that can determine if a point lies on a line that does not contain the point by thinking about similar triangles and their properties (MP7). Proportional reasoning based only on the graphed line no longer applies here, but proportional reasoning using similar slope triangles does. 

Monitor for different expressions students write in the second question. For example, for line students may write:

Note that the rule or equation for the line is unlikely to come in the form . Students do not need to know this for now as it will be addressed in future work. 

The goal of this discussion is for students to see how different equations relate to different lines. 

Invite selected students, as described in the Activity Narrative, to share their equations for line , making sure to note different forms. For example, one equation is , though some students might rewrite this as or even . 

For line , one equation is , and some students may also write  or .

There is no reason to manipulate the equations or , as these two equations contain all of the proportional information from the similar slope triangles. Once the equations are manipulated, this structure is lost, and it is this structure that is of central importance here (MP7).

Here are some questions about those two equations for discussion:

• “What are the slopes of lines and ?” (Both lines have a slope of .)

• “How can we see the slope of the lines in their equations?” (Both equations contain the fraction .)

• “How are the equations for lines and alike and how are they different?” (They can both be written as some quotient involving and equal to . They have different values subtracted from and on the left side of the equations.)

If needed, note that the equation does not make sense if the denominator is equal to zero. For example, does not make sense if and . The reason for this is that the equation is based on using the point as one point. So, a slope triangle cannot be drawn using  again as the second point.