In this activity, students continue to examine equations of the form as they consider combinations of fruit that keep a total cost constant. The solutions are limited to non-negative integers and the set of all solutions is finite. 

Monitor for students who write the equation to describe the $10 combinations of fruit. There is no reason to solve for one variable in terms of the other, since a graph has not been requested and neither variable is dependent on the other.

The goal of this discussion is for students to see that the combinations of fruit that can be purchased for a total cost of $10 are solutions to this situation and also to the equation (or some equivalent).

Invite students to share the combinations of avocados and pineapples that cost a total of $10. (There are 6 combinations.) Record their responses for all to see. Next, invite previously selected students to share their equation representing the situation and record them for all to see. 

Explain to students that the 6 combinations of fruit that cost $10 are solutions to this situation and also to the equation . Emphasize that each solution has two values — one for avocados and one for pineapples. 

Ask students: 

• “Were there any patterns that helped you find the combinations?” (Buying 1 less pineapple means you can buy 2 more avocados.)

• “According to the equation you wrote, buying a pineapple and 9 avocados would also cost $10. Do you think this situation is realistic?” (Probably not, you would only buy whole pieces of fruit.)

In this activity, students write an equation representing a relationship between two quantities. Note that the relationship stated here matches the relationship describing avocados and pineapples that can be purchased for $10, but without the constraints of requiring whole-number values. 

Students find pairs of numbers that make the equation and stated relationship true and not true. By graphing both sets of points, they use repeated reasoning to observe that the graph of a linear equation is the set of its solutions, and any point not on this line is not a solution (MP8).

As students plot points that make the equation and stated statement true, monitor for students who create graphs with these features, sequenced in order from more common to less common: 

• Points in the first quadrant only

• Points on the axes (there are only two of these!)

• Points in the second or fourth quadrants

• Points with non-integer coordinate values

The purpose of this discussion is for students to see that values that make the relationship and equation true all lie in a line, and numbers not on the line will not make the equation or relationship true. 

Invite previously selected students to share their graphs of points that make the equation and statement true. Sequence the discussion of the graphs in the order listed in the Activity Narrative. Create a classroom display with a graph that includes all of the points shared by students, adding additional points as each student shares. As each graph is shared, ask students:

• “What do you notice about this graph?” (The points all lie on the same line.)

• “What is something on this graph that hasn’t been seen yet?” (a point that lies one of the axes; negative - or -values; values for or that are not integers)

If not brought up in students’ graphs, ask if the equation could be true if was a negative value, such as  ? If was a negative value, such as  ? If was not an integer such as  ?” (Yes, the corresponding - or -values would be  , , and respectively.) If necessary, add these points to the classroom display. 

After all previously selected graphs have been shared, connect the different responses to the learning goals by asking questions such as:

• “Based on your observations, what is the relationship between the solutions of an equation and its graph?” (The graph of the equation is the set of all solution pairs plotted as points in the coordinate plane.)

• “Is there any number for or where this equation would not be true?” (No. For every value of or there will be a corresponding - or -value that will make the equation true.)

• “What does the graph tell about the number of solutions of your equation?” (There are an infinite number of solutions.) 

• “What did you notice about the points with - and -values that did not make the statement true?” (Those points did not lie on the same line as the points with values that did make the statement true.)

• “What does it mean if a point does not lie on the line for the equation ?” (It means that pair of values for and is not a solution to the equation.)