The purpose of this discussion is for students to generalize their strategies for solving two-number riddles and connect them to what they know about inequalities. The discussion focuses on preparing students to use a graph to interpret solutions to systems of inequalities.

Ask previously identified students to share their strategies in this order:

• Guessing values and checking them for the conditions
• Rewriting verbal descriptions into equations or inequalities

Here are some questions for discussion:

• “Did both strategies arrive at the same answer?” (Yes.)
• “Does anyone feel one strategy takes more time than another?” (In some cases, it may be easy to guess and check quickly, but sometimes it's hard to think of numbers that work, then writing equations or inequalities can end up being quicker.)

The rest of this discussion will focus on the riddle about finding two numbers whose sum is more than 15 and whose difference is more than 1.

Display the image of a blank graph for all to see.

Ask students to share the pairs of values they found that had a sum greater than 15. Record those in one color. Call on a student to graph the inequality for numbers with a sum greater than 15. Invite students to share what they notice. Ask students to share the pairs of values they found that had a difference greater than 1. Record those in another color. Call on a student to graph the inequality for numbers with a difference greater than 1. Invite students to share what they notice. Ask students to share pairs of values they found that had a sum greater than 15 and a difference greater than 1. Record those in a third color. Ask students what they notice. If no student mentions it, point out that the solutions to the riddle are in the area of the graph where the two linear inequalities overlap.

The purpose of the discussion is to recognize how lines can determine regions of the coordinate plane and methods for describing the regions.

Direct students’ attention to the reference created using Collect and Display. Ask students to share how they described the chosen region. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases. (For example, the display may have “the region to the left” already on it and can be updated with the more precise phrase “the region containing the point .”)

Here are some questions for discussion:

• "What do the graphs of equations that have 4 regions have in common?" (There are 2 lines that intersect at one point.)
• "Will the plane always be split into 3 regions when there are 2 parallel lines graphed?" (Yes, unless the two lines are actually the same line and there are only 2 regions.)
• "Into how many regions could 3 lines divide the plane?" (2 regions if all three lines are the same. 3 regions if two of the lines are the same and one is parallel, but different. 4 regions if two of the lines are the same and the third intersects it at one point or if all 3 lines are distinct, parallel lines. 6 regions if all three lines intersect at the same point or there are two parallel lines and one that is not parallel. 7 regions if each line intersects the other two in distinct points.)