The purpose of this activity is to reinforce that some conditions define a unique polygon while others do not. Students build polygons given only a description of the polygons' side lengths. They articulate that this is not enough information to guarantee that a pair of quadrilaterals are identical copies. On the other hand, triangles have a special property that three specific side lengths result in a unique triangle. Students should notice that their recreation of Jada’s triangle is rigid—the side lengths and angles are all fixed.

Monitor for students who try putting the side lengths together in different orders to build different polygons, and invite them to share during the whole-class discussion.

In the digital version of the activity, students use an applet to create polygons dynamically. The digital version may reduce barriers for students who need support with fine-motor skills and for students who benefit from extra processing time.

Select previously identified students to share their constructions and explanations. Display each student’s example for all to see.

If desired, reveal Diego and Jada’s shapes, and display for all to see alongside students’ work. 

• “Is this what you thought Jada and Diego’s shapes looked like?”
• “Which shape did you make an identical copy of?” (Jada’s triangle.)
• “Why did you not make an identical copy of Diego’s shape?” (Because you can make quadrilaterals with the same side lengths but different angle measure.)

The purpose of this activity is for students to experience that the sum of the lengths of the two shorter sides of a triangle must be greater than the length of the longest side. Students continue working with the cardboard strips and fasteners from a previous lesson to see how many different triangles they can build given two of the three side lengths. In the Activity Synthesis, the possible triangles are arranged in a way that helps students see the structure of an unknown angle between two known side lengths as a hinge (MP7). This prepares students for using compasses to draw triangles with given side lengths. They also continue to work at recognizing when two triangles are identical copies that are oriented differently.

As students work, monitor for those who:

• Find different lengths for the third side of the triangle.
• Use precise language to describe how the two side lengths can move in relation to each other.
• Make a connection to the circle of Lin’s possible positions from the previous activity.

In the digital version of the activity, students use an applet to create polygons dynamically. The digital version may reduce barriers for students who need support with fine-motor skills and for students who benefit from extra processing time.

Select previously identified groups to share a triangle that they created. Establish whether each new triangle shared is the same as a triangle previously shared or is a different triangle. Collect one example of each possible triangle and display them for all to see, in order of increasing side length for the third side. Continue until students agree that all possible triangles are displayed.

To help students generalize about all the possible triangles that could be built with sides of 4 inches and 5 inches, ask questions like the following:

• “What do you notice about the triangles?”
• “Why was it impossible to use the 9-inch side to create a triangle?”
• “What is the longest the third side of the triangle could be?” (more than 8, but less than 9, e.g., 8.5, 8.75, 8.9)
• “What is the shortest the third side of the triangle could be?” (less than 2, but more than 1, e.g., 1.5, 1.25, 1.1)
• “What happens when the third side is 1 inch or 9 inches?” (You get a straight line instead of a triangle.)

Display a 5-inch strip fastened to a 4-inch strip for all to see. Demonstrate rotating the 4-inch strip around to line up with each of the displayed triangles, as well as to show the idea that the third side could have a fractional side length. Invite students to share how this relates to the previous activity about Lin’s distance from the swings. (If we hold one strip fixed, then all the possible locations where the other strip could end form a circle.)