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.)

The purpose of this activity is to relate the process for building a triangle given 3 side lengths (using cardboard strips and metal fasteners) to the process for drawing a triangle given 3 side lengths (using a compass). Students use the cardboard strips as an informal compass for drawing all the possible locations where the given segments could end. They are reminded of their work with circles in a previous unit: That a circle is the set of all the points that are equally distant from a center point and that a compass is a useful tool, not just for drawing circles, but also for transferring lengths in general. This prepares them for using a compass to draw triangles in future lessons.

In this activity, students also consider what their drawing would look like if the two shorter sides were too short to make a triangle with the third given side length.

Left-handed students may find it easier to start with drawing the 3-inch circle on the left side of the 4-inch segment.

This activity works best when each student has access to the strips and fasteners. If physical manipulatives are not available, consider using the digital version of the activity. In the digital version, students use an applet to manipulate fixed side lengths of polygons. 

The goal of this discussion is for students to connect building figures with fixed lengths with constructing figures using a compass.
 
Display 2–3 representations from previously selected students for all to see. If time allows, invite students to briefly describe their approach, then use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:

• “What do using the strips and using the compass have in common? How are they different?”
• “Did anyone create their triangles in the same way, but would explain it differently?”
• “How does the side length of each triangle show up in each method?”
• “Are there any benefits or drawbacks to one representation compared to another?”

If time allows, highlight patterns that students noticed in previous activities with questions such as:

• “How many different triangles could we draw when we had only traced a circle on one side? Why?” (Lots of different triangles, because we were only using two of the given side lengths.)
• “What is the longest the third side could have been? And the shortest?” (Less the 7 inches; More than 1 inch)
• “How many different triangles could we draw once we had traced a circle on each side?” (It looked like there were 2 different triangles, but they are identical copies, so there’s really only 1 unique triangle.)