Lesson 7

The purpose of this warm-up is to remind students that when you have a fixed starting point, all the possible endpoints for a segment of a given length form a circle (centered around the starting point). The context of finding Lin’s position in the playground helps make the geometric relationships more concrete for students. Since there are many possible distances between Lin and the swings (but not infinitely many), this activity serves as an introduction to formalizing rules about what lengths can and cannot be used to form a triangle.

Monitor for students who come up with different locations for Lin, as well as students who recognize that there are many possible locations, to share during the whole-class discussion.

Some students might assume that the swings, the slide, and Lin are all on a straight line, and that she must be 8 meters away. Ask these students if the problem tells us which direction Lin is from the slide.

Some students may confuse the type of compass discussed in the Launch and the type of compass discussed in the Activity Synthesis. Consider displaying a sample object or image of each of them and explain that the same name refers to two different tools.

First, have students compare answers and share their reasoning with a partner until they reach an agreement.

Next, ask selected students to share their diagrams of where Lin is located. Discuss the following questions with the whole class:

• “Do we know for sure where Lin is?” (No, because we don’t know what direction she is from the swings.)
• “What shape is made by all the possible locations where Lin could be?” (a circle)
• “What is the closest Lin could be to the swings?” (2 m)
• “What is the farthest Lin could be away from the swings?” (8 m)

Consider using the applet at https://ggbm.at/qkHk6TpJ to show all the locations where Lin could be. Based on their work with drawing circles in a previous unit, some students may suggest that a compass could be used to draw all the possible locations where Lin could be. Consider having a student demonstrate how this could be done. If not mentioned by students, it is not necessary for the teacher to bring it up at this point.

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 the 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 unknown angle between two known side lengths as a hinge. 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

Some students may think that there are more than 5 possible triangles they can build, because they don’t realize that some of the triangles they have listed are identical copies of other triangles on their list, with the side lengths written in a different order.

Some students may think that the third side of the triangle cannot be 4 or 5 inches, because then the triangle would have two sides of that length instead of the one asked for in the question. Explain that the triangle is acceptable as long as at least one side is 5 inches long and at least one side is 4 inches.

Some students may think that 8 is the longest the third side can be and 2 would be the shortest (if they were given a strip of that length), because they don’t realize that there could be fractional side lengths.

Select previously identified groups to share a triangle 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 all possible triangles are displayed.

To help students generalize about all the possible triangles that could be built with sides 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 \(180^\circ\) 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.

Display a 4-inch strip connected to two 3-inch strips, positioned parallel to each other as pictured in the Launch. To help students connect the process of building with cardboard strips to drawing on paper, ask questions like:

• “If you want to build a triangle with these side lengths, how do you know at what angle to position the cardboard strips?” (Turn the sides until their unattached endpoints are touching.)
• “If you want to draw a triangle with these side length, how can you know at what angle to draw the sides?” (Find the point where both circles intersect.)
• “We have seen with the cardboard strips that an unknown angle works like a hinge. How is that represented in your drawing?” (with a circle centered on the endpoint of one segment and a radius the length of the other segment)

Select students to share their drawings with the class. To reinforce the patterns that students noticed in the previous activity, consider asking questions like these:

• “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.)