Lesson 15

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 to reinforce that some conditions define a unique polygon while others do not. Students build polygons given only a description of their 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.

Students may think that their triangle is different from Jada’s because hers is “upside down.” Ask the student to turn their triangle around and ask them if it is now a different triangle. While there is a good debate to be had if they continue to insist they are different, let the students know that, for this unit, we will consider shapes that have been turned or flipped or moved as identical copies and thus “not different.”

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 along side students’ work. 

Ask students:

• “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.)

Consider explaining to students how this finding is applied in construction projects. For stability, the internal structures of many buildings (and bridges) will include triangles. Rectangles or other polygons with more than three sides often include triangular supports on the inside, to make the construction more rigid and less floppy.

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.

In IM 6–8 Math Accelerated, the unit on circles is in the previous year instead of the previous unit as stated in the narrative. Remind students of their work with circles and that a circle is the set of all the points that are equally distant from a center point.

During the activity synthesis, use the term “congruent” instead of “identical copies”. After asking the question “What is the longest the third side could have been? And the shortest?”, if students do not bring up what happens when the third side length is less than 1 inch, invite them to consider this case with their partner. If possible, demonstrate for students using the applet and the trace feature or, if time allows, have a student demonstrate or share a drawing they have made for the situation.

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.)