# Lesson 15

Building Polygons

## 15.1: Where Is Lin? (5 minutes)

### Warm-up

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.

Teacher Notes for IM 6–8 Math Accelerated
If students have yet to work with compasses this year, take time during the activity synthesis to demonstrate their use in preparation for future activities. Make sure students understand that a compass is useful not just for drawing circles, but also for transferring lengths in general.

### Launch

Arrange students in groups of 2. If necessary, remind students of the directions north, south, east, and west and their relative position on a map. Provide access to geometry toolkits. Give students 2 minutes of quiet work time, followed by a partner and whole-class discussion.

Representation: Develop Language and Symbols. Display or provide charts with symbols and meanings. Display a chart of a compass showing the directions north, south, east and west.
Supports accessibility for: Conceptual processing; Memory

### Student Facing

At a park, the slide is 5 meters east of the swings. Lin is standing 3 meters away from the slide.

1. Draw a diagram of the situation including a place where Lin could be.

2. How far away from the swings is Lin in your diagram?

3. Where are some other places Lin could be?

### Launch

Arrange students in groups of 2. If necessary, remind students of the directions north, south, east, and west and their relative position on a map. Provide access to geometry toolkits. Give students 2 minutes of quiet work time, followed by a partner and whole-class discussion.

Representation: Develop Language and Symbols. Display or provide charts with symbols and meanings. Display a chart of a compass showing the directions north, south, east and west.
Supports accessibility for: Conceptual processing; Memory

### Student Facing

At a park, the slide is 5 meters east of the swings. Lin is standing 3 meters away from the slide.

1. Draw a diagram of the situation including a place where Lin could be.

2. How far away from the swings is Lin in your diagram?
3. Where are some other places Lin could be?

### Anticipated Misconceptions

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.

### Activity Synthesis

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.

## 15.2: Building Diego’s and Jada’s Shapes (10 minutes)

### Activity

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.

### Launch

Arrange students in groups of 2. Remind students that a quadrilateral is a four-sided polygon. If necessary, demonstrate how to use the fasteners to connect the strips.

Distribute one set of strips and fasteners to each group. Provide access to geometry toolkit, including rulers and protractors. Encourage students to think about whether there are different shapes that would fulfill the given conditions. Give students 5–6 minutes of quiet work time followed by a whole-class discussion.

For classes using the digital materials, there is an applet for students to use to build polygons with the given side lengths. If necessary, demonstrate how to create a vertex by overlapping the endpoints of two segments. It may work best for positioning each segment to put the green endpoint in place first and then adjust the yellow endpoint as desired.

Representation: Internalize Comprehension. Begin the activity with concrete or familiar contexts. Remind students that a polygon is a closed figure with straight sides. Demonstrate how to use fasteners to connect the slips.
Supports accessibility for: Conceptual processing; Memory

### Student Facing

1. Diego built a quadrilateral using side lengths of 4 in, 5 in, 6 in, and 9 in.
1. Build such a shape.
2. Is your shape an identical copy of Diego’s shape? Explain your reasoning.
2. Jada built a triangle using side lengths of 4 in, 5 in, and 8 in.
1. Build such a shape.

### Launch

Arrange students in groups of 2. Remind students that a quadrilateral is a four-sided polygon. If necessary, demonstrate how to use the fasteners to connect the strips.

Distribute one set of strips and fasteners to each group. Provide access to geometry toolkit, including rulers and protractors. Encourage students to think about whether there are different shapes that would fulfill the given conditions. Give students 5–6 minutes of quiet work time followed by a whole-class discussion.

For classes using the digital materials, there is an applet for students to use to build polygons with the given side lengths. If necessary, demonstrate how to create a vertex by overlapping the endpoints of two segments. It may work best for positioning each segment to put the green endpoint in place first and then adjust the yellow endpoint as desired.

Representation: Internalize Comprehension. Begin the activity with concrete or familiar contexts. Remind students that a polygon is a closed figure with straight sides. Demonstrate how to use fasteners to connect the slips.
Supports accessibility for: Conceptual processing; Memory

### Student Facing

1. Diego built a quadrilateral using side lengths of 4 in, 5 in, 6 in, and 9 in.

1. Build such a shape.

2. Is your shape an identical copy of Diego’s shape? Explain your reasoning.

2. Jada built a triangle using side lengths of 4 in, 5 in, and 8 in.

1. Build such a shape.

### Anticipated Misconceptions

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.”

### Activity Synthesis

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.

• “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.

Speaking: MLR7 Compare and Connect. Use this routine to support whole-class discussion. Ask students to consider what is the same and what is different about their quadrilateral as compared to Diego’s. Draw students' attention to the association between the order of side lengths in each quadrilateral and the angle measures in each shape. In this discussion, demonstrate the language used to make sense of the conditions required to make identical figures. Talking about and comparing the quadrilaterals will help draw students' attention to the conditions that define a unique polygon.
Design Principle(s): Maximize meta-awareness

## 15.3: Swinging the Sides Around (15 minutes)

### Activity

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.

Teacher Notes for IM 6–8 Math Accelerated

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.

### Launch

Arrange students in groups of 2.

Explain that students will be using the Trace feature to see the path of the point. It is accessed by right-clicking on the point.

Action and Expression: Internalize Executive Functions. Begin with a small-group or whole-class demonstration of how to use the cardboard strips as an informal compass for drawing all the possible locations where the given segments could end.
Supports accessibility for: Memory; Conceptual processing

### Student Facing

We'll explore a method for drawing a triangle that has three specific side lengths.  Use the applet to answer the questions.

1. Follow these instructions to mark the possible endpoints of one side:

1. For now, ignore segment $$AC$$ , the 3-inch side length on the left side

2. Let segment $$BD$$ be the 3-unit side length on the right side.  Right-click on point $$D$$, check Trace On.  Rotate the point, drawing all the places where a 3-inch side could end.

2. What shape have you drawn while moving $$BD$$ around?  Why?  Which tool in your geometry toolkit can do something similiar?

3. Use your drawing to create two unique triangles, each with a base of length 4 inches and a side of length 3 inches.  Use a different color to draw each triangle.

4. Repeat the previous instructions, letting segment $$AC$$ be the 3-unit side length.

5. Using a third color, draw a point where the two traces intersect. Using this third color, draw a triangle with side lengths of 4 inches, 3 inches, and 3 inches.

### Launch

Arrange students in groups of 2. Distribute one copy of the blackline master to each group. Make sure each group has one complete set of strips and fasteners from the previous activity. Provide access to geometry toolkits and compasses.

Tell students to take one 4-inch piece and two 3-inch pieces and connect them so that the 4-inch piece is in between the 3-inch pieces as seen in the image. If necessary, display the image for all to see. Students should not connect the 3-inch pieces to each other.

Explain to students that the sheet distributed to them is the 4-inch segment that is mentioned in the task statement and they will be drawing on that sheet.

If using the digital version of the activity, students will be using the Trace feature  to see the path of the point. It is accessed by right-clicking on the point.

Action and Expression: Internalize Executive Functions. Begin with a small-group or whole-class demonstration of how to use the cardboard strips as an informal compass for drawing all the possible locations where the given segments could end.
Supports accessibility for: Memory; Conceptual processing

### Student Facing

We’ll explore a method for drawing a triangle that has three specific side lengths. Your teacher will give you a piece of paper showing a 4-inch segment as well as some instructions for which strips to use and how to connect them.

1. Follow these instructions to mark the possible endpoints of one side:

1. Put your 4-inch strip directly on top of the 4-inch segment on the piece of paper. Hold it in place.
2. For now, ignore the 3-inch strip on the left side. Rotate it so that it is out of the way.
3. In the 3-inch strip on the right side, put the tip of your pencil in the hole on the end that is not connected to anything. Use the pencil to move the strip around its hinge, drawing all the places where a 3-inch side could end.
4. Remove the connected strips from your paper.
2. What shape have you drawn while moving the 3-inch strip around? Why? Which tool in your geometry toolkit can do something similar?
3. Use your drawing to create two unique triangles, each with a base of length 4 inches and a side of length 3 inches. Use a different color to draw each triangle.
4. Reposition the strips on the paper so that the 4-inch strip is on top of the 4-inch segment again. In the 3-inch strip on the left side, put the tip of your pencil in the hole on the end that is not connected to anything. Use the pencil to move the strip around its hinge, drawing all the places where another 3-inch side could end.
5. Using a third color, draw a point where the two marks intersect. Using this third color, draw a triangle with side lengths of 4 inches, 3 inches, and 3 inches.

### Activity Synthesis

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.)
Speaking, Listening, Representing: MLR7 Compare and Connect. Use this routine to help students compare their processes for building and drawing their triangles. Ask students “What is the same and what is different?” about the approaches. Draw students' attention to the connections between building and drawing such as: opening the hinge between the cardboard strips and drawing the circle using the compass. These exchanges strengthen students' language use and reasoning from concrete to representational approaches.
Design Principle(s): Maximize meta-awareness

## Lesson Synthesis

### Lesson Synthesis

Highlight some important takeaways from today’s lesson:

• “How is building a triangle with three given side lengths different from building a quadrilateral with four given side lengths?” (The triangle must be a specific one, but the quadrilateral might be a lot of different things by changing the angles.)
• “When you are given side lengths and asked to draw a triangle, how can you get started?” (Hold one length fixed and swing the other around in a circle.)
• “If you draw one side of the triangle with circles (of the correct radius for the other two side lengths) on each end, what does it look like when it is impossible to make a triangle?” (The two circles do not intersect, or they intersect at a point on the first line segment.)
• “If you draw one side of the triangle with circles on each end, and the circles do intersect, they may intersect twice. Why do we say there’s only one possible triangle instead of two?” (The two triangles are congruent.)

## Student Lesson Summary

### Student Facing

If we want to build a polygon with two given side lengths that share a vertex, we can think of them as being connected by a hinge that can be opened or closed:

All of the possible positions of the endpoint of the moving side form a circle:

You may have noticed that sometimes it is not possible to build a polygon given a set of lengths. For example, if we have one really, really long segment and a bunch of short segments, we may not be able to connect them all up. Here's what happens if you try to make a triangle with side lengths 21, 4, and 2:

The short sides don't seem like they can meet up because they are too far away from each other.

If we draw circles of radius 4 and 2 on the endpoints of the side of length 21 to represent positions for the shorter sides, we can see that there are no places for the short sides that would allow them to meet up and form a triangle.

In general, the longest side length must be less than the sum of the other two side lengths. If not, we can’t make a triangle!

If we can make a triangle with three given side lengths, it turns out that the measures of the corresponding angles will always be the same. For example, if two triangles have side lengths 3, 4, and 5, they will have the same corresponding angle measures.