Lesson 17

The purpose of this warm-up is to remind students that a compass is useful for transferring a length in general, and not just for drawing circles. As students discuss answers with their partners, monitor for students who can clearly explain how they can use the compass to compare the length of the third side.

Arrange students in groups of 2. Give students 2 minutes of quiet work time followed by time to discuss their answers with their partner. Follow with a whole-class discussion. Provide access to geometry toolkits and compasses.

Ask previously identified students to share their responses to the final question. Display their drawing of the angle for all to see. If not mentioned in students’ explanations, demonstrate for all to see how to use the compass to estimate the length of the third side of the triangle.

In this activity, students are asked to draw as many different triangles as they can with the given conditions. The purpose of this activity is to provide an opportunity for students to see the three main results for this unit: a situation in which only a unique triangle can be made, a situation in which it is impossible to create a triangle from the given conditions, and a situation in which multiple triangles can be created from the conditions.

Students are not expected to remember which conditions lead to which results, but should become more familiar with some methods for attempting to create different triangles. They will practice including various conditions into the triangles, including the conditions in different combinations, and recognizing when the resulting triangles are identical copies or not.

The activity synthesis references a previous activity which is not included in IM 6–8 Math Accelerated. If time allows, demonstrate for students that one way to draw a triangle (Lin's method) with one angle measuring \(60^\circ\), one angle measuring \(90^\circ\), and one side measuring 4 cm is to:

1. Draw the 4 cm segment.
2. Draw the \(60^\circ\) angle on one end of the segment, with a very long ray.
3. Place a protractor along the ray.
4. Line up a ruler at the \(90^\circ\) measure on the protractor.
5. Keeping the ruler and protractor together, slide them along the ray until the edge of the ruler intersects with the other end of the 4 cm segment.
6. Keeping the ruler in place on the paper, remove the protractor from underneath.
7. Draw a line along the ruler from the ray to the segment.

Here is an image showing what this would look like for a triangle with one angle measuring \(75^\circ\), one angle measuring \(45^\circ\), and one side measuring 5 cm:

Some students may draw two different orientations of the same triangle for the third set of conditions, with the 4 cm side in between the \(60^\circ\) and \(90^\circ\) angles. Prompt them to use tracing paper to check whether their two triangles are really different (not identical copies).

Some students may say the third set of measurements determines one unique triangle, because they assume the side length must go between the two given angle measures. Remind them of the discussion about Lin’s triangle in the previous activity.

Ask students to indicate how many different triangles (triangles that are not identical copies) they could draw for each set of conditions. Select students to share their drawings and reasoning about the uniqueness of each problem. Discuss methods students used to try to think about other triangles that might fit the conditions.

Consider asking some of the following questions:

• “Which conditions produced a unique triangle?” (the first set of conditions)
• “Were there conditions that produced more than one triangle?” (the third set of conditions)
• “Were there conditions you could not draw a triangle for?” (the second set of conditions)
• “Why could you not draw a triangle for the second set of conditions?” (because two sides are parallel and will never intersect)

If not mentioned by students, explain to students that for the third set of conditions it is possible that all students drew identical copies using the 4 cm length as the side between the \(60^\circ\) and \(90^\circ\) angles. Consider asking them to think of the previous activity and to try to draw the triangle the way Lin would.

In grade 7, students do not need to know that the angles within a triangle sum to \(180^\circ\). Tell them that next year they will learn more about why these different conditions determine different numbers of triangles.

If students struggle to create more than one triangle from the first set of conditions, prompt them to write down the order they already used for their measurements and then brainstorm other possible orders they could use.

If students struggle to get started, remind them of the technique shared at the end of the previous activity of using the protractor and a ruler to make an angle that can move along a line.

Select students to share how many different triangles they were able to draw with each set of conditions.

If not brought up in student explanations, point out that for the first problem, one possible order for the measurements (\(40^\circ\), 5 cm, 4 cm) can result in two different triangles (the bottom two in the solution image). One way to show this is to draw a 5 cm segment and then use a compass to draw a circle with a 4 cm radius centered on the segment’s left endpoint. Next, draw a ray at a \(40^\circ\) angle centered on the segment’s right endpoint. Notice that this ray intersects the circle twice. Each one of these points could be the third vertex of the triangle. While it is helpful for students to notice this interesting aspect of their drawing, it is not important for students to learn rules about the number of possible triangles given different sets of conditions.

If students do not touch on the scaled triangles created by the same three angles, ask, “Why is there more than one triangle that can be made with the measurements in the first problem?” (Because there are no side lengths mentioned, so we can create smaller or larger copies of the triangles with the same angles but with shorter or longer side lengths.) Students begin their study of scaled copies in the next unit, so their work on this question is a sneak peak of the work ahead.

MLR 1 (Stronger and Clearer Each Time): Before discussing the second set of conditions as a whole class, have student pairs share their reasoning for why there were no triangles that could be drawn with the given measures, with two different partners in a row. Have students practice using mathematical language to be as clear as possible when sharing with the class, when and if they are called upon.