Lesson 10

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.

Students continue to practice drawing triangles from given conditions and categorizing their results. This activity focuses on the inclusion of a single angle and two sides. Again, they do not need to memorize which conditions result in unique triangles, but should begin to notice how some conditions (such as the equal side lengths) result in certain requirements for the completed triangle.

There is an optional blackline master that can help students organize their work at trying different configurations of the first set of measurements. If you provide students with a copy of the blackline master, ask them to determine whether any of the configurations result in the same triangle, as well as whether any one configuration results in two possible triangles.

Some students may draw two different orientations of the same triangle for the first set of conditions, with the \(40^\circ\) angle in between the 4 cm and 5 cm sides. Prompt them to use tracing paper to check whether their two triangles are really different (not identical copies).

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.

Ask one or more students to share how many different triangles they were able to draw with each set of conditions. Select students to share their solutions.

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 the optional blackline master was used, ask students:

• “Which configurations made identical triangles?” (the top left and bottom left)
• “Which configurations made more than one triangle?” (the bottom right)

If not mentioned by students, explain to students that the top left and bottom left configurations result in the same triangle, because in both cases the \(40^\circ\) angle is in between the 4 cm and 5 cm sides and that the bottom right configuration results in two different triangles, because the arc intersects the ray in two different places.

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 more 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.

This activity focuses on including three angle conditions. The goal is for students to notice that some angle conditions result in a large number of possible triangles (all scaled copies of one another) or are impossible to create. Students are not expected to learn that the angles must sum to 180 degrees in a triangle, but are not barred from noticing this fact.

If students struggle to get started, remind them of Lin’s technique of using the protractor and a ruler to make an angle that can move along a line.

Select students to share their drawings and display them for all to see. Ask students:

• “Were there any sets of measurements that produced a unique triangle?” (no)
• “Which combinations of angles could not be drawn?” (the angles in the second problem, \(50^\circ, 60^\circ, 100^\circ\))
• “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 scaled copies of the triangles with the same angles but with shorter or longer side lengths)

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.