Lesson 8

This warm-up prompts students to use what they know about operations to create related expressions. While many warm-ups in the unit encourage students to work mentally and verbally, students will write their responses to this prompt. Since many different responses are possible, the task is accessible to all students and provides an opportunity to hear how each student reasons about the operations.

Arrange students in groups of 2. Display problem. Give students 1 minute of quiet think time and ask them to give a signal when they have three expressions. Follow with a 1 minute partner discussion and then a whole-class discussion.

If students only rely on the associative and commutative properties, suggest that they try to include at least one expression containing parentheses.

Ask selected students to share their expressions. Record and display their responses for all to see. Ask students if or how the factors in the problem impacted their strategy. To involve more students in the conversation, consider asking:

• “Did anyone have a similar expression?”
• “Did anyone have a different expression related to this one?”
• “Do you agree or disagree? Why?”

In this activity, students create triangles with given angles and compare them to classmates’ triangles to see that they are not necessarily congruent, but they are similar. 

Each angle measure in a triangle imposes a constraint and students are examining in this activity how many constraints there are in similar triangles. There are only two. This makes sense since we can place one side of the triangle and then the triangle is determined by the length of that side and the two angles made with that side. (Note: the length of that side can be scaled with an appropriate dilation to give any particular similar triangle with these angles.)

Watch for how students deal with measurement error. Remind students who are looking for proportional side lengths in similar triangles that they may not be exactly proportional. Also, students may find that their rounded angle measures do not add up to \(180^\circ\). Remind them that the measurements are only approximate. 

Tell students that they are going to build triangles using pasta and some of their given angles. Then they will find classmates who used the same angle(s) and compare their triangles. Tell them that they will need to move around the classroom to identify partners who use the same angle(s). For large classrooms, consider asking students to find three matching partners instead of two. Demonstrate how to measure angles in a pasta triangle.

In this picture, the approximate angle measures are \(90^\circ\), \(50^\circ\),  and \(40^\circ\). Tell students that they can trace their angle(s) on the sheet of paper they are going to use to build their pasta triangle.

Provide students with 1 strip of 3 angles (A, B, and C) pre-cut from the blackline master. Each student should also have access to tape, and extra paper to tape their triangles to. Finally, each student will need a ruler (which should be available in their geometry toolkits).

Pause students after they have made and compared triangles with one angle (first problem) and three angles (second problem). Make sure that students have found some non-similar triangles on the first problem and similar triangles for the second. Consider doing a gallery walk to see how the first sets of triangles differ and how the second sets of triangles are alike. Then have students work on the third problem.

Students may need a reminder that the sum of angles in any triangle is 180 degrees. 

Ask students if they needed all three angle measurements to build a triangle in the second problem. Some students may notice that it was not necessary: once they knew two angles, that was enough to build the pasta triangle, the third angle automatically being the right measure. Other students may find that having the third angle was helpful to get better accuracy for all three angle measures. Ask students how this relates to their findings on the last question: the last question shows that two shared angle measures would be enough to guarantee that two triangles are similar. This makes sense because the sum of the angle measures in a triangle is 180 degrees so triangles that share two pairs of congruent angles actually share three pairs of congruent angles. 

Some important discussion questions include

• “Did your angles in the triangle always add up to 180 degrees?” (Answers may vary for the first triangles because the angles were rounded to the nearest 5 degrees and the rounding can influence the sum of the angles.)
• “How did you decide whether or not the sides of your triangle were proportional to other triangles?” (Answers vary, the key being that measurement error means that quotients computed from measured side lengths may not be exactly equal even if the triangles are similar.)
• “How did you check whether or not your triangle was similar to another?” (Answers should include aligning an angle and then dilating when the triangles were similar, and observing that the angles were different or the sides were not proportional when they were not similar.)

A big conclusion from this activity is that if triangles share two pair of congruent angles then they are similar. For the quadrilaterals studied in the previous activity, there was another instance where a pair of congruent angles was enough to decide that quadrilaterals are similar, namely rhombuses. But the triangle result is more surprising because there are three side lengths and no restrictions are made on these. 

This activity presents a complex figure with many triangles and asks students to find triangles similar to a given triangle. From the previous activity, students know that finding two pair of congruent angles is sufficient to show similarity. Students may also measure all three angles and check that they are congruent. In addition to using angle measures, students can describe transformations that take one triangle to another.

Monitor for students who use these methods to show that their chosen triangles are similar to triangle \(DJI\):

• finding a sequence of translations, rotations, reflections, and dilations
• checking that two (or three) corresponding angles are congruent

Select students who use these methods and invite them to present during the discussion.

Tell students that they may use anything in their geometry toolkits as well as a copy of triangle \(DJI\) to find triangles similar to it.  

If students eyeball the triangles in predicting similar triangles, make sure they justify their decisions by including specific measurements.

Invite selected students to share some of the triangles they found and explain how they determined the triangles to be similar. Sequence them so students who used rigid transformations and dilations go first, followed by those who use angle measures. Concluding that two triangles are similar using the angle criterion from the previous task is a quick, efficient argument. On the other hand, finding explicit transformations and dilations taking one triangle to another provides a more tactile, concrete experience. Both are important methods and students can choose based on their own personal comfort and the nature of the problem at hand. 

Make a list of different ways to show that two triangles are similar, including:

• using transformations (cutting out the triangle and placing it appropriately)
• finding two congruent corresponding angles in the triangles
• finding three congruent corresponding angles in the triangles

Among those triangles that are not similar to \(DJI\), ask if some of these may be similar to each other.