This Warm-up prompts students to make sense of a figure by considering properties of triangles and similar polygons. This figure is also used in the optional activity.
This activity uses the Co-Craft Questions math language routine to advance reading and writing as students make sense of a context and practice generating mathematical questions.
Student Lesson in Spanish
Launch
Tell students to close their books or devices (or to keep them closed). Arrange students in groups of 2. Introduce the context image from the task. Use Co-Craft Questions to orient students to the context and elicit possible mathematical questions.
Give students 1–2 minutes to write a list of mathematical questions that could be asked about the situation before comparing questions with a partner.
Activity
None
Student Task Statement
Student Response
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Building on Student Thinking
Activity Synthesis
Invite several partners to share 1 question with the class and record responses. Ask the class to make comparisons among the shared questions and their own. Ask, “What do these questions have in common? How are they different?”
If questions about similarity or scale factor are not mentioned by students, ask students "What is a mathematical question about similar figures that you could ask using this diagram?"
8.2
Activity
30 mins
Making Pasta Angles and Triangles
Standards Alignment
Building On
7.RP.A.2.a
Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
In this activity, students create triangles with given angles using dried pasta. When comparing triangles constructed with just one angle measure in common, they notice that the triangles are all different, usually only sharing the one given angle. When comparing triangles constructed with two angle measures in common, they notice that the triangles are all similar (MP7).
This is true because a triangle is determined by the length of one side and the two angles made with that side. The length of that side can then be scaled with a dilation to give any particular similar triangle. This means that triangles with two pairs of congruent corresponding angles must be similar, commonly known as the angle-angle (AA) triangle similarity theorem. Students do not need to know the AA triangle similarity theorem by name at this time, but they will see this idea again in a later activity.
Launch
Tell students that they are going to build triangles using dried pasta and a specific set of given angles. Then they will find classmates who used the same angle(s) and compare triangles. Display the image for all to see.
In the image the approximate angle measures are , , and . Demonstrate how students can trace their angle(s) on a sheet of paper before they 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, extra paper to tape their triangles, and a ruler from their geometry toolkits.
Pause students after they have made and compared triangles for the first problem, making sure they find some non-similar triangles. Pause students again after the second problem, making sure they find similar triangles. 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.
MLR2 Collect and Display. Direct attention to words collected and displayed from a previous lesson or activity. Invite students to borrow language from the display as needed, and update it throughout the lesson. Circulate, listen for, and collect the language students use as they find others who use the same angles and decide whether their triangles are similar or not. Record words and phrases such as “The triangles are not similar because the angles do not match.” Advances: Conversing, Reading
Engagement: Provide Access by Recruiting Interest. Use visible timers or audible alerts to help learners anticipate and prepare to transition between different parts of this activity. Supports accessibility for: Organization, Attention
Activity Synthesis
The goal of this discussion is to make sure students understand that if triangles share 2 pairs of congruent angles then they are similar. Discuss with students:
“Did the angles in your triangles always add up to 180 degrees?” (Yes, but because the angles were rounded to the nearest 5 degrees sometimes they weren’t exactly 180 degrees.)
“What did you notice in the first problem when you compared your triangle with other classmates who used the same angle?” (The triangles were all different.)
“What did you notice in the problem where you had to use 2 angles to make a triangle?” (The triangles all looked the same but were different sizes. They were similar.)
“Did you need all 3 angle measurements to build a triangle in the second problem?” (No, once we knew 2 angles, that was enough to build the pasta triangle, the third angle automatically being the right measure.)
“How did you decide whether or not the sides of your triangle were proportional to other triangles?” (By dividing 2 side lengths of one triangle and comparing that to the quotient of the 2 corresponding sides in the other triangle. Measurement error may mean that quotients computed may not be exactly equal even if the triangles are similar.)
“How did you check whether or not your triangle was similar to another?” (Lining up 1 angle and then dilating would tell me if the triangles were similar, and observing that the angles were different or the sides were not proportional would tell me they were not similar.)
8.3
Activity
Optional
10 mins
Similar Figures in a Regular Pentagon
Standards Alignment
Building On
Addressing
8.G.A
Understand congruence and similarity using physical models, transparencies, or geometry software.
This optional activity presents students with a complex figure and asks them to find triangles similar to a given triangle. This activity provides an additional opportunity for students to practice explaining why two triangles are similar (MP3).
Monitor for students who use these methods to show that their chosen triangles are similar to triangle :
Finding a sequence of translations, rotations, reflections, and dilations
Checking that two (or three) corresponding angles are congruent
Launch
Arrange students in groups of 2. Provide access to geometry toolkits.
If students created a mathematical question about similarity or scale factors during the Warm-up, highlight those questions now. Give students 5 minutes of quiet work time followed by a partner then whole-class discussion.
Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example, use the same color to illustrate which triangles are similar to . Annotate side and angle measurements in the appropriate places on the image. Supports accessibility for: Visual-Spatial Processing
Activity Synthesis
The goal of this discussion is to compare different strategies for showing that 2 triangles are similar. Invite students to share some of the triangles they found and explain how they determined the triangles to be similar.
MLR7 Compare and Connect. Lead a discussion comparing, contrasting, and connecting the different representations. Ask, “How are these strategies the same?” “How are they different?” and “Are there any benefits or drawbacks to one strategy compared to another?” Advances: Representing, Conversing
Lesson Synthesis
The goal of this discussion is for students to understand that when 2 triangles share 2 pairs of congruent angles, they are similar. Begin by asking students what they know about a triangle that has a angle. Discuss with students:
“Could this triangle be isosceles?” (Yes, it could have two angles and one angle.)
“Could this triangle not be isosceles?” (Yes, it could have one angle, one angle, and one angle.)
“Are all triangles with a angle similar?” (No, because the other two pairs of angles could be different and the sides might not be proportional.)
Next, ask students what they know about a triangle that has 2 angles that measure and . Then show this image.
Discuss with students:
“Are these two triangles similar?” (Yes, we can dilate triangle with center and scale factor 3, and then apply rigid transformations so that angle matches up with angle and angle matches up with angle . The vertices also match up and so the 2 triangles are similar.)
“Did we need to know the measure of the third angle in each triangle to determine they were similar?” (No, since the sum of the interior angles of a triangle is always , if two corresponding angles are congruent, then the third angle will also be congruent.)
"Consider a third triangle that is also similar to triangle . What do we know about this third triangle?" (It has 2 angles that measure and . It is also similar to triangle .)
Add to or create a new class display listing different ways to show that 2 triangles are similar, such as:
Describe a series of transformations that take one figure to the other.
Find 2 congruent corresponding angles in triangles.
Find 3 congruent corresponding angles in triangles.
If 2 triangles are both similar to the same triangle, then they are similar to each other.
This display should be posted in the classroom for the remaining lessons within this unit.
Student Lesson Summary
Two polygons are similar when there is a sequence of translations, rotations, reflections, and dilations taking one polygon to the other. When the polygons are triangles, we only need to check that both triangles have two corresponding angles to show they are similar.
For example, triangle and triangle both have a 30-degree angle and a 45-degree angle.
We can translate to and then rotate around point so that the two 30-degree angles are aligned, giving this picture:
Then a dilation with center and appropriate scale factor will move to . This dilation also moves to , showing that triangles and are similar.
Have feedback on the curriculum?
Help us improve by sharing suggestions or reporting issues.
Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
Your teacher will give you dried pasta, a set of 3 angles labeled , , and , blank paper, and tape.
Create a triangle using 3 pieces of pasta and angle . Your triangle must include the angle you were given, but you are otherwise free to make any triangle you like. Tape your pasta triangle to a sheet of paper so it won’t move.
After you have created your triangle, measure each side length with a ruler and record the length on the paper next to the side. Then measure the angles to the nearest using a protractor and record these measurements on your paper.
Find 2 others in the room who have the same angle and compare your triangles. What is the same? What is different?
Are the triangles congruent? Are the triangles similar? Explain your reasoning.
Now use more pasta and all 3 angles , , and to create 1 new triangle. Tape this pasta triangle on a separate blank sheet of paper.
After you have created your triangle, measure each side length with a ruler and record the length on the paper next to the side. Then measure the angles to the nearest using a protractor and record these measurements on your paper.
Find 2 others in the room who used your same 3 angles and compare your triangles. What is the same? What is different?
Are the triangles congruent? Are the triangles similar? Explain your reasoning.
Student Response
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Building on Student Thinking
Students may need a reminder that the sum of angles in any triangle is 180 degrees.
Some students may think that their triangle side lengths are not proportional or that their angle measures do not add up to 180 degrees. Remind them that their measurements are only approximate.
Activity
None
Student Task Statement
This diagram has several triangles that are similar to triangle .
Three different scale factors were used to make triangles similar to . In the diagram, find at least 1 triangle of each size that is similar to .
Explain how you know each of these 3 triangles is similar to .
Find a triangle in the diagram that is not similar to .
Student Response
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Building on Student Thinking
If students predict similar triangles just by looking at the triangles, make sure they justify their decisions by including specific measurements.
