We can use congruent angles and ratios of corresponding sides to identify similar right triangles and explain why their ratios stay constant.
Why do you think it is important that the angles in similar right triangles are the same, even if the corresponding side lengths are different?
I think it is important that the angles in similar right triangles are the same, even if the corresponding side lengths are different because...
By studying this visual, students might:
- The triangles have the same angle measures
- The triangles are different sizes but the same shape
- The side length values follow a pattern
- The ratios between the side length values are the same
- The triangles have the same shape, but different sizes.
- How do we know the triangles are proportional
- Why do the ratios stay the same
- How do corresponding sides match
- Do all triangles with the same angles have the same ratios
- Why angles matter for similarity
Extending the Discussion
- After randomly calling on students, if there is anything from this list that was not mentioned, then ask the class, "Did anyone notice...?"
- After students have shared what they notice, ask the class, "Did anyone wonder...?" using the suggestions above or anything else you might think is interesting or relevant to the lesson.
Structured Conversation Prompts
What is a similar right triangle?
A similar right triangle is...
How is a similar right triangle related to congruence?
A similar right triangle is related to congruence because...
Why do you think it is important that the angles in similar right triangles are the same, even if the corresponding side lengths are different?
I think it is important that the angles in similar right triangles are the same, even if the corresponding side lengths are different because...
Example Student Responses to the Observational Question
A similar right triangle has the same shape but different side length values.
A similar right triangle has congruent angles and proportional corresponding sides with equal ratios, even if the side length values are different.
Responding to Responses
Emphasize and celebrate each student's use of the key vocabulary to support a culture of "no wrong answers."
Structuring Student Conversations
Have students list observations from the visual as a warm-up, then use the Q-SSS-A process to guide small-group conversations. In the slide decks, brackets can be moved to prepare the structured conversation. In the example to the right, students will be instructed: Q-SSS-A.
- To put a thumb up, then lower their hand when they are ready to answer the question
- To share with their elbow/shoulder partner, and that the student with the darkest shoe will share first
- That they will be randomly called on after the conversation
Here is an example of structuring a conversation with Q-SSS-A.
Note: the inferential question is the same as the language objective. It is recommended that students answer the inferential question in a small-group discussion before answering it individually as the closure or exit ticket of the lesson.
Structured Reading
The purpose for reading is to understand how angles and ratios are used to determine if triangles form a similar right triangle.
- Angles in each triangle
- How corresponding sides match
- The ratio between side length values
- When ratios stay the same
- When triangles are proportional
How can you use angles and ratios to determine if the new ramps form a similar right triangle?
I can use the angles and ratios to determine if the ramps form a similar right triangle because...
Structuring the Reading
Communicate the purpose of reading to the students and instruct them to make a note every time they see something on the PAT ("Pay Attention To") list. How you have students note items on the PAT list is up to you. This could include:
- Putting an asterisk in the margin
- Underlining text that supports the PAT list
- Putting a comment in the margin
Follow the reading with the post-reading discussion. Structure this discussion using the Q-SSS-A process just like the structured conversations in this lesson.
Note: you might find the relational question is better discussed before or after the reading. This depends on whether the relational question is directly related to the reading or might make connections across units.
Differentiating the Reading
You will notice that three different reading passages are provided with this lesson. Look at the shapes in the top-left of each passage to determine the grade level.
In a class with students at diverse reading level proficiencies, you can give the appropriate reading passage to different students, while having all students follow the same PAT list and post-reading discussion.