Triangles with 3 Common Measures
10 min
Teacher Prep
Setup
Narrative
The purpose of this Warm-up is to begin looking at the different triangles that can be drawn when three measures are specified. The first set of triangles in this activity all share the same 3 side lengths. The second set of triangles all share the same 3 angle measures. Students consider which set of conditions mean that the created triangles must be identical.
Launch
Provide access to geometry toolkits. Give students 1 minute of quiet think time, followed by a whole-class discussion.
Student Task
Examine each set of triangles. What do you notice? What is the same about the triangles in the set? What is different?
Set 1:
Set 2:
Sample Response
- All of the side lengths and angles are the same size. These triangles are identical copies. The triangles face different directions.
- These triangles all have the same angles, but different side lengths. They could be scaled copies that are oriented differently.
Activity Synthesis (Teacher Notes)
Invite students to share things they notice—things that are the same and things that are different about the triangles. Record and display these ideas for all to see.
If these discussion points do not come up in students’ explanations, make them explicit:
In the first set:
- All the triangles are identical copies, just in different orientations.
- They have the same 3 side lengths.
- They have the same 3 angle measures (can be checked with tracing paper or a protractor).
In the second set:
- The triangles are not identical copies.
- Note: Students may recognize that these triangles are scaled copies of each other, since they have the same angle measures. However, this is the first time students have seen scaled copies in different orientations, and it is not essential to this lesson that students recognize that these triangles are scaled copies.
- They have the same 3 angle measures.
- They have different side lengths (can be checked with tracing paper or a ruler).
The goal is to make sure students understand that the second set has 3 different triangles (because they are different sizes) and that the first set really shows only 1 triangle in many different orientations. Tracing paper may be helpful to convince students of this.
Anticipated Misconceptions
Some students may say that all the triangles in the second set are “the same shape.” This statement can result from two very different misconceptions. Listen to the students’ reasoning and explain as needed:
- Just because they are all in the same category, “triangles,” doesn’t mean they are all the same shape. If we can take two shapes and position one exactly on top of the other, so all the sides and corners line up, then they are identical copies.
- These triangles are scaled copies of each other, but that does not make them “the same” because their side lengths are still different. Only scaled copies made using a scale factor of 1 are identical copies.
Standards
- 7.G.2·Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
- 7.G.A.2·Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
15 min
10 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Triangles with 3 Common Measures
10 min
Teacher Prep
Setup
Narrative
The purpose of this Warm-up is to begin looking at the different triangles that can be drawn when three measures are specified. The first set of triangles in this activity all share the same 3 side lengths. The second set of triangles all share the same 3 angle measures. Students consider which set of conditions mean that the created triangles must be identical.
Launch
Provide access to geometry toolkits. Give students 1 minute of quiet think time, followed by a whole-class discussion.
Student Task
Examine each set of triangles. What do you notice? What is the same about the triangles in the set? What is different?
Set 1:
Set 2:
Sample Response
- All of the side lengths and angles are the same size. These triangles are identical copies. The triangles face different directions.
- These triangles all have the same angles, but different side lengths. They could be scaled copies that are oriented differently.
Activity Synthesis (Teacher Notes)
Invite students to share things they notice—things that are the same and things that are different about the triangles. Record and display these ideas for all to see.
If these discussion points do not come up in students’ explanations, make them explicit:
In the first set:
- All the triangles are identical copies, just in different orientations.
- They have the same 3 side lengths.
- They have the same 3 angle measures (can be checked with tracing paper or a protractor).
In the second set:
- The triangles are not identical copies.
- Note: Students may recognize that these triangles are scaled copies of each other, since they have the same angle measures. However, this is the first time students have seen scaled copies in different orientations, and it is not essential to this lesson that students recognize that these triangles are scaled copies.
- They have the same 3 angle measures.
- They have different side lengths (can be checked with tracing paper or a ruler).
The goal is to make sure students understand that the second set has 3 different triangles (because they are different sizes) and that the first set really shows only 1 triangle in many different orientations. Tracing paper may be helpful to convince students of this.
Anticipated Misconceptions
Some students may say that all the triangles in the second set are “the same shape.” This statement can result from two very different misconceptions. Listen to the students’ reasoning and explain as needed:
- Just because they are all in the same category, “triangles,” doesn’t mean they are all the same shape. If we can take two shapes and position one exactly on top of the other, so all the sides and corners line up, then they are identical copies.
- These triangles are scaled copies of each other, but that does not make them “the same” because their side lengths are still different. Only scaled copies made using a scale factor of 1 are identical copies.
Standards
- 7.G.2·Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
- 7.G.A.2·Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.