Formula for the Area of a Triangle
10 min
Teacher Prep
Setup
Narrative
In this activity, students think about the meaning of base and height in a triangle by studying examples and non-examples. Then, they examine some statements about bases and heights and determine if the statements are true. The goal is for students to see that in a triangle:
- Any side can be a base.
- A segment that represents a height must be drawn at a right angle to the base, but can be drawn in more than one place. The length of this perpendicular segment is the distance between the base and the vertex opposite it.
- A triangle can have three possible bases, each with a corresponding height.
Monitor for students who notice similarities between the bases and heights in a triangle and those in a parallelogram. Ask them to share their observations later.
As students justify how they know whether the given statements are true and consider others’ justifications, they practice constructing logical reasoning and critiquing the reasoning of others (MP3).
Students will have many opportunities to make sense of bases and heights in this lesson and an upcoming one, so they do not need to know how to draw a height correctly at this point.
Launch
Remind students that recently they looked at bases and heights of parallelograms. Tell students that they will now examine bases and heights of triangles.
Display the examples and non-examples of bases and heights for all to see. Read aloud the first paragraph of the activity and the description of each set of images. Give students a minute to observe the images. Then tell students to use the examples and non-examples to determine what is true about bases and heights in a triangle.
Arrange students in groups of 2. Give them 2–3 minutes of quiet think time and then a minute to discuss their responses with a partner.
Student Task
Here are six copies of a triangle. In each copy, one side is labeled base.
In the first three drawings, the dashed segments represent heights of the triangle.
In the next three drawings, the dashed segments do not represent heights of the triangle.
Select all the statements that are true about bases and heights in a triangle.
- Any side of a triangle can be a base.
- There is only one possible height.
- A height is always one of the sides of a triangle.
- A height that corresponds to a base must be drawn at a right angle to the base.
- Once we choose a base, there is only one segment that represents the corresponding height.
- A segment representing a height must go through a vertex.
Sample Response
Only statements 1 and 4 are true.
Activity Synthesis (Teacher Notes)
For each statement, ask students to indicate whether they think it is true. For each statement, ask one or two students to explain how they know. Encourage students to use the examples and counterexamples to support their argument. (Ror instance, “The last statement is not true because the examples show dashed segments or heights that do not go through a vertex.”) Make sure that students agree about each statement before moving on. Display the true statements for all to see.
Students should see that only Statements 1 and 4 are true—that any side of a triangle can be a base, and a segment for the corresponding height must be drawn at a right angle to the base. What is missing—an important gap to fill during discussion—is the length of any segment representing a height.
Ask students, “How long should a segment that shows a height be? If we draw a perpendicular line from the base, where do we stop?” Solicit some ideas from students. Then, highlight the following:
- The length of each perpendicular segment is the shortest distance between the base and its opposite vertex. The opposite vertex is the vertex that is not an endpoint of the base. (Point out the opposite vertex for each base.)
- The segment representing height does not have to be drawn through the vertex (although that would be a natural place to draw it). It does need to maintain that distance between the base and the opposite vertex.
If any students noticed connections between the bases and heights in a triangle with those in a related parallelogram, invite them to share their observations. Otherwise, draw students’ attention to it. Consider duplicating a triangle that shows a base and a height (by tracing on patty paper or creating a paper cutout). Use the original and the copy to compose a parallelogram. Ask students: “Suppose we choose the same side as the base of both the parallelogram and the triangle. What do you notice about the height of each shape?” (The two shapes have the same height.)
Anticipated Misconceptions
If students are unsure how to interpret the diagrams, ask them to point out parts of the diagrams that might be unclear. Clarify as needed.
Students may not remember from their experience with parallelograms that a height needs to be perpendicular to a base. Consider posting a diagram of a parallelogram—with its base and height labeled—in a visible place in the room so that it can serve as a reference.
Standards
- 6.G.1·Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
- 6.G.A.1·Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
15 min
10 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Formula for the Area of a Triangle
10 min
Teacher Prep
Setup
Narrative
In this activity, students think about the meaning of base and height in a triangle by studying examples and non-examples. Then, they examine some statements about bases and heights and determine if the statements are true. The goal is for students to see that in a triangle:
- Any side can be a base.
- A segment that represents a height must be drawn at a right angle to the base, but can be drawn in more than one place. The length of this perpendicular segment is the distance between the base and the vertex opposite it.
- A triangle can have three possible bases, each with a corresponding height.
Monitor for students who notice similarities between the bases and heights in a triangle and those in a parallelogram. Ask them to share their observations later.
As students justify how they know whether the given statements are true and consider others’ justifications, they practice constructing logical reasoning and critiquing the reasoning of others (MP3).
Students will have many opportunities to make sense of bases and heights in this lesson and an upcoming one, so they do not need to know how to draw a height correctly at this point.
Launch
Remind students that recently they looked at bases and heights of parallelograms. Tell students that they will now examine bases and heights of triangles.
Display the examples and non-examples of bases and heights for all to see. Read aloud the first paragraph of the activity and the description of each set of images. Give students a minute to observe the images. Then tell students to use the examples and non-examples to determine what is true about bases and heights in a triangle.
Arrange students in groups of 2. Give them 2–3 minutes of quiet think time and then a minute to discuss their responses with a partner.
Student Task
Here are six copies of a triangle. In each copy, one side is labeled base.
In the first three drawings, the dashed segments represent heights of the triangle.
In the next three drawings, the dashed segments do not represent heights of the triangle.
Select all the statements that are true about bases and heights in a triangle.
- Any side of a triangle can be a base.
- There is only one possible height.
- A height is always one of the sides of a triangle.
- A height that corresponds to a base must be drawn at a right angle to the base.
- Once we choose a base, there is only one segment that represents the corresponding height.
- A segment representing a height must go through a vertex.
Sample Response
Only statements 1 and 4 are true.
Activity Synthesis (Teacher Notes)
For each statement, ask students to indicate whether they think it is true. For each statement, ask one or two students to explain how they know. Encourage students to use the examples and counterexamples to support their argument. (Ror instance, “The last statement is not true because the examples show dashed segments or heights that do not go through a vertex.”) Make sure that students agree about each statement before moving on. Display the true statements for all to see.
Students should see that only Statements 1 and 4 are true—that any side of a triangle can be a base, and a segment for the corresponding height must be drawn at a right angle to the base. What is missing—an important gap to fill during discussion—is the length of any segment representing a height.
Ask students, “How long should a segment that shows a height be? If we draw a perpendicular line from the base, where do we stop?” Solicit some ideas from students. Then, highlight the following:
- The length of each perpendicular segment is the shortest distance between the base and its opposite vertex. The opposite vertex is the vertex that is not an endpoint of the base. (Point out the opposite vertex for each base.)
- The segment representing height does not have to be drawn through the vertex (although that would be a natural place to draw it). It does need to maintain that distance between the base and the opposite vertex.
If any students noticed connections between the bases and heights in a triangle with those in a related parallelogram, invite them to share their observations. Otherwise, draw students’ attention to it. Consider duplicating a triangle that shows a base and a height (by tracing on patty paper or creating a paper cutout). Use the original and the copy to compose a parallelogram. Ask students: “Suppose we choose the same side as the base of both the parallelogram and the triangle. What do you notice about the height of each shape?” (The two shapes have the same height.)
Anticipated Misconceptions
If students are unsure how to interpret the diagrams, ask them to point out parts of the diagrams that might be unclear. Clarify as needed.
Students may not remember from their experience with parallelograms that a height needs to be perpendicular to a base. Consider posting a diagram of a parallelogram—with its base and height labeled—in a visible place in the room so that it can serve as a reference.
Standards
- 6.G.1·Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
- 6.G.A.1·Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.