Bases and Heights of Triangles
10 min
Teacher Prep
Setup
Required Preparation
Narrative
In this Warm-up, students are given an area measure and are asked to create several triangles with that area. This work involves reversing the reasoning process used in previous lessons, in which students were given triangles with measurements and asked to find the area.
Students are likely to gravitate toward right triangles first (or to halve rectangles that have factors of 12 as their side lengths). This is a natural and productive starting point. Prompting students to create non-right triangles encourages them to apply their understanding of the area of non-right parallelograms.
As students work alone and discuss with partners, notice the strategies they use to draw their triangles and to verify their areas. Identify a few students with different strategies and, later, ask them to share.
In the digital version of the activity, students use an applet to draw triangles on a grid and reason about their area. The applet allows students to adjust the vertices of line segments, measure lengths, and make annotations.
Launch
Arrange students in groups of 2. Provide access to geometry toolkits. Give students 2–3 minutes of quiet think time and 2 minutes to share their drawings with their partner afterwards. Encourage students to refer to previous work as needed. If students finish their first drawing early, tell them to draw a different triangle with the same area.
During partner discussion, each partner should convince the other that the triangle drawn is indeed 12 square units.
Student Task
On the grid, draw a triangle with an area of 12 square units. Try to draw a non-right triangle. Be prepared to explain how you know the area of your triangle is 12 square units.
Sample Response
Sample responses:
-
This right triangle has a base of 8 units and a height of 3 units. The area is half of 3⋅8 or half of 24, which is 12.
-
This triangle has a side of 6 units. This can be the base. Draw a height segment that is perpendicular to the base and is 4 units long. The area of the triangle is b⋅h÷2, so it is 6⋅4÷2, which is 12.
-
Draw a parallelogram with a base of 12 and a height of 2, and then draw a diagonal line to create two identical triangles. Each of the triangles has an area of 12 because it is half of a parallelogram with an area of 24.
Activity Synthesis (Teacher Notes)
Invite a few students to share their drawings and ways of reasoning with the class. For each drawing shared, ask the creator for the base and height and record them for all to see. Ask the class:
- “Did anyone else draw an identical triangle?”
- “Did anyone draw a different triangle but with the same base and height measurements?”
To reinforce the relationship between base, height, and area, discuss:
- “Which might be a better way to draw a triangle: by starting with the base measurement or with the height? Why?”
- “Can you name other base-height pairs that would produce an area of 12 square units without drawing? How?”
Anticipated Misconceptions
If students have trouble getting started, ask:
- “Can you draw a quadrilateral with an area of 12?”
- “Can you use what you know about parallelograms to help you?”
- “Can you use any of the area strategies—decomposing, rearranging, enclosing, subtracting—to arrive at an area of 12?”
Students who start by drawing rectangles and other parallelograms may use factors of 12, instead of factors of 24, for the base and height. If this happens, ask them what the area of the their quadrilateral is and how it relates to the triangle they are trying to draw.
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.
25 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Bases and Heights of Triangles
10 min
Teacher Prep
Setup
Required Preparation
Narrative
In this Warm-up, students are given an area measure and are asked to create several triangles with that area. This work involves reversing the reasoning process used in previous lessons, in which students were given triangles with measurements and asked to find the area.
Students are likely to gravitate toward right triangles first (or to halve rectangles that have factors of 12 as their side lengths). This is a natural and productive starting point. Prompting students to create non-right triangles encourages them to apply their understanding of the area of non-right parallelograms.
As students work alone and discuss with partners, notice the strategies they use to draw their triangles and to verify their areas. Identify a few students with different strategies and, later, ask them to share.
In the digital version of the activity, students use an applet to draw triangles on a grid and reason about their area. The applet allows students to adjust the vertices of line segments, measure lengths, and make annotations.
Launch
Arrange students in groups of 2. Provide access to geometry toolkits. Give students 2–3 minutes of quiet think time and 2 minutes to share their drawings with their partner afterwards. Encourage students to refer to previous work as needed. If students finish their first drawing early, tell them to draw a different triangle with the same area.
During partner discussion, each partner should convince the other that the triangle drawn is indeed 12 square units.
Student Task
On the grid, draw a triangle with an area of 12 square units. Try to draw a non-right triangle. Be prepared to explain how you know the area of your triangle is 12 square units.
Sample Response
Sample responses:
-
This right triangle has a base of 8 units and a height of 3 units. The area is half of 3⋅8 or half of 24, which is 12.
-
This triangle has a side of 6 units. This can be the base. Draw a height segment that is perpendicular to the base and is 4 units long. The area of the triangle is b⋅h÷2, so it is 6⋅4÷2, which is 12.
-
Draw a parallelogram with a base of 12 and a height of 2, and then draw a diagonal line to create two identical triangles. Each of the triangles has an area of 12 because it is half of a parallelogram with an area of 24.
Activity Synthesis (Teacher Notes)
Invite a few students to share their drawings and ways of reasoning with the class. For each drawing shared, ask the creator for the base and height and record them for all to see. Ask the class:
- “Did anyone else draw an identical triangle?”
- “Did anyone draw a different triangle but with the same base and height measurements?”
To reinforce the relationship between base, height, and area, discuss:
- “Which might be a better way to draw a triangle: by starting with the base measurement or with the height? Why?”
- “Can you name other base-height pairs that would produce an area of 12 square units without drawing? How?”
Anticipated Misconceptions
If students have trouble getting started, ask:
- “Can you draw a quadrilateral with an area of 12?”
- “Can you use what you know about parallelograms to help you?”
- “Can you use any of the area strategies—decomposing, rearranging, enclosing, subtracting—to arrive at an area of 12?”
Students who start by drawing rectangles and other parallelograms may use factors of 12, instead of factors of 24, for the base and height. If this happens, ask them what the area of the their quadrilateral is and how it relates to the triangle they are trying to draw.
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.