Bases and Heights of Parallelograms
10 min
Teacher Prep
Setup
Required Preparation
For the digital version of the activity, acquire devices that can run the applet.
In the “Doing Math” teacher section of the Math Community Chart, add 2–5 commitments you have for what your teaching practice “looks like” and “sounds like” this year.
Narrative
In this Warm-up, students compare and contrast two ways of decomposing and rearranging a parallelogram on a grid such that its area can be found. This work allows students to practice communicating their observations and prompts them to notice features of a parallelogram that are useful for finding area—a base and its corresponding height.
The flow of key ideas—to be uncovered during discussion and gradually throughout the lesson—is as follows:
- There are multiple ways to decompose a parallelogram (with one cut) and rearrange it into a rectangle whose area we can determine.
- The cut can be made in different places, but to compose a rectangle, the cut has to be at a right angle to two opposite sides of the parallelogram.
- The length of one side of this newly composed rectangle is the same as the length of one side of the parallelogram. We use the term base to refer to this side.
- The length of the other side of the rectangle is the length of the cut we made to the parallelogram. We call this segment a height that corresponds to the chosen base.
- We use these two lengths to determine the area of the rectangle, and thus also the area of the parallelogram.
As students work and discuss, identify those who recognize that both Elena and Tyler decomposed the parallelogram by making a cut that is perpendicular to one side and then rearranged the pieces into a rectangle. Ask them to share their observations later. Be sure to leave enough time to discuss the first four key ideas as a class.
In the digital version of the warm-up, students use applets to animate the moves that Elena and Tyler made (decomposing and rearranging) to find the area of the parallelogram.
Launch
Arrange students in groups of 2. Give students 2 minutes of quiet think time and access to geometry toolkits. Ask them to share their responses with a partner afterward.
Student Task
Elena and Tyler were finding the area of this parallelogram:
Here is how Elena did it:
Here is how Tyler did it:
How are the two strategies for finding the area of a parallelogram the same? How they are different?
Sample Response
Sample responses:
- Similar: They both cut off a piece from the left of the parallelogram and moved it over to the right to make a rectangle. The rectangles they made are identical.
- Different: They cut the parallelogram at different places. Elena cut a right triangle from the left side and Tyler cut off a trapezoid. The rectangles they made are not in the same place.
Activity Synthesis (Teacher Notes)
Select previously identified students to share what was the same and what was different about Elena’s and Tyler’s methods.
If not already mentioned by students, highlight the following points on how Elena’s and Tyler's approaches are the same, though do not expect students to use the language as written here. Clarify each point by gesturing, pointing, and annotating the images.
- The rectangles are identical. They have the same side lengths. (Label the side lengths of the rectangles.)
- The cuts were made in different places, but the length of the cuts was the same. (Label the lengths along the vertical cuts.)
- The horizontal sides of the parallelogram have the same length as the horizontal sides of the rectangle. (Point out how both segments have the same length.)
- The length of each cut is the distance between the two horizontal sides of the parallelogram. It is also the vertical side length of the rectangle. (Point out how that distance stays the same across the horizontal length of the parallelogram.)
Begin to connect the observations to the terms “base” and “height.” For example, explain:
- “The two measurements that we see here have special names. The length of one side of the parallelogram—which is also the length of one side of the rectangle—is called a base. The length of the vertical cut segment—which is also the length of the vertical side of the rectangle—is called a height that corresponds to that base.”
- “Here, the side of the parallelogram that is 7 units long is also called a base. In other words, the word ‘base’ is used for both the segment and the measurement.”
Tell students that we will explore bases and heights of a parallelogram in this lesson.
Math Community
After the Warm-up, display the Math Community Chart with the “doing math” actions added to the teacher section for all to see. Give students 1 minute to review. Then share 2–3 key points from the teacher section and your reasoning for adding them. For example,
- If “questioning vs. telling,” a shared reason could focus on your belief that students are capable mathematical thinkers and your desire to understand how students are making meaning of the mathematics.
- If “listening,” a shared reason could be that sometimes you want to sit quietly with a group just to listen and hear student thinking and not because you think the group needs help or is off-track.
After sharing, tell students that they will have the opportunity to suggest additions to the teacher section during the Cool-down.
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
Bases and Heights of Parallelograms
10 min
Teacher Prep
Setup
Required Preparation
For the digital version of the activity, acquire devices that can run the applet.
In the “Doing Math” teacher section of the Math Community Chart, add 2–5 commitments you have for what your teaching practice “looks like” and “sounds like” this year.
Narrative
In this Warm-up, students compare and contrast two ways of decomposing and rearranging a parallelogram on a grid such that its area can be found. This work allows students to practice communicating their observations and prompts them to notice features of a parallelogram that are useful for finding area—a base and its corresponding height.
The flow of key ideas—to be uncovered during discussion and gradually throughout the lesson—is as follows:
- There are multiple ways to decompose a parallelogram (with one cut) and rearrange it into a rectangle whose area we can determine.
- The cut can be made in different places, but to compose a rectangle, the cut has to be at a right angle to two opposite sides of the parallelogram.
- The length of one side of this newly composed rectangle is the same as the length of one side of the parallelogram. We use the term base to refer to this side.
- The length of the other side of the rectangle is the length of the cut we made to the parallelogram. We call this segment a height that corresponds to the chosen base.
- We use these two lengths to determine the area of the rectangle, and thus also the area of the parallelogram.
As students work and discuss, identify those who recognize that both Elena and Tyler decomposed the parallelogram by making a cut that is perpendicular to one side and then rearranged the pieces into a rectangle. Ask them to share their observations later. Be sure to leave enough time to discuss the first four key ideas as a class.
In the digital version of the warm-up, students use applets to animate the moves that Elena and Tyler made (decomposing and rearranging) to find the area of the parallelogram.
Launch
Arrange students in groups of 2. Give students 2 minutes of quiet think time and access to geometry toolkits. Ask them to share their responses with a partner afterward.
Student Task
Elena and Tyler were finding the area of this parallelogram:
Here is how Elena did it:
Here is how Tyler did it:
How are the two strategies for finding the area of a parallelogram the same? How they are different?
Sample Response
Sample responses:
- Similar: They both cut off a piece from the left of the parallelogram and moved it over to the right to make a rectangle. The rectangles they made are identical.
- Different: They cut the parallelogram at different places. Elena cut a right triangle from the left side and Tyler cut off a trapezoid. The rectangles they made are not in the same place.
Activity Synthesis (Teacher Notes)
Select previously identified students to share what was the same and what was different about Elena’s and Tyler’s methods.
If not already mentioned by students, highlight the following points on how Elena’s and Tyler's approaches are the same, though do not expect students to use the language as written here. Clarify each point by gesturing, pointing, and annotating the images.
- The rectangles are identical. They have the same side lengths. (Label the side lengths of the rectangles.)
- The cuts were made in different places, but the length of the cuts was the same. (Label the lengths along the vertical cuts.)
- The horizontal sides of the parallelogram have the same length as the horizontal sides of the rectangle. (Point out how both segments have the same length.)
- The length of each cut is the distance between the two horizontal sides of the parallelogram. It is also the vertical side length of the rectangle. (Point out how that distance stays the same across the horizontal length of the parallelogram.)
Begin to connect the observations to the terms “base” and “height.” For example, explain:
- “The two measurements that we see here have special names. The length of one side of the parallelogram—which is also the length of one side of the rectangle—is called a base. The length of the vertical cut segment—which is also the length of the vertical side of the rectangle—is called a height that corresponds to that base.”
- “Here, the side of the parallelogram that is 7 units long is also called a base. In other words, the word ‘base’ is used for both the segment and the measurement.”
Tell students that we will explore bases and heights of a parallelogram in this lesson.
Math Community
After the Warm-up, display the Math Community Chart with the “doing math” actions added to the teacher section for all to see. Give students 1 minute to review. Then share 2–3 key points from the teacher section and your reasoning for adding them. For example,
- If “questioning vs. telling,” a shared reason could focus on your belief that students are capable mathematical thinkers and your desire to understand how students are making meaning of the mathematics.
- If “listening,” a shared reason could be that sometimes you want to sit quietly with a group just to listen and hear student thinking and not because you think the group needs help or is off-track.
After sharing, tell students that they will have the opportunity to suggest additions to the teacher section during the Cool-down.
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.