8.1: Composing Parallelograms (10 minutes)
This warm-up has two aims: to solidify what students learned about the relationship between triangles and parallelograms and to connect their new insights back to the concept of area.
Students are given a right triangle and the three parallelograms that can be composed from two copies of the triangle. Though students are not asked to find the area of the triangle, they may make some important observations along the way. They are likely to see that:
- The triangle covers half of the region of each parallelogram.
- The base-height measurements for each parallelogram involve the numbers 6 and 4, which are the lengths of two sides of the triangle.
- All parallelograms have the same area of 24 square units.
These observations enable them to reason that the area of the triangle is half of the area of a parallelogram (in this case, any of the three parallelograms can be used to find the area of the triangle). In upcoming work, students will test and extend this awareness, generalizing it to help them find the area of any triangle.
Display the images of the triangle and the three parallelograms for all to see. Give students a minute to observe them. Ask them to be ready to share at least one thing they notice and one thing they wonder. Give students a minute to share their observations and questions with a partner.
Give students 2–3 minutes of quiet time to complete the activity, and provide access to their geometry toolkits. Follow with a whole-class discussion.
Here is Triangle M.
Han made a copy of Triangle M and composed three different parallelograms using the original M and the copy, as shown here.
For each parallelogram Han composed, identify a base and a corresponding height, and write the measurements on the drawing.
- Find the area of each parallelogram Han composed. Show your reasoning.
When identifying bases and heights of the parallelograms, some students may choose a non-horizontal or non-vertical side as a base and struggle to find its length and the length of its corresponding height. Ask them to see if there's another side that could serve as a base and has a length that can be more easily determined.
Ask one student to identify the base, height, and area of each parallelogram, as well as how they reasoned about the area. If not already brought up by students in their explanations, discuss the following questions:
- “Why do all parallelograms have the same area even though they all have different shapes?” (They are composed of the same parts—two copies of the same right triangles. They have the same pair of numbers for their base and height. They all call be decomposed and rearranged into a 6-by-4 rectangle.)
- “What do you notice about the bases and heights of the parallelograms?” (They are the same pair of numbers.)
- “How are the base-height measurements related to the right triangle?” (They are the lengths of two sides of the right triangles.)
- “Can we find the area of the triangle? How?” (Yes, the area of the triangle is 12 square units because it is half of the area of every parallelogram, which is 24 square units.)
8.2: More Triangles (25 minutes)
In this activity, students apply what they have learned to find the area of various triangles. They use reasoning strategies and tools that make sense to them. Students are not expected to use a formal procedure or to make a general argument. They will think about general arguments in an upcoming lesson.
Here are some anticipated paths students may take, from more elaborate to more direct. Also monitor for other approaches.
- Draw two smaller rectangles that decompose the given triangle into two right triangles. Find the area of each rectangle and take half of its area. Add the areas of the two right triangles. (This is likely used for B and D.)
For Triangle C, some students may choose to draw two rectangles around and on the triangle (as shown here), find half of the area of each rectangle, and subtract one area from the other.
- Enclose the triangle with one rectangle, find the area of the rectangle, and take half of that area. (This is likely used for right triangle A.)
- Duplicate the triangle to form a parallelogram, find the area of the parallelogram, and take half of its area. (Likely used with any triangle.)
Monitor the different strategies students use. Consider asking each student that uses a unique strategy to create a visual display of their work and to share it with the class later.
Tell students that they will now apply their observations from the past few activities to find the area of several triangles. Arrange students in groups of 2–3. Give students 6–8 minutes of quiet work time and a few more minutes to discuss their work with a partner. Ask them to confer with their group only after each person has attempted to find the area of at least two triangles. Provide access to their geometry toolkits (especially tracing paper).
Supports accessibility for: Visual-spatial processing; Conceptual processing
Design Principle(s): Support sense-making; Maximize meta-awareness
Find the areas of at least two of these triangles. Show your reasoning.
At this point students should not be counting squares to determine area. If students are still using this approach, steer them in the direction of recently learned strategies (decomposing, rearranging, enclosing, or duplicating).
Students may not recognize that the vertical side of Triangle D could be the base and try to measure the lengths the other sides. If so, remind them that any side of a triangle can be the base.
Though students may have conferred with one or more partners during the task, take a few minutes to come together as a class so that everyone has a chance to see a wider range of approaches.
Select previously identified students to explain their approach and display their reasoning for all to see. Start with the most-elaborate strategy (most likely a strategy that involves enclosing a triangle), and move toward the most direct (most likely duplicating the triangle to compose a parallelogram). After each student presents, ask the class:
- “Did anyone else reason the same way?”
- “Did anyone else draw the same diagram but think about the problem differently?”
- “Can this strategy be used on another triangle in this set? Which one?”
- “Is there a triangle for which this strategy would not be helpful? Which one, and why not?”
8.3: Decomposing a Parallelogram (25 minutes)
By now students have more than one path for finding the area of a triangle. This optional activity offers one more lens for thinking about the relationship between triangles and parallelograms. Previously, students duplicated triangles to compose parallelograms. Here they see that a different set of parallelograms can be created from a triangle, not by duplicating it, but by decomposing it.
Students are assigned a parallelogram to be cut into two congruent triangles. They take one triangle and decompose it into smaller pieces by cutting along a line that goes through the midpoints of two sides. They then use these pieces to compose a new parallelogram (two parallelograms are possible) and find its area.
Students notice that the height of this new parallelogram is half of the original parallelogram, and the area is also half of that of the original parallelogram. Because the new parallelogram is composed of the same parts as a large triangle, the area of triangle is also half of that of the original parallelogram. This reasoning paves another way to understand the formula for the area of triangles.
Of the four given parallelograms, Parallelogram B is likely the most manageable for students. When decomposed, its pieces (each with a right angle) resemble those seen in earlier work on parallelograms. Consider this as you assign parallelograms to students.
Tell students that they will investigate another way in which triangles and parallelograms are related. Arrange students in groups of 2–4. Assign a different parallelogram from the blackline master to each student in the group. Give each student two copies of the parallelogram and access to a pair of scissors and some tape or glue.
Each parallelogram shows some measurements and dotted lines for cutting. For the first question, students who have parallelograms C and D should not cut off the measurements shown outside of the figures.
Give students 10 minutes to complete the activity, followed by a few minutes to discuss their work (especially the last three questions). Ask students who finish early to find someone with the same original parallelogram and compare their work.
Supports accessibility for: Organization; Attention
Your teacher will give you two copies of a parallelogram. Glue or tape one copy of your parallelogram here and find its area. Show your reasoning.
Decompose the second copy of your parallelogram by cutting along the dotted lines. Take only the small triangle and the trapezoid, and rearrange these two pieces into a different parallelogram. Glue or tape the newly composed parallelogram on your paper.
- Find the area of the new parallelogram you composed. Show your reasoning.
- What do you notice about the relationship between the area of this new parallelogram and the original one?
- How do you think the area of the large triangle compares to that of the new parallelogram: Is it larger, the same, or smaller? Why is that?
Glue or tape the remaining large triangle to your paper. Use any part of your work to help you find its area. Show your reasoning.
Are you ready for more?
Can you decompose this triangle and rearrange its parts to form a rectangle? Describe how it could be done.
Students may struggle to form a new parallelogram because the two composing pieces are not both facing up (i.e. either the triangle or the trapezoid is facing down). Tell them that the shaded side of the cut-outs should face up.
Students may struggle to use the appropriate measurements needed to find the area of the parallelogram in the first question. They may multiply more numbers than necessary because the measurements are given. If this happens, remind them that only two measurements (base and height) are needed to determine the area of a parallelogram.
For each parallelogram, invite a student to share with the class their new parallelogram (their answer to the second question). Then, ask if anyone who started with the same original figure created a different, new parallelogram. If a student created a different one, ask them to share it to the class. After all four parallelograms have been presented, discuss:
- “How many possible parallelograms can be created from each set of trapezoid and triangle?”
- “Do they all yield the same area? Why or why not?”
- “How does the area of the new parallelogram relate to the area of the original parallelogram?” (It is half the area of the original.) Why do you think that is?” (The new parallelogram is decomposed and rearranged from a triangle that is half of the original parallelogram. The original and new parallelograms have one side in common (they have the same length), but the height of the new parallelogram is half of that of the original.)
- “Can the area of the large triangle be determined? How?” (Yes. It has the same area as the new parallelogram because it is composed of the same pieces.)
If not already observed by students, point out that, just as in earlier investigations, we see that:
- The area of a triangle is half of that of a related parallelogram that share the same base.
- The triangle and the related parallelogram have at least one side in common.
Design Principle(s): Optimize meta-awareness; Support sense-making
In this lesson, we practiced using what we know about parallelograms to reason about areas of triangles. We duplicated a triangle to make a parallelogram, decomposed and rearranged a triangle into a parallelogram, or enclosed a triangle with one or more rectangles.
“What can we say about the area of a triangle and that of a parallelogram with the same height?” (The area of the triangle is half of the area of the related parallelogram.)
“In the second activity, we cut a triangle along a line that goes through the midpoints of two sides and rearranged the pieces into a parallelogram. What did we notice about the area and the height of the resulting parallelogram?” (It has the same area as the original triangle but half its height.)
“How might we start finding the area of any triangle, in general?” (Start by finding the area of a related parallelogram whose base is also a side of the triangle.)
8.4: Cool-down - An Area of 14 (5 minutes)
Student Lesson Summary
We can reason about the area of a triangle by using what we know about parallelograms. Here are three general ways to do this:
Make a copy of the triangle and join the original and the copy along an edge to create a parallelogram. Because the two triangles have the same area, one copy of the triangle has half the area of that parallelogram.
The area of Parallelogram B is 16 square units because the base is 8 units and the height 2 units. The area of Triangle A is half of that, which is 8 square units. The area of Parallelogram D is 24 square units because the base is 4 units and the height 6 units. The area of Triangle C is half of that, which is 12 square units.
Decompose the triangle into smaller pieces and compose them into a parallelogram.
In the new parallelogram, \(b = 6\), \(h = 2\), and \(6 \boldcdot 2 = 12\), so its area is 12 square units. Because the original triangle and the parallelogram are composed of the same parts, the area of the original triangle is also 12 square units.
Draw a rectangle around the triangle. Sometimes the triangle has half of the area of the rectangle.
The large rectangle can be decomposed into smaller rectangles. The one on the left has area \(4 \boldcdot 3\) or 12 square units; the one on the right has area \(2 \boldcdot 3\) or 6 square units. The large triangle is also decomposed into two right triangles. Each of the right triangles is half of a smaller rectangle, so their areas are 6 square units and 3 square units. The large triangle has area 9 square units.
Sometimes, the triangle is half of what is left of the rectangle after removing two copies of the smaller right triangles.
The right triangles being removed can be composed into a small rectangle with area \((2 \boldcdot 3)\) square units. What is left is a parallelogram with area \(5 \boldcdot 3 - 2 \boldcdot 3\), which equals \(15-6\) or 9 square units. Notice that we can compose the same parallelogram with two copies of the original triangle! The original triangle is half of the parallelogram, so its area is \( \frac12 \boldcdot 9\) or 4.5 square units.