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Activity Synthesis

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.)

9.2

Activity

Standards Alignment

Building On

Addressing

Instructional Routines

MLR8: Discussion Supports

Materials

To Gather

Geometry toolkits

Activity Narrative

In this activity, students generalize a formula for the area of triangles. They build on their observations about the relationship between the area of a triangle and the area of a parallelogram with the same base and height.

Students first find the areas of several triangles given base and height measurements. They notice regularity in repeated reasoning and arrive at an expression for finding the area of any triangle (MP8). Students might write , , or another equivalent expression.

At the end of the activity, consider giving students a chance to reason more abstractly and think about why the expression would hold true for all triangles. See the activity synthesis for prompts and diagrams that support such reasoning.

Launch

Arrange students in groups of 2–3. Explain that they will now find the area of some triangles using what they know about base-height pairs in triangles and the relationships between triangles and parallelograms.

Give students 5 minutes to complete the activity. Provide access to geometry toolkits, especially tracing paper. Ask them to find the area of at least two triangles independently before discussing with their partner(s).

If needed, remind students how they reasoned about the area of triangles in the previous lesson (such as by composing a parallelogram, enclosing a triangle with one or more rectangles, and so on.). Encourage them to refer to their previous work and to use tracing paper as needed.

Activity

None

For each triangle:

Identify a base and a corresponding height, and record their lengths in the table.
Find the area of the triangle and record it in the last column of the table.

Four triangles labeled A--D on a grid.

triangle	base (units)	height (units)	area (square units)
A
B
C
D
any triangle

In the last row, write an expression for the area of any triangle, using and .

Activity Synthesis

Select a few students to share their expression for finding the area of any triangle. Record each expression for all to see.

To give students a chance to reason logically and deductively about their expression, ask, “Can you explain why this expression is true for any triangle?”

Display the following diagrams for all to see. Give students a minute to observe the diagrams. Ask them to choose one that makes sense to them and use that diagram to explain or show in writing that the expression works for finding the area of any triangle. (Consider giving each student an index card or a sheet of paper on which to write their reasoning so that their responses could be collected, if desired.)

<p>3 triangles on a grid</p>

When dealing only with the variables and and no numbers, students are likely to find Jada's and Lin's diagrams more intuitive to explain. Those choosing to use Elena's diagram are likely to suggest moving one of the extra triangles and joining it with the other to form a non-rectangular parallelogram with an area of . Expect students to be less comfortable reasoning in abstract terms than in concrete terms. Prepare to support them in piecing together a logical argument using only variables.

If time permits, select students who used different diagrams to share their explanation, starting with the most commonly used diagram (most likely Jada's). Ask other students to support, refine, or disagree with their arguments. If time is limited, consider collecting students' written responses now and discussing them in an upcoming lesson.

MLR8 Discussion Supports. Provide students with the opportunity to rehearse with a partner their explanation for why the expression works for finding the area of any triangle before they share with the whole class or explain it in writing.
Advances: Speaking. Writing

Representation: Develop Language and Symbols. Maintain a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of finding the area of a triangle. Terms may include: opposite vertex, base (of a triangle), height (of a triangle), and formula for finding area.
Supports accessibility for: Conceptual Processing, Language

9.3

Activity

Standards Alignment

Building On

Addressing

Instructional Routines

None

Materials

None

Activity Narrative

In this activity, students apply the expression they previously generated to find the areas of various triangles. Each diagram is labeled with two or three measurements. Before calculating, students think about which lengths can be used to find the area of each triangle.

As students work, monitor for students who choose different bases for Triangles B and D. Later, invite them to contribute to the discussion about finding the areas of right triangles.

Launch

Explain to students that they will now practice using their expression to find the area of triangles without a grid. For each triangle, ask students to be prepared to explain which measurement they chose for the base and which one for the corresponding height and why.

Keep students in groups of 2–4. Give students 3 minutes of quiet think time and 1 minute to discuss their responses with their group.

Activity

None

For each triangle, circle a base measurement that you can use to find the area of the triangle. Then, find the area of any three triangles. Show your reasoning.

A triangle labeled A. — A

A triangle labeled B. — B

A triangle labeled C. — C

A triangle labeled D. — D

A triangle labeled E. — E

Student Response

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Building on Student Thinking

The extra measurement in Triangles C, D, and E may confuse some students. If they are unsure how to decide which measurement to use, ask what they learned must be true about a base and a corresponding height in a triangle. Urge them to review the work from the Warm-up activity.

Some students may omit the step of dividing by two when finding the area of a triangle. Remind them of the idea that a triangle takes up half as much space as does a parallelogram with the same base and height, post images for reference, or point out the importance of the in the formula for the area of a triangle.

Activity Synthesis

The aim of this whole-class discussion is to deepen students’ awareness of the base and height of triangles. Discuss questions such as:

“For Triangle A, can we say that the 6-cm segment is the base and the 5-cm segment is the height? Why or why not?” (No, the base of a triangle is one of its sides.)
“For Triangle C, can the 3.5-cm segment serve as the base? Why or why not?” (Yes, it is a side of the triangle.) “What about the 3-cm segment?” (No, that segment is not a side of the triangle.)
“More than two measurements are given for Triangles C, D, and E. Which ones are helpful for finding area?” (We need a base and a corresponding height, which means the length of one side of the triangle and the length of a perpendicular segment between that side and the opposite vertex.)

Student Lesson Summary

We can choose any of the three sides of a triangle to call the base. The term “base” refers to both the side and its length (the measurement).
The corresponding height is the length of a perpendicular segment from the base to the vertex opposite it. The opposite vertex is the vertex that is not an endpoint of the base.

Here are three pairs of bases and heights for the same triangle. The dashed segments in the diagrams represent heights.

Three images of a triangle, each with a different side labeled “base” and an accompanying dashed line perpendicular to the base indicating the height.

A segment showing a height must be drawn at a right angle to the base, but it can be drawn in more than one place. It does not have to go through the opposite vertex, as long as it connects the base and a line that is parallel to the base and goes through the opposite vertex, as shown here.

Triangle with 3 perpendicular heights drawn.

The base-height pairs in a triangle are closely related to those in a parallelogram. Recall that two copies of a triangle can be composed into one or more parallelograms. Each parallelogram composed of the triangle and its copy shares at least one base with the triangle.

Two identical triangles, each with a copy composing the triangle into two different parallelograms.

For any base that they share, the corresponding height is also shared, as shown by the dashed segments.

We can use the base-height measurements and our knowledge of parallelograms to find the area of any triangle.

The formula for the area of a parallelogram with base and height is .
A triangle takes up half of the area of a parallelogram with the same base and height. We can therefore express the area, , of a triangle as:

The area of Triangle A is 15 square units because .
The area of Triangle B is 4.5 square units because .
The area of Triangle C is 24 square units because .

In each case, one side of the triangle is the base but neither of the other sides is the height. This is because the angle between them is not a right angle.

In right triangles, however, the two sides that are perpendicular can be a base and a height.

The area of this triangle is 18 square units whether we use 4 units or 9 units for the base.

A right triangle with legs of length 4 and 9.