Lesson 8

In this activity, students think about the meaning of base and height in a triangle by studying examples and non-examples. The goal is for them 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 and three corresponding heights.

Students may draw on their experience with bases and heights in a parallelogram and observe similarities. Encourage this, as it would help them conceptualize base-height pairs in triangles.

As students discuss with their partners, listen for how they justify their decisions or how they know which statements are true (MP3).

Display the examples and non-examples of bases and heights 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 the class a minute to share some of their observations and questions. 

Tell students they will now 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 to share their response with a partner.

Some students may struggle to interpret the diagrams. Ask them to point out parts of the diagrams that might be unclear and 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.

For each statement, ask students to indicate whether they think it is true. For each “true” vote, ask one or two students to explain how they know (MP3). Do the same for each “false” vote. Encourage students to use the examples and counterexamples to support their argument (e.g., "The last statement is not true because the examples show dashed segments or heights that do not go through a vertex.") which means more than one height." Agree on the truth value of each statement before moving on. Record and display the true statements for all to see.

Students should see that only statements 1 and 5 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.

Explain that the length of each perpendicular segment is the distance between the base and the vertex opposite of it. The opposite vertex is the vertex that is not an endpoint of the base. Point out the opposite vertex for each base. Clarify that the segment does not have to be drawn through the vertex (although that would be a natural place to draw it), as long as it maintains that distance between the base and the opposite vertex.

It is helpful to connect this idea to that of heights in a parallelogram. Consider duplicating the triangle and use the original and the copy to compose a parallelogram. The height for a chosen base in the triangle is also the height of the parallelogram with the same base.

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.

This task culminates in writing a formula for the area of triangles. By now students are likely to have developed the intuition that the area of a triangle is half of that of a parallelogram with the same base and height. This activity encapsulates that work in an algebraic expression.

Students first find the areas of several triangles given base and height measurements. They then generalize the numerical work to arrive at an expression for finding the area of any triangle (MP8).

If needed, remind students how they reasoned about the area of triangles in the previous lesson (i.e. by composing a parallelogram, enclosing with one or more rectangles, etc.). Encourage them to refer to their previous work and use tracing paper as needed. Students might write \(b \boldcdot h \div 2\) or \(b \boldcdot h \boldcdot \frac12\) as the expression for the area of any triangle. Any equivalent expression should be celebrated.

At the end of the activity, consider giving students a chance to reason more abstractly and deductively, i.e., to think about why the expression \(b \boldcdot h \div 2\) would hold true for all triangles. See Activity Synthesis for prompts and diagrams that support such reasoning.

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 5–6 minutes to complete the activity and access to geometry toolkits, especially tracing paper. Ask them to find the area at least a couple of triangles independently before discussing with their partner(s).

Students may not be inclined to write an expression using the variables \(b\) and \(h\) and instead replace the variables with numbers of their choice. Ask them to reflect on what they did with the numbers for the first four triangles. Then, encourage them to write the same operations but using the letters \(b\) and \(h\) rather than numbers.

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 \(b \boldcdot h \div 2\) 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.)

When dealing only with the variables \(b\) and \(h\) 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 \(b \boldcdot h\). Expect students 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. It time is limited, consider collecting students' written responses now and discussing them in an upcoming lesson.

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, notice students who choose different bases for Triangles B and D. Invite them to contribute to the discussion about finding the areas of right triangles later.

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 choose for the base and which one for the corresponding height and why. 

Keep students in groups of 2–4. Give students 5 minutes of quiet think time, followed by 1–2 minutes for discussing their responses in their group.

The extra measurement in Triangles C, D, and E may confuse some students. If they are unsure how to decide the 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.

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.)
• What about for Triangle C:  Can the 3-cm segment serve as the base? Why or why not? (No, that segment is not a side of the triangle.)
• Can the 3.5-cm side in Triangle C serve as the base? Why or why not? (Yes, it is a side of the triangle, but because we don't have the height that corresponds to it, it is not helpful for finding the area here.)
• 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.)
• When it comes to finding area, how are right triangles—like B and D—unique? (Either of the two sides that form the right angle could be the base or the height. In non-right triangles—like A, C, and E—the height segment is not a side of the triangle; a different line segment has to be drawn.)