Lesson 10

So far, students have determined area given a triangle and some measurements. In this warm-up, students are invited to reverse the process. They are given an area measure and are asked to create several triangles with that area.

Expect students 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 insights from their experiences with 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 to share later.

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.

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?”

Students may be able to recognize a measurement that can be used for height when they see it, but identifying and drawing an appropriate segment is more challenging. This activity, and the demonstration needed to launch it, gives students a concrete strategy for identifying a height accurately. When students use a strategy of drawing an auxiliary line to solve problems, they are looking for and making use of structure (MP7). Explicit instruction, as in this activity, is often needed before students can be expected to use this strategy spontaneously.

Explain to students that they will try to draw a height that corresponds to each side of a triangle. Arrange students in groups of 2. Give each student an index card and 1–2 minutes to complete the first question. Remind them that there is more than one correct way to draw the corresponding height for a base. Ask them to pause after the first question. As students work, notice how students are using the index cards (if at all).

Afterwards, solicit a few quick comments on the exploration. Ask questions such as:

• “How did you know where to draw the segments?”
• “How did you draw them?”
• “Why were you given index cards? How might they help?”

Explain that you will now demonstrate a way to draw heights effectively. (If any students used the index card correctly, acknowledge that they were on the right track.)

Remind students that any line we draw to show the height of a triangle must be drawn perpendicular to the base. Having a tool with a right angle and with straight edges can help us make sure the line we draw is both straight and perpendicular to the base. This is what the index card is for. 

Ask: “How do we know where to stop this line we are drawing? How long should it be?”

Explain that the easiest way is to draw the line so it would pass through the vertex opposite of the chosen base. Draw or display a triangle for all to see. Demonstrate the following.

• Choose one side of the triangle as the base. Identify the opposite vertex.
• Line up one edge of the index card with that base.
• Slide the card along the base until a perpendicular edge of the card meets the opposite vertex.
• Use that edge to draw a line segment from that vertex to the base. The measure of that segment is the height.

Ask: What if the opposite vertex is not directly over the base? Explain that sometimes we need to extend the line of the base and demonstrate the process.

Demonstrate the process with another example in which the card needs to slide from right to left (e.g., by rotating the obtuse triangle above clockwise). Left-handed students may find this particularly helpful.

Prompt students to use this method to check the heights they drew in the first question, revise the drawings if they were incorrect, and share their revisions with their partners. Circulate and support students as they draw.  Those who finish verifying the heights in the first question can move on to complete the rest of the activity with their partners.

Some students may use the index card simply as a straightedge and therefore draw heights that are not perpendicular to the given base. Remind them that a height needs to be perpendicular or at a right angle to the base.

Students may mistakenly think that a base must be a horizontal side of a triangle (or one closest to being horizontal) and a height must be drawn inside of the triangle. Point to some examples from earlier work to remind students that neither is true. Remind them to align their index card to the side labeled "base." 

Some students may find it awkward to draw height segments when the base is not horizontal. Encourage students to rotate their paper as needed to make drawing easier.

If time permits, consider selecting one student to share the height drawing for each triangle, or display the solutions in the Student Response for all to see. To help students reflect on their work, discuss questions such as:

• “For which triangles was it easy to find the corresponding height for the given base?”
• “For which triangles was it harder?”
• “How was the process of finding the height of triangle D different from that of the others?” (The height of a right triangle is already drawn: it is the other segment framing the right angle.)
• “When might we need to extend the line of the base, or draw a height line outside of the triangle?” (When dealing with obtuse triangles, or when the opposite vertex is not directly over the base.)

This activity allows students to practice identifying the base and height of triangles and using them to find areas.

Because there are no directions on which base or height to use, and because not all sides would enable them to calculate area easily, students need to think structurally and choose strategically. All triangles in the problems have either a vertical or a horizontal side. Choosing such a side as the base makes it easier to identify the corresponding height.

In some cases, students may opt to use a combination of area-reasoning strategies rather than finding the base and height of the shaded triangles and applying the formula. For instance, they may enclose a shaded triangle with a rectangle and subtract the areas of extra triangles (with or without using the formula on those extra triangles). Notice students who use such strategies so they could share later.

Keep students in groups of 2. Explain that they will now practice locating or drawing heights and using them to find area of triangles. Give students 8–10 minutes of quiet think time and time to share their responses with a partner afterwards. Provide access to their geometry toolkits (especially index cards).

If time is limited, consider asking students find the area of two or three triangles instead of all four.

Students may think that a vertical side of a triangle is the height regardless of the segment used as the base. If this happens, have them use an index card as a straightedge to check if the two segments they are using as base and height are perpendicular.

Some students may not immediately see that choosing a side that is either vertical or horizontal would enable them to find the corresponding height very easily. They may choose a non-vertical or non-horizontal side and not take advantage of the grid. Ask if a different side might make it easier to determine the base-height lengths without having to measure.

Focus the whole-class discussion how students went about identifying bases and heights. Discuss:

• “Which side did you choose as the base for triangle B? C? Why?”
• “Aside from choosing a vertical or horizontal side as the base, is there another way to find the area of the shaded triangles without using their bases and heights?” (Invite a couple of students who use the enclose-and-subtract method to find the area of B, C, or D to share.)
• “Which strategy do you prefer or do you think is more efficient?”
• “Can you think of an example where it might be preferable to find the base and height of the triangle of interest?” (Students may point to any of the triangles in the task.)
• “Can you think of an example where it might be preferable to enclose the triangle of interest and subtract other areas?” (Students may point to the triangle shown in "Are you ready for more?", where none of the shaded triangle's sides are horizontal or vertical.)