# Discovering the Area Formula for Triangles

Lesson 1 of 4

6th grade

60–70 minutes

**Description**

Students develop the area of triangles formula using the area of rectangles and by comparing triangles with equal bases and heights.

**Materials**

### Introduce

As an introductory activity, have students draw at least one square and at least one **rectangle** on grid paper. Have students determine the **area** of each shape, using multiple strategies and justifying their reasoning. Be sure to allow time for student to share strategies with the class. If necessary, reinforce the **area** formula for **rectangles**: A = b × h.

### Explore

Divide students into partners or small groups. Distribute the Squares and Rectangles Activity Sheet and the Squares and Rectangles Recording Sheet (download from Materials section above). Have each group measure and record the dimensions and **area** of each **quadrilateral**.

Using rulers, have students draw a **diagonal** in shapes A, B, and C, and cut each shape along the **diagonal**. Have students estimate the **area** of each **triangle** formed by dividing shapes A, B, and C in half along the **diagonal**.

Students can estimate the **areas** using any methods they choose. Discuss strategies with the class as a whole. Teachers may want to create a chart or visual display to record student strategies. Highlight student strategies which are focused on developing precision.

Next, have students find the **area** of shape D. As instructed on the Activity Sheet, students should mark a point on the top edge of shape D (not choosing a vertex as a possible point). Each member of the group should pick a different point on the top of the shape. Using a straightedge, each student should draw a line from each bottom** vertex** to the point marked and cut out the three **triangles** formed. As with shapes A, B, and C, students estimate the **area** of the largest triangle using various strategies.

Have each student in the group share the point that he or she chose, along with their estimates. If needed, help students manipulate the two smaller **triangles** to cover the largest **triangle**. Students should also recognize that although the shape of the largest **triangle** may be different than other students' **triangles**, the **base**, **height**, and **area** are the same.

### Synthesize

In their partners or small groups, have students create a formula for determining the **area** of a **triangle**. Have them explain their reasoning and justify that their formula works. To promote discussion, you may need to ask questions about the relationship between the **area** of a **triangle** and the **area** of a **rectangle**. Be sure to include a discussion about the **base** and **height** of the **triangle** and **rectangle**. These ideas will help students discover the formula for **area **of **triangles** on their own.

### Assessment

Distribute the Unknown Triangle Activity Sheet (download from Materials section above), in which the dimensions of two triangles are known, and ask students to determine the area of both triangles. Allow them to use any method they like, but encourage them to use what they just discovered to find the area.

Students will likely realize that the first shape is a right triangle, so it is congruent to one-half a rectangle that was divided along the diagonal. However, they may have more difficulty realizing that the area of the second triangle is equal to one-half the area of a 3 × 4 rectangle. Students should be able to demonstrate their understanding by justifying their work.

### Extension

**Activity 1** (Technology Option)

Use the **triangle** tab in the Area Tool. When students share their points and estimates, they should realize that, although the shape of the **triangle** may change, the **base**, **height**, and **area** do not. To emphasize this point, ask them to drag point D so that it lies directly on top of point B; this forms a right **triangle** with the right angle at A, as shown above. Then, ask students to drag point D so that it lies directly above point C; this forms a right **triangle **with the right angle at C. Students should recognize that these **triangles** are congruent, so they must have the same **area**.

Discuss the results with the class. Students should realize that, in each case, the **area** of the **triangle** is equal to one-half the **area** of the **rectangle.** (At this point, you may be tempted to give students the formula A = ½(bh), but it will be more valuable to let them formalize the rule on their own in the ensuing discussion.)

**Activity 2**

The Bermuda Triangle is the triangular region defined by San Juan, Puerto Rico; Miami, Florida; and Bermuda. Using a map, students should determine the dimensions of the Bermuda Triangle, measure the distances using the scale on the map, and calculate the area of the triangle. Students can use the Bermuda Triangle Activity Sheet (download from Materials section) to record their work.

### Teacher Reflection

- What strategies did students use to find the
**areas** of the **triangles**? How did you sequence student’s strategies to promote conceptual understanding and student discourse?
- What were some of the ways that the students illustrated they were actively engaged in the learning process?
- How did activities build towards student understanding of determining a general rule or formula for
**area** of **triangles**?
- What scaffolding and/or differentiation was needed during the lesson to ensure all students experienced success?
- What were common misconceptions that students had and how did you address them?

Leave your thoughts in the comments below.