Prior to this lesson, students will need experience in measuring rectangles and triangles and calculating their areas.
Students will also need to be able to identify trapezoids. It may be beneficial to review the properties of trapezoids in relation to other quadrilaterals. For example, students could be given pictures of various quadrilaterals and identify which ones are trapezoids and which ones are not. Require students to explain their classifications. Be sure that students are given examples of trapezoids in various orientations. It might also be useful to present the following Venn diagram to help students understand how quadrilaterals are related to one another:
As a warm-up, allow students to estimate the areas of trapezoids. Using the Trapezoids activity sheet, students working in groups can overlay centimeter grid paper and estimate the total area in square centimeters. (If possible, provide centimeter grid paper on a transparency sheet; alternatively, copy the Centimeter Grid Paper onto thin paper.) Later in the lesson, students will use measurements and a formula to calculate the area, and they will compare their estimates with their calculations.
Students should work in groups of three for the main part of the lesson. All students are responsible for the work, but the following tasks could be assigned:
- Recorder: Records all important information on the record sheet.
- Measurer: Double checks all measurements and calculations.
- Reporter: Shares all pertinent information with the class.
These roles are important because they hold each student in the group accountable. Each student needs to be an active participant, and no one student should do all of the work.
Once students are divided into groups, display a standard trapezoid, like the one shown below, on the overhead projector or chalkboard:
The main portion of this lesson involves the derivation of an area formula for trapezoids. Prior to that, allow students some exploration time. Ask them to suggest methods for finding the area of the trapezoid shown above. As a prompt, ask, "What other shapes could you use to help you? Are there any shapes for which you already know how to find the area?" After students have had some time to discuss suggestions in their groups, ask the reporter from some groups to share their ideas.
Students may suggest several possibilities. One likely idea is to divide the trapezoid into three pieces—a rectangle and two triangles—as shown below:
Using any of the methods that they identify, have students determine the area of a trapezoid with bases of 24 cm and 10 cm and with legs of 15 cm and 13 cm. Allow all groups an opportunity to determine the area, and be sure to get class consensus on what the area is. It is important that all groups determine the area correctly. Later in the lesson, students will compare the area they find here with the area of a rectangle formed by dividing the trapezoid, and it is critical that the answers match.
Another option is to divide the trapezoid as shown below, using a segment that connects the midpoints of the legs. Although the decomposition in the figure above will allow students to calculate the area, the figure below demonstrates how any trapezoid can be transformed into a rectangle, leading students to understand the derivation of the area formula for trapezoids.
When triangles are removed from each corner and rotated, a rectangle will be formed. It’s important for kids to see that the midline is equal to the average of the bases. This is the basis for the proof—the midline is equal to the base of the newly formed rectangle, and the midline can be expressed as ½(b1 + b2), so the proof falls immediately into place. To be sure that students see this relationship, ask, "How is the midline related to the two bases?" Students might suggest that the length of the midline is "exactly between" the lengths of the two bases; more precisely, some students may indicate that it is equal to the average of the two bases, giving the necessary expression.
Remind students that the area of a rectangle is base × height; for the rectangle formed from the original trapezoid, the base is ½(b1 + b2) and the height is h, so the area of the rectangle (and, consequently, of the trapezoid) is A = ½h(b1 + b2). This is the traditional formula for finding the area of the trapezoid.
After this proof has been demonstrated, students can measure the dimensions of the rectangle and calculate its area. Ask the class, "How does the area of this new rectangle relate to the area of the trapezoid from which it was formed?" When calculating the area of the rectangle, students should get the same answer as obtained above for the area of the trapezoid. More importantly, students should realize that only the shape changed, not the area.
Allow students to work in groups to calculate the area of the trapezoids on the Trapezoids activity sheet. Circulate as students work, offering help as necessary. Also note successful approaches that students use to calculate the area of each trapezoid, and be sure to have students share those approaches during a class discussion. After ample time, gather the class together, and discuss the results.
To push the discussion further, you might want to return to the original trapezoid shape and use other decompositions to prove the area formula for a trapezoid. If your students are prepared for the algebra involved in generating such proofs, seeing more than one derivation can be very valuable. For instance, if students suggested that the trapezoid be divided into two triangles and a rectangle, then label the pieces as shown below, and discuss the labels with students:
The proof of the area formula then falls into place as follows:
- The area of the rectangle is b1h.
- The area of the triangle on the left side is ½xh.
- The area of the triangle on the right side
is ½(b2 – b1 – x) × h = ½b2h – ½b1h – ½xh.
- The combined area of the three pieces, then,
is b1h + ½xh + ½b2h – ½b1h – ½xh, which simplifies to ½b1h + ½b2h. This further simplifies to ½(b1 + b2) h, which is the standard area formula.
You may wish to explore the proof for special trapezoids, such as right and isosceles trapezoids. Although the proofs will be similar, students may benefit from multiple opportunities to verify the formula.
Students can also use the Area Tool for Trapezoids to investigate the relationship of the height and the length of the bases to the area.
Return to the warm-up activity during which students estimated the areas of the trapezoids on the Trapezoids activity sheet. Have students compare their estimates with the calculated areas. How close did they come? Were students able to predict the area within one square centimeter?
To conclude the lesson and allow for practice, rearrange students into pairs. Each student should create a trapezoid, measure its dimensions, and determine its area. They should then swap trapezoids, measure and calculate the area, and discuss the results.
