In this lesson, students will use the area formula for rectangles to discover the area formula for parallelograms. As necessary, review the concept of area as well as the area formula,
A =
bh.
It may also be beneficial to review the properties of parallelograms as they relate to other quadrilaterals. You might ask students questions such as, "Is every square a parallelogram?" or "Is any parallelogram also a rhombus?" To help students answer these questions, you may want to use the diagram below; or, you may want to take some time to create the diagram with your students.
To begin the lesson, have students look at a U.S. map. Ask students, "What state is in the shape of a parallelogram?" Although not exactly, Tennessee is roughly a parallelogram. Working in groups of three, students should discuss methods that could be used to determine the area of Tennessee. (At the end of the lesson, students will return to this problem and discuss their findings with the class.)
Distribute the Rectangles and Parallelograms and the Quadrilateral Area Record activity sheets. Give students time to determine the area of shapes A-E, and have them record the information on the record sheet.
For each rectangle, students can simply count the number of squares, or they can multiply the length by the width. Alternatively, they could practice either metric or customary measurements by measuring the length and width using a ruler, and then multiplying to find the area. For each parallelogram, students will likely need to count the squares to determine the area; they will need to combine partial squares to form full squares when making their estimates.
Then, have student cut out shapes A, B, and C.
With rectangle A, have students cut from the lower left corner to a point on the top edgt that is three units in from the upper left vertex; this cut will form a 45-degree angle, which divides each of the squares through which it passes exactly in half. Then, for rectangles B and C on the activity sheet, have students remove a triangle by cutting from the lower left corner diagonally to any point along the top edge. One such cut is shown below. (It might be helpful for students to first draw a straight line with a ruler. You should also encourage them to choose a point along the top where the edge and one of the grid lines meet.) As shown below, students should then place the removed triangle at the other end of the rectangle. Encourage students to make a different cut than other members of their group.
Students should then determine the area of the resulting parallelograms and record the results on the record sheet.
Working in the other direction, have students cut out shapes D and E. From these shapes, have students remove the right triangle on either the right or left side, as shown below, and move it to the other side. Students should realize that this modification changed the parallelogram to a rectangle with the same area. As above, students should determine the area of the resulting rectangle and record the information on the record sheet.
As students complete the record sheet, verify the accuracy of their measurements and calculations. Without correct values, students will not discover that the formulas for rectangles and parallelograms are identical.
To explore other shapes, students can use the Shape Cutter tool.
Students can create either a rectangle or parallelogram, make an appropriate cut, and then rearrange the pieces. Further, students could even make other cuts to show that two non-rectangular parallelograms with the same base and height have the same area, as shown below. (If the red piece were moved to the other side, notice that a different parallelogram would be formed.)
After the explorations, discuss what happened as a class, and pose the following questions to students:
- Has the shape changed? [Yes.]
- Have the dimensions changed? [Side lengths have changed, but the base and height have not changed.]
- Has the area changed? [No.]
- Did other students in your group make a different cut? If so, did they get a different area than you did? [The area should be the same, regardless of the cut that was made.]
At the conclusion of this discussion, students should realize that the area formula for a rectangle, A = bh, is also the area formula for parallelograms.
As a final piece, students should also recognize the relationship between the area formulas for trapezoids and parallelograms. The formula for a trapezoid is A = ½(b1 + b2) × h. For a parallelogram, the bases are equal, so b1 = b2. Therefore, using the trapezoid formula to calculate the area of a parallelogram results in the following:
A = ½(b1 + b2) × h = ½(b1 + b1) × h = b1h
As in previous lessons in this unit, students can also use the Area Tool for Parallelograms to investigate the relationship of the height and the length of the base to the area of a parallelogram.
At the end of the lesson, return to the motivating problem: What is the area of the state of Tennessee? Students should now use measurements from the map to determine the height and base, and then they should use the formula to find the area.
