Discovering the Area Formula for Parallelograms Lesson 2
Students use prior knowledge of the area formula for rectangles and triangles to discover the formula for the area of parallelograms.
In partners or small groups, provide students with Pattern Blocks Activity Sheet (download from Materials section above) or pattern blocks of these shapes: square, rhombus, parallelogram, and rectangle. Teachers could also use Shape Tool for students to create the shapes on their own devices.
Have students compare the attributes of these shapes.
To begin the lesson, have students look at a U.S. map. Ask students, "What state is in the approximate shape of a parallelogram?" Although not exactly, Tennessee is roughly a parallelogram. Review the area formula for rectangles, A = bh. Working in groups of three, students should discuss strategies that could be used to determine the area of Tennessee.
Distribute the Rectangles and Parallelograms Activity Sheets. Give students time to determine the area of shapes A-E.
For each rectangle, students can use multiple methods for finding area. Some methods can include counting the number of squares, multiplying the length by the width, or cutting shapes into multiple pieces and rearranging the pieces. For each parallelogram, students may need to count the squares to determine the area; they will need to combine partial squares to form full squares when making their estimates. Students should share their strategies with each other or as a whole class.
In their partners or small groups, have students create a formula for determining the area of a parallelogram. 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 parallelogram and the area of rectangle. Be sure to include a discussion about the base and height of the parallelogram and rectangle. These ideas will help students discover the formula for area of parallelogram on their own.
At the end of the lesson, return to the motivating problem: Students should now use measurements from the map to determine the height and base, and then they should use the formula they've discovered to find the area.
Activity 1 (Technology Option)
Shape Tool using a tablet or computer. 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.)
Students can create parallelograms by giving coordinates on the coordinate plane; another student can draw the parallelogram and determine its area. Students can use the distance formula to calculate the base and height if the parallelogram is not in a typical orientation.
Students can create parallelograms of varying sizes using geoboards. In pairs, one student can create a parallelogram, and the other student can determine its area using the area formula. To verify the result, students can use Pick's theorem (requires use of a geoboard or grid paper): I + (½P) - 1, where I is the number of points in the interior of the polygon and P is the number of points on the perimeter of the polygon.
Note that an online geoboard is also available to make this technology-based.
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Students develop the area of triangles formula using the area of rectangles and by comparing triangles with equal bases and heights.
Students explore several strategies for calculating the area of a trapezoid while discovering the area formula for trapezoids.
Students will estimate the area of irregular shapes and use a process of decomposition to calculate the areas of irregular polygons.
CCSS, Content Standards to specific grade/standard
CCSS, Standards for Mathematical Practices
PtA, highlighted Effective Teaching Practice and/or Guiding Principle CCSS