Materials: For teachers: whiteboard and markers; per small group: whiteboard or chart paper, markers, ruler, paper
1. Introduction (5 minutes, small groups)
Divide the class into small groups of two to three students. Distribute whiteboards and markers. Ask students in their groups to list the properties of the following polygons and draw a diagram identifying how they are related: quadrilateral, square, rectangle, rhombus, and parallelogram.
2. Properties of Two-Dimensional Shapes (10 minutes, whole group)
Ask students, What are the properties of a square, rectangle, rhombus, and parallelogram? Have students share their answers. Discuss how all of the shapes are quadrilaterals; how a rectangle, rhombus, and square have the properties of a parallelogram; and how the square has the properties of both a rhombus and rectangle.
Answers should include:
●Quadrilateral: four-sided closed figure
●Parallelogram: quadrilateral, opposite sides are parallel and congruent, opposite angles are congruent
●Rectangle: quadrilateral, parallelogram, four right angles, opposite sides are parallel and congruent, diagonals drawn from opposite corners bisect each other and are congruent
●Rhombus: quadrilateral, parallelogram, opposite angles are congruent, four congruent sides, diagonals are perpendicular
●Square: quadrilateral, parallelogram, rhombus, four right angles, four congruent sides (has all of the properties of a rectangle and rhombus)
Ask students, How are any of these shapes related to a triangle? (Answer: If we divide any of the quadrilaterals in half by drawing a line from one angle to the opposite angle, we get two triangles. In addition, a parallelogram, rectangle, rhombus, and square, divided in half, are two equal triangles. A triangle is a figure with three sides and three angles.) How are they different? (A triangle does not have parallel sides but can have congruent sides.)
Ask students to draw a parallelogram and identify the triangles within the shape. (Demonstrate by drawing two parallelograms on the board. In one, bisect it in half by drawing a line from one angle to the opposite angle. In the other, draw lines within the parallelogram to identify two triangles and a rectangle.)
Ask, What is the area of a square or rectangle? (Answer: base x height) Discuss with students how area is measured in square units, identifying both the base and the height in a diagram you draw on the board. Draw a rectangle that has a height of 4 square units and a base of 12 square units. Then, multiply to get an answer of 48 square units.
3. Rule for the Area of a Triangle (5 minutes, small groups)
Draw a diagonal line segment through the rectangle from one vertex to the opposite side so that you bisect the shape into two triangles. Direct student attention to one of the triangles and ask them, in their groups, to discuss what strategies they can come up with to find the area of one of the triangles. Then, write down a rule that supports their thinking. (Answers will vary; however, students should notice that the triangle is half the size of the rectangle.)
Ask students to share their rules, recording them on the board or on chart paper.
4. Area of a Triangle Video (10 minutes, whole group)
Explain, We are going to watch a video that tells us how to find the area of a triangle. Show students the video.
Review the rules that students recorded and discuss, comparing the rules that students developed with the way the area of a triangle was calculated in the video.
5. Areas of Triangles Activity (15 minutes, small groups)
Ask students, What are examples of triangles in our classroom and school? (Answers will vary but may include items in the classroom, spaces in the actual building, and structures on the playground.)
Distribute rulers. Have students in their small groups walk around the classroom (and hallway or another area of the school, such as a lunchroom) and find at least three different triangles to measure and then calculate the areas of.
6. Conclusion (5 minutes, whole group)
Ask students, How does the area of a triangle relate to that of a quadrilateral? (Answer: The area of a triangle is one half of the area of a parallelogram.)
Explain to students that in the video, Julia explained that measuring area is important in many different professions, such as architecture, engineering, transportation, and construction. Ask students, Why do you think it is important? (Answers will vary, but should state that measurement is involved in designing various spaces. Calculating area is important for determining the costs of construction supplies. You may wish to show students a map of the United States and ask them how they would find the areas of various states, knowing their dimensions.)