Prior to this lesson, students need experience in measuring squares and rectangles that are not squares and calculating their areas. As a warm-up, have students measure at least square and one rectangle found within the classroom, record its dimensions, and calculate the area of each. For example, they could measure floor tiles, windows, chalkboards, bulletin boards, desktops, shelves, and so forth. Challenge them to measure as many different shapes as possible and report back to the class.
Divide students into groups of three. All three are responsible for the work, but the following tasks could be assigned:
- Recorder: Keeps a record of all important information.
- Measurement Verifier: Confirms all measurements and calculations.
- Reporter: Shares all pertinent information with the class.
Distribute the Squares and Rectangles activity sheet. Each member of the group should measure the dimensions of each shape on the sheet and calculate its area. Allow students to share within their groups, and check the measurements and calculations before allowing them to continue. If necessary, review the area formula for rectangles: A = b × h.
Using rulers, students should draw one diagonal in each of the shapes A, B, and C, and then cut each shape along the diagonal into two parts. In their groups, 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. One method is to simply count the number of squares, half-squares, and partial squares that are formed when the shapes are divided. Another method is to realize that each shape has an area equal to half the area of the original shape. (Students can see this by placing one half over the other.)
Discuss the results with the class as a whole.
Similarly, have students estimate the area of the largest triangle formed when shape D is divided into three parts as shown below. As with A, B, and C, students can estimate the area by counting squares, or they can fit the two smaller triangles together to form the larger triangle.
Within each group, students will likely have chosen several different points along the top of the rectangle. However, each student should notice that the area of the largest triangle is equal to half the area of the original rectangle. More importantly, students within each group should realize that this is the case, regardless of the point chosen along the top.
To allow students further opportunity to explore this idea, you may ask them to use the Area Tool for Triangles.
Students should realize that though the shape of the triangle may change, the base, height, and area do not. To emphasize the point, ask them to drag point B so that point D lies directly on top of point A; this forms a right triangle with the right angle at A, as shown below. Then, ask students to drag point B so that point D lies directly on top of point C; this forms a right triangle with the right angle at C. Students quickly recognize that these triangles are congruent, so they must have the same area.
Discuss the results with the class. Ask students how the area of each triangle relates to the area of the original shape. 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 = ½bh, but it will be more valuable to let them formalize the rule on their own, through the next activity and the ensuing discussion.)
Distribute the Unknown Triangle activity sheet, 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. As students work, circulate and ask questions to lead them to this conclusion.
Ask students to create a formula for determining the area of a triangle. Have them explain their reasoning and prove that their formula works. To prompt discussion, you may need to ask leading questions, such as, "How is the area of a triangle related to the area of a rectangle?" and "What is the formula for finding the area of a rectangle?" Be careful not to ask too many questions too soon in the discussion, as learning will be more effective if students generate the formula on their own.
