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. Have each group measure the dimensions
of each shape on the sheet and calculate its area. If necessary, review the area formula
for rectangles: A = b × h.
Squares and Rectangles Activity Sheet
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. 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. Have each group share the point they chose along with their estimates. Eventually, 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.
Area Tool for Triangles
By having students share their points and estimates, they should realize that although the shape of the triangle
may change, the base, height, and area do not. To emphasize this point,
ask them to drag point D so that it lies directly on top of
point B; this forms a right triangle with the right angle at A, as
shown below. Then, ask students to drag point D so that it lies
directly above point C; this forms a right triangle with the right
angle at C. Students should 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.
Unknown Triangles Activity Sheet
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.
Questions for Students
1. Do two triangles with the same height have the same area? Why or why not? Give some examples.
[Two triangles with the same height have the same area only if they
also have the same base. Consider two triangles, both of which have a
height of 4 inches. If one of the triangles has a base of 3 inches, its
area is A = ½ × 3 × 4 = 6 in.2. If the other triangle has a base of 5 inches, its area is A = ½ × 5 × 4 = 10 in.2. The areas are clearly not equal.
On the other hand, the two triangles below have the same area because the base and height have equal measures.
It doesn't matter that the shapes of the triangles are different.]
2. Explain how other shapes (besides rectangles and squares) could been have used to derive the area formula for triangles.
[The formula for finding the area of a parallelogram is A = bh,
the same formula used for rectangles. By dividing a parallelogram along
the diagonal, two congruent triangles are formed, which would lead to
the same conclusion; namely, that the area formula for a triangle is A = ½bh.]
Teacher Reflection
- Did students come up with alternative methods for finding the areas
of their triangles? If so, how did you react to their explanations?
- What were some of the ways that the students illustrated that they were actively engaged in the learning process?
- How did students demonstrate an understanding of how and why we use the formula A = ½bh?
- Did you notice any positive or negative effects of asking students to check each others’ work?
- Did you find it necessary to make adjustments while teaching
the lesson? If so, what adjustments, and were these adjustments
effective?
