Same Base and Height, Variation 1
Task
Make the following triangles on a geoboard. Remember that the pegs are equally spaced in a square grid. Compare the areas of the triangles.
Which has the greatest area?
Which has the least area?
Do any of the triangles have the same area?
Are some areas impossible to compare?
Explain why you answered as you did.
IM Commentary
This is the first version of a task asking students to find the areas of triangles that have the same base and height, and is the most concrete. The next two versions ask the same question but use increasingly more abstract representations.
Students may try to determine the area of each triangle by counting the square units or using the “surround and subtract” method. Students may think that $\Delta ABC$ has the largest area because the others appear thinner.
This task is a good precursor to students developing the formula for the area of a triangle. The fact that each triangle has the same area can be used to highlight the meaning of the components of the area formula, as well as the meaning of the altitude of a triangle (an issue since the given triangles are not acute.)
Solution

The first triangle has half the area of a 1 unit by 4 unit rectangle, so the area is 2 square units.

The area of the second triangle is part of a rectangle that is a 2 unit by 4 unit rectangle with an area of 8 square units. The other two regions are two right triangles. Each of these has an area that is half a rectangle; the area of one is 4 square units and the other is 2 square units. Subtracting, we see that the area is 8  4  2 = 2 square units.

The area of the third triangle is part of a rectangle that is a 3 unit by 4 unit rectangle with an area of 12 square units. The other two regions are two right triangles. Each of these has an area that is half a rectangle; the area of one is 6 square units and the other is 4 square units. Subtracting, we see that the area is 12  6  4 = 2 square units.

The area of the fourth triangle is part of a rectangle that is a 4 unit by 4 unit rectangle with an area of 16 square units. The other two regions are two right triangles. Each of these has an area that is half a rectangle; the area of one is 8 square units and the other is 6 square units. Subtracting, we see that the area is 16  8  6 = 2 square units.
All the triangles have the same area.
Same Base and Height, Variation 1
Make the following triangles on a geoboard. Remember that the pegs are equally spaced in a square grid. Compare the areas of the triangles.
Which has the greatest area?
Which has the least area?
Do any of the triangles have the same area?
Are some areas impossible to compare?
Explain why you answered as you did.