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Same Base and Height, Variation 1


Alignments to Content Standards: 6.G.A.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.

Task_1_6d84af63f1bd7c5c510f82ce04837b63

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.

Michael Nakamaye says:

over 2 years

thanks for catching this-- it has been fixed now!

Fawn says:

over 2 years

The third and fourth bullets in solution should refer to the third and fourth triangles, respectfully.