What Is Area?

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New York State Common Core Math Geometry, Module 3, Lesson 1

Student Outcomes

Students review the area formula for rectangles with rational side lengths and prove the area formula for an arbitrary rectangle.
Students use a square grid to estimate the area of a curved region using lower approximations, upper approximations, and average approximations.

What Is Area?

Classwork

Exploratory Challenge 1

a. What is area?
b. What is the area of the rectangle below whose side lengths measure 3 units by 5 units?
c. What is the area of the 3/4 × 5/3 rectangle below?

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Exploratory Challenge 2

a. What is the area of the rectangle below whose side lengths measure √3 units by √2 units? Use the unit squares on the graph to guide your approximation. Explain how you determined your answer. b. Is your answer precise?

Discussion

Use Figures 1, 2, and 3 to find upper and lower approximations of the given rectangle.

Discussion

If it takes one can of paint to cover a unit square in the coordinate plane, how many cans of paint are needed to paint the region within the curved figure?

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