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Cairo wants to create a graph representing the heights and bases of all rectangles that have an area of square units. He started by drawing the rectangles A, B, C, and D below. Examine the dimensions (length and width) of each rectangle. 

Rectangle A has 36 tiles arranged 4 by 9. Rectangle B has a row of 8 half tiles followed by 4 rows of 8 tiles each. Rectangle C has 36 tiles arranged 6 by 6. Rectangle D has 36 tiles arranged 3 by 12.

  1. Copy the graph at right onto graph paper. Then match the letter of each rectangle above with a point on the graph. Which point is not matched?

    Rectangle A has a height of  units and a base of units. Its corresponding point should be .

    It corresponds to number .

    Repeat for each rectangle.

    You should find that is not matched.

  2. What are the base, height, and area for the unmatched point?

    Point 5 is over and up on the graph. Knowing this information, what are the rectangle's base, height, and area? Remember that the area is the base multiplied by the height.

  3. Why should the unmatched point not be on Cairo’s graph?

    Check the area of the unmatched rectangle that is represented by point .
    Does that area meet the qualifications of Cairo's graph that only contains rectangles with areas that are equal to ?

  4. Determine the dimensions of three more rectangles that have areas of square units. At least one of your examples should have dimensions that are not integers. Place a new point on the graph for each new rectangle you find.

    Answers will vary.

  5. Connect all of the points representing an area of square units. Describe the resulting graph.

    The graph should display a curve similar to the one at left.

    A first quadrant graph of a curve with x-axis labeled base and the y-axis labeled height, with the following points on a curve: The point (3, comma 16). The point (3.5, comma 12). The point (4, comma 9). The point (6, comma 6). The point (9, comma 4). The point (12, comma 3). The point (16, comma 2).

A first quadrant graph with x-axis labeled base and the y-axis labeled height with the following points labeled as follows: The point (3, comma 12) is labeled 1. The point (4, comma 9) is labeled 2. The point (6, comma 6) is labeled 3. The point (8, comma 4.5) is labeled 4. The point (6, comma 4) is labeled 5.