Question

from unit 2, lesson 9
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a. how are the diagrams alike? how are they different?
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b. how is finding the area of the shaded region in each diagram alike? how is it different?
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Explanation:

Part a

• Alike: Both diagrams use a unit - length (1) to define a larger shape. The first diagram is a square (or rectangle) divided into smaller squares, and the second is a rectangle divided into smaller rectangles. They both use an area - model concept where the total area is composed of unit - related sub - areas.
• Different: The first diagram is a \(2\times4\) grid of squares (all shaded), and the second is a \(1\times4\) grid of rectangles where only \(\frac{1}{3}\) of the total height is shaded. The first has all sub - regions shaded, while the second has only a fraction of the vertical dimension shaded.

Part b

• Alike: For both, we can use the formula for the area of a rectangle \(A = l\times w\). In the first diagram, if we consider the side - length of the small square as 1, the total area is \(2\times4=8\) (since all are shaded). In the second, the length of the large rectangle is 4 (unit - length) and the width is 1. The area of the shaded part can be calculated using the formula where we consider the fraction of the width.
• Different: In the first diagram, since all sub - squares are shaded, we just calculate the total number of unit - squares (using \(l\times w\)). In the second, we need to multiply the length (\(l = 4\)) by the fraction of the width that is shaded. If we assume the total width of the second rectangle is divided into 3 equal parts (each of length 1 in the vertical direction, but the shaded part is \(\frac{1}{3}\) of the total width). So the area of the shaded part is \(4\times\frac{1}{3}=\frac{4}{3}\), while in the first it is a simple whole - number multiplication (\(2\times4 = 8\)).

Answer:

Part a

• Alike: Both use area - model with unit - length reference.
• Different: First is fully - shaded \(2\times4\) square/rectangle grid, second is \(\frac{1}{3}\) - shaded \(1\times4\) rectangle.

Part b

• Alike: Use \(A=l\times w\) formula.
• Different: First is total \(l\times w\) (whole - number), second is \(l\times\) (fraction of \(w\)).