answer:To find the area of the remaining paper, we first need to calculate the total area of the original square and then subtract the areas of the four smaller squares that were cut out. The area of the original square is found by multiplying the side length by itself, so it's 6 * 6 = 36 square inches. The area of each smaller square is 3 * 3 = 9 square inches, and since four such squares were cut out, the total area of the cut-out squares is 4 * 9 = 36 square inches. However, notice that cutting out a 3-inch square from each corner is equivalent to removing a 6-inch square (since 3 + 3 = 6) from the area of the original square, but with the added step of also removing four small right-angled triangles (formed by the 3-inch cut and the edges of the original square) and then adding back a 3-inch by 3-inch square (which was originally removed twice, once from each side). A simpler approach to think about it is to calculate the remaining shape's area directly. After cutting the squares from each corner, the remaining shape is a smaller square (3 inches smaller on each side) with 4 added right-angled triangles (each with 3-inch legs). The smaller square has sides of 6 - 2*3 = 0 inches on each side but this is an internal square, not visible from outside the shape, so let's focus on a simpler decomposition. Alternatively, think of the shape as a square with a side length of 6 inches, but with 3 inches removed from each side. This creates a square shape in the middle with 4 rectangles attached to it, and then the four removed corners. The remaining shape has sides that are 3 inches less on each end, hence 6 inches total (3 from each end) but only removed 3 inches total from the total width, to get an internal smaller square. A better perspective is to view it as a square (having 0-inch internal side) which can also be perceived as a square with 4, 3-inch wide arms (sides) extending out from that internal square. The 0-inch internal square does not make sense in this context; therefore, that perspective is not effective in this problem. The simpler perspective is this shape can be broken into an internal square and four arms, each arm being 3 inches wide, and 3 inches long. Now let's focus on an effective way of solving this problem. Instead, let's correctly view the cut-out paper as creating 4 rectangles. The original square was 6 inches by 6 inches, but the cut created four rectangles each with 3-inch width and 3-inch height, which together with the internal (smaller) square can also be perceived as one larger square with 4 of the corners cut out. Notice this description is similar to the given problem statement. In essence, this shape is simply a square (of 0 inch by 0 inch) plus four rectangles, each with a 3-inch by 3-inch area but a more useful perspective is to view this shape as one, large 6 inch by 6 inch square with the 4 smaller squares removed. Then, focusing on the simpler decomposition of the shape, we stretch the internal square from 0 by 0 to 6 by 6 and remove 4 corners. The following explanation avoids considering an internal square. Another approach is to notice that the width of each of the 4 rectangles created is 3 inches (6 inches minus 3 inches). These four 3-inch by 3-inch rectangles would form a square with a total area of 6 * 3 = 18 square inches. You may be thinking, but isn't there another square? Yes, there is another square, 3 inches by 3 inches. To be precise, and combining the descriptions above into a correct analysis: the four cut-out squares removed 4 * 9 = 36 square inches of area from the original square, but also removed (piecewise) four 3-inch by 3-inch sections from the original square that overlap in that inner square region, hence we must add back a single 3 by 3-inch square to accurately measure the total area of paper removed from the original square. However, looking at the problem more simply: After cutting out the 4 smaller squares from each corner of the original square (which created the four rectangles), the resulting shape has an area equal to the original square minus the area of the four removed squares, but the corners were overlaps - a simpler perspective would be to remove the four smaller squares but then add back one 3-inch by 3-inch square because that particular 3-inch square was removed twice from the area total for the corners. Now let's calculate that more simply. This removed area from the four corners (but

answer:To find the area of the remaining paper, we first need to calculate the total area of the original square and then subtract the areas of the four smaller squares that were cut out, and then add back the overlap area. The area of the original square is found by multiplying the side length by itself, so it's 6 * 6 = 36 square inches. The area of each smaller square is 3 * 3 = 9 square inches, and since four such squares were cut out, the total area of the cut-out squares is 4 * 9 = 36 square inches. However, the four removed squares overlap at their innermost corners. Essentially, the removed area includes a 3 by 3-inch square twice (from each side), so we add back a 3 by 3-inch square to compensate for the overlap. This overlap area is 3 * 3 = 9 square inches. Thus, the area of the remaining paper is the original square's area minus the area of the cut-out squares plus the overlap area, which is 36 - 36 + 9 = 9 square inches. To visualize this a bit differently, we notice that we have 4 right-angled triangles (3-inch sides), and one, 3-inch square (from the overlap), and four, 3-inch by 3-inch rectangles on each side of that central square - all of these shapes fill the remaining space and they can be rearranged to a total area equivalent to one, 3 by 3 square and four, 3-inch by 3-inch rectangles which combine into a total of one 6 inch by 3 inch rectangle and one 3 inch by 6 inch rectangle minus the overlap (which is the 3x3 inch square) which equals the remaining area. However, this explanation obscures the fact that the simplest perspective is to think of the shape in terms of one square (itself split into four corners and four 3-inch side lengths) minus the four removed corners and then adding back their overlap. That alternative visualization confirms the calculation above of the remaining area of the paper: 36 (original area) - 36 (removed area) + 9 (overlap area).

answer:When cutting out the smaller squares from the four corners of the original square, the resulting shape can be thought of as the original square minus the area of the four removed squares, plus the area where they overlap. The area of the original square is 6 * 6 = 36 square inches. The area of each smaller square is 3 * 3 = 9 square inches. Since four such squares were cut out, the total area removed is 4 * 9 = 36 square inches. However, these squares overlap in the center, essentially removing a 3 by 3 square twice, so we need to add back this overlap area of 3 * 3 = 9 square inches. The area of the remaining paper is the original square's area minus the area removed, plus the overlap area: 36 - 36 + 9 = 9 square inches.

answer:Dear James, I hope this message finds you well. I am writing to propose a collaborative project that I believe could effectively leverage our combined expertise. The project involves developing a comprehensive lesson plan that encourages students to analyze historical social movements and draw connections to contemporary environmental challenges. Utilizing data from NASA and other real-world examples, this curriculum would aim to illustrate the profound impact of social change on environmental conditions. I would greatly appreciate your thoughts on this idea. If you are interested, perhaps we could arrange a meeting to discuss the project in more detail. Best regards, Sarah