Question

Form squares using tiles and rectangles. How will you represent the total area of each figure? Using the sides of the tiles, write all the dimensions of the squares. What did you notice about the dimensions of the squares? Did you find any pattern in their dimensions? How can unknown quantities in geometric problems be solved?

Options:
A. The total area of the figure is calculated by adding the areas of the tiles and rectangles. The dimensions of the squares are 1x1, 2x2, 3x3, 4x4, 5x5. The dimensions form a sequence of perfect squares. Unknown quantities in geometric problems can be solved by observing patterns and using algebraic expressions.
B. The total area is determined by measuring the length and width of each tile and rectangle. The dimensions of the squares are random. There doesn't appear to be a pattern in their sizes. Unknown quantities can be solved through trial and error.
C. The area of each figure is the sum of the areas of tiles and rectangles. The dimensions of the squares are 2x2, 4x4, 6x6, 8x8. The dimensions form an even number sequence. Unknown quantities in geometry are solved through mathematical formulas.
D. Calculating the total area involves adding the areas of the tiles and rectangles. The dimensions of the squares are 3x3, 6x6, 9x9, 12x12. There's a clear pattern of increasing by 3. Unknown quantities can be solved through algebraic equations.

Answer 1

To represent the total area of figures created from squares and rectangles, sum the areas of individual figures. The dimensions of squares formed follow a pattern of perfect squares, and unknown geometric quantities can be resolved by using algebraic equations and understanding the relationship between dimensions and areas.

Step-by-step explanation:

To form squares using tiles and rectangles and represent their total area, calculate the area of each individual figure and add them together. If we use the sides of the tiles to form squares, the dimensions of these squares could be multiples of the tile's side. For example, if we have a tile with a 1-inch side, we can form squares with dimensions 1x1, 2x2, 3x3, and so on. In this context, the dimensions indicate the length of one side of the square, which are essentially perfect squares when multiplied by themselves.

Marta has a square tile with a side length of 4 inches. She has another square that is twice as large as the first one, which means the side length of the larger square is 4 inches x 2 = 8 inches. The area of the larger square is 8 inches x 8 inches = 64 square inches, which is 4 times the area of the smaller square (4 inches x 4 inches = 16 square inches).

This illustrates an important concept in geometry: when you scale the dimensions of a figure by a factor, the area is scaled by the square of that factor. Thus, unknown geometric quantities can often be solved through algebraic equations and understanding the relationships between linear dimensions and areas.