how many ways can you divide a square into four congruent shapes? Remember that the answer cannot be infinity because rotations of one pattern still result in the same four congruent shapes.
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As pointed out in greenturtle3141's comment, there are indeed
ways to split the square into 4 congruent shapes without rotations.
Draw two lines crossing through the square's center, splitting it into 4 congruent shapes. If you were to pivot the lines around the square's center by any angle between 0 and 90 degrees (not inclusive), you end up with a new unique set of 4 congruent shapes. The range of unique rotations is just under 90 degrees; pivoting at least 90 degrees will result in a repeated split.
The answer is "infinity" because the rotation angle can be infinitesimally small. Pivoting by just a tiny fraction of a degree will produce new shapes that are not rotations. The range of rotation (just under 90 degrees) is continuous, so there are infinite possible angles.
It is possible that there are other ways to draw these two lines to divide the square into 4 congruent shapes. However, it doesn't matter, because infinity plus any positive number is still infinity.