Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2006 Grade 5 Geometry
Problem
1) Each smaller square is made by joining the midpoints of the sides of the larger surrounding square. What fraction of the area of the square ABCD is shaded? Give your answer as a reduced fraction.
2) An 8-inch square of thin paper is folded in half vertically and then horizontally to create a new square. These 2 folds are repeated 5 times each, including the first two folds, to create a tiny square. What is the perimeter of the new square? (An 8-inch square is a square that is 8 inches on each side.)
3) The figure is missing a measurement for one line segment. What is the missing measurement?

Problem
4) William cut a rectangle with a perimeter of 36 inches into 2-inch squares with no leftover scraps. What are possible dimensions for William's rectangle? List 3 sets of dimensions giving the length and width. (Note: A rectangle of 2 by 3 inches is the same as a rectangle of 3 by 2 inches.)
5) A cube sculpture is created from cubes stacked on top of each other and next to each other. Cubes that are stacked meet along their square faces. The top view, front view, and side view of a cube sculpture are given below, where only the squares directly facing you from each view point are shown. What is the least number of cubes you would need to add to the sculpture to make it a solid rectangular prism of cubes?