Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2006 Grade 6 Geometry
Problem
Solutioon
1) A teacher places a plastic right triangle on the overhead projector to enlarge it so that the whole class can see it. It has a base of 5 cm and a height of 15 cm. If the image on the screen has a base of 20 cm, what is the area of the image triangle? l. Compute the ratio of the projected base to the one on the plastic triangle = 20/5 = 4
2. Apply this ratio to find the projected height =
    4 x 15 = 60 cm.
3. Compute the projected triangle's area =
    Area = (60 x 20) / 2 = 1200 / 2 = 600 sq. cm.
2) 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?
The figure to the right is what this stack looks like.

Please note that this problem is asking you to complete a rectangular prism, not a 3x3x3 cube.

1. Figure out the number of cubes in the figure, noting that everything is in the front except for 1 cube.
    # cubes in the figure = 7.
2. Since the front is 3 across and
    (by the right side figure) only 2 deep,
    a solid rectangular prism would have
    3 x 3 x 2 = 18 cubes.
3. Subtract these two to get what you need to add =
    18 - 7 = 11 cubes.

Problem
Solution
3) Jill decides to rearrange her room. She will rotate the bed and table clockwise 90 degrees around the center of her room. The center of the room is given by the origin. The original coordinates of the table were
(0, -4), (2, -4), (2, -6), (0, -6).
What will be the new coordinates of the table? 		"Clockwise" means this way: ↷
This problem is asking for the coordinates of the table, not the bed!

  1. To rotate coordinates means to negate the x coordinate and exchange the x and the y, like this:
    (x,y) = (y,-x)
  2. Perform the rotations:
    (0,-4) = (-4,0)
    (2,-4) = (-4,-2)
    (2,-6) = (-6,-2)
    (0,-6) = (-6,0)
See figure to the left.
4) 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 2 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.) 1. Each fold halves the area of the square and there are 5 of these folds, so the final area is
    1/2 x 1/2 x 1/2 x 1/2 x 1/2 = 1/32 of the original paper's area.
2. Compute the side length of the folded paper =
    8 x (1/32) = 1/4 inch.
3. Compute the perimeter of the small resulting square =
    4 x 1/4 = 1
5) Andy wants to make circular flower beds that will be outlined with 4 meter lengths of plastic edging. What is the radius of the biggest circular flower bed he can make with one length of plastic edging? Give your answer to the nearest tenth of a meter. 1. Given the circumference of the flower bed, what is its radius?
    Radius = C / (2) =
    4 / (2 X 3.14) = 0.63662 m.
2. Round to the nearest tenth of a meter = 0.6 m.