Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2013 Grade 7 Geometry
Problem
Solution
1) Write the letters M, T, and H on Figure 2 so that when the figure is folded into a box, it will spell MATH around the sides of the box. Figure 1 shows the correct placement of the letters.
  1. Well, the first 2 letters (the M and the T) should be pretty easy
  2. The problem is where does the "H" go and what is it's orientation?
  3. Try (in your head) some folding of the blank spaces to see which one produces a space next to the "T" that you placed. Make sure you have it's orientation correct.
2) The figure shows a drawing of a pyramid tower climbing toy located in a park. It is constructed with large congruent wooden cubes solidly nailed together. The volume of one wooden cube is 1000 cu. inch. What is the visible surface area in square inches? This includes the exposed north, south, east and west faces and looking top down.
 Since the volume of each cube is 1000 cu. inches, the side length is ____ inches.
    so, the area of one face is _____ sq. inches.
  • Looking down from the top, count the number of surfaces you see: ____ surfaces.
  • The only other exposed surfaces are the sides. Compute the number of exposed sides for each layer and add to the top surfaces


  • Total exposed surfaces = _____
  • Total surface area exposed = _____

Problem
Solution
3) In the diagram of nested squares each next square is inscribed in the previous square with its vertices bisecting the sides of the previous square. What fraction of the area of the largest outside square is shaded?
  1. Each inscribed square is 1⁄2 the area of the square it is inscribed in.

  2. The innermost square is _____ of the area of the outermost square.

  3. The 2 smaller shaded triangles, together, are ____ of the area of the innermost square = ______ the area of the outermost square

  4. The bottom shaded triangle is _____ of the area of the second inscribed square = _______ the area of the outer square

  5. Together, the 2 smaller triangles + the bottom triangle =
    _______ of the area of the outermost square.
4) A dog food company wants a bigger can size and a new look. It considers doubling the diameter or doubling the height. What is the ratio of the volume of the doubled diameter can to the volume of the doubled height can?
(Volume of a cylinder is r2 h )
  1. Volume of the doubled diameter can =
    _____________

  2. Volume of the doubled height can =
    _____________



  3. Their ratio = ________
5) The triangle with vertices (0,0), (3,0) and (0,4) is rotated by 90 degrees counterclockwise around the origin, the image is rotated by another 90 degrees counterclockwise around the origin, and that image is rotated by another 90 degrees counterclockwise around the origin to produce four triangles. Create a single figure by shading all four triangles. How many points that have integer values for both the x and y coordinate are in the interior of the shaded figure?
  1. Put the points on the graph to the right and perform the 3 rotations (counterclockwise is this way:↶),
    putting the new points on the same graph.
  2. Draw the lines of each of the 4 triangles with a ruler
  3. Carefully count the integer x and y points that are inside the combined figure.
  4. There are ____ of them.