Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2014 Grade 7 Geometry
Problem
Solution
1) The net shown folds to a cube with a missing face. Mark with an x two edges where the final square can be placed so that the net will fold to a cube with no missing faces.
 Start from face #1 and start folding.
  1. Face 1 folds up under face 2
  2. Face 3 folds around the side of face 2
  3. After face 3 is folded, face 4 then folds down to make a top
  4. Face 5 then folds down, making the face opposite face 2
  5. This leaves the face to the left of face 2 open, so the final face can be placed in any of 4 positions:
    1. To the left of face 2
    2. To the left of face 1
    3. On top of face 4
    4. On top of face 5


    Here's what that folded net looks like from several perspectives:

Problem
Solution
2) The point P (2, 3) is plotted for you. Reflect this point across y = x. Reflect P and its image across x = 0. Keep alternating reflecting the new image points across either y = x and across x = 0 until no new image points can be created. The points are the vertices of a polygon. How many lines of symmetry does the polygon have?
 Here is the figure with all reflected points and the following lines of symmetry:
  1. y = x
  2. y= -x
  3. x = 0
  4. y = 0
There are 4 lines of symmetry.
3) Treena is making cone hats for the play out of felt. She has a circle of felt with diameter 2 ft and is planning on making radial cuts to get slices of the circle. She will sew or use fabric glue to join the two radial edges together. The circumference of the opening needs to be 21 inches to fit the actors' heads. How many hats can she make from one circle?
  1. The circumference of the circle is
    C = 24 = 75.36 in (approximately)
  2. 75.36 / 21 = 3.588
  3. She can make 3 hats from one circle.
4) The length of a rectangle is increased to 2 times its original size and its width is increased to 3 times its original size. If the area of the new rectangle is equal to 1,800 square units, how many square units is the area of the original rectangle? Use L for the original rectangle's length and W for it's width.
The equation for the larger rectangle is:
    (2L)(3W) = 1800 sq. units
    6 LW = 1800
    LW = 1800 / 6 = 300 sq. units.
5) Initially the rectangular prism on the left was full of water. Then water was poured into the right cylindrical container so that the heights of water in both containers are equal. Find the height (h) of water in both containers. Round your answer to the nearest tenth of a centimeter.
(Drawing not to scale.)
  1. The volume of water removed from the prism is:
        Vp = 10x4x2 - 2x4xh
  2. The volume of water in the cylinder is:
        V = (1)2 h
  3. Set them equal to each other and solve for h:
    80 - 8 h = h
    (8 + ) h = 80
    11.14 h = 80
    h = 80/11.14 = 7.181 cm, rounded = 7.2 cm