Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2008 Grade 5 Geometry
Problem
Solution
1) A pinwheel is made of four identical shapes each with a different color arranged counterclockwise in color order red, blue, green, and gold. They are evenly spaced. The wind spins the pinwheel counterclockwise, and when it stops, the red shape is now where the blue shape used to be. If the pinwheel went through two complete rotations before red ended up where blue used to be, how many total degrees did the red shape travel? One complete rotation is 360 degrees.
Two complete rotations are 720 degrees.
Each shape is how many degrees from the one next to it? 90 degrees.
The red shape and the blue shape are this many degrees apart.
Total rotation = two complte rotations 720o + 90o red rotation to blue = 810 total degrees.
2) John is trying to make a rectangle on the given coordinate grid. He has already labeled the point A at (-1, 3), the point B at (1, -1), and point C at (-2, 2). Where should he plot point D to complete the rectangle?
 1. To get from point B to A go down 2 and left 2 spaces.
2. Find point D from point C by these same offsets: (-4,0)
See figure to the left.

Problem
Solution
3) Toleen's math teacher makes a model of a solid rectangular prism out of white one centimeter cubes to explore surface area and volume. She paints the entire outside surface red. If the dimensions of the prism are 3 cm by 4 cm by 6 cm, how many of the cubes have exactly two sides painted red? The 3 cm edges (4 of them) have 1 cube
    with 2 painted sides = 4 cubes
The 4 cm edges (4 of them) have 2 cubes
    with 2 painted sides = 8 cubes
The 6 cm edges (4 of them) have 4 cubes
    with 2 painted sides = 16 cubes.
    4 + 8 + 16 = 28 cubes with 2 painted sides.
4) In Latoya's class they are cutting out snowflakes from square paper folded in half and then in half again to make a folded square. Latoya cuts along the three lines shown in the picture of the folded square and removes the smaller piece. When she unfolds this smaller piece what shape will it be?
The shape is an octagon
5) In right triangle AMC (see diagram) triangles ANP and PLC are congruent. MNPL is a square and segment LC has length 2 cm. What is the perimeter of the square MNPL?
Since they are congruent and LC is 2 cm, then NP is 2 cm.
It is a square, so the perimeter of the square is 4 x 2 = 8 cm.