Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2015 Grade 5 Geometry
Problem
Solution
1) Ahmed is trying to make a parallelogram on the given coordinate grid. A, B, C are three corners of the parallelogram. If he wants to complete the figure in quadrant III, where should the final point go?
1. Quadrant III is the lower left quadrant where X and Y are both negative.
2. To make a parallelogram, the offset of A to the new point D should be the same as the offset of point B to point C which is (-2,-7).
3. Apply this offset to A to get the 4th point =
    (-3 -2 = -5,+1 -7 = -6).
    The new point is (-5,-6).
2) What is the angle between the hour and minute hands of an analog clock at 3:30 pm?


 1. Minute hand: is at 30 minutes which is 180 from 0 degrees.
2. In one hour the hour hand moves 30 degrees from 0.
3. In 3.5 hours it moves 105 degrees.
4. Subtract these two angles =
180 - 105 = 75 degrees.

Problem
Solution
3) The rectangle shown is divided evenly into squares. Each square has an area of 25 square centimeters. What is the perimeter of the rectangle?
 1. If each of the squares has an area of 25 sq. cm, then one side of the square is 5 cm.
2. Compute the perimeter given this measurement =
    14 x 5 = 70 cm.

4) Lenore is filling a box measuring 14 inches by 6 inches by 10 inches with cubes measuring 2 inches to a side. With the box standing at its tallest, she has filled one layer with the cubes. How many more cubes will she need to fill the box?		 1. The tallest side of the box is 14 inches.
2. If the box is standing at its tallest, then the base it is standing on is 6 by 10 inches.
3. Compute the area of one cube face = 2 x 2 = 4 sq. in.
4. Compute the area of the base = 6 x 10 = 60 sq. in..
5. Divide this area by the area of one cube face = 60/4 = 15 cubes.
6. Compute the volume of one little cube = 2 x 2 x 2 = 8 cu. in.
7. Compute the volume of the box = 14 x 6 x 10 = 840 cu. in.
8. Divide this volume by the volume of one cube to get the total number of cubes in the box = 840/8 = 105 cubes.
9. Subtract the number of cubes in that bottom row to get the number she will need to fill the box = 105 - 15 = 90 additional cubes.
5) A 16 in x 16 in square piece of paper is folded in half to form a triangle. That triangle is folded in half again three more times. If the original paper has a side length of 16 inches, what is the area of the last triangle after folding? 1. After the square piece of paper is folded it makes a triangle whose area is 128 sq. in.
2. With each fold after the first fold the area of the triangle is halved.
3. The number of halvings is 3.
4. The area of the last triangle is therefore 1/8 of the original triangles area = 16 sq. in.