Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2008 Grade 8 Algebra
Problem
Solution
1) Study the sequence of shapes below built out of squares. How many squares would be in the nth term of the sequence?
 Let n = the term number (1, 2, 3, ...)
  1. The top row is 3n
  2. The bottom vertical squares are n
  3. The ones in between are n(n-1)
  4. So the total number of squares in the nth term is:
        3n + n + n(n-1) =
        4n + n2 - n =
        3n + n2
  5. Test this on the first 3:
        for n=1: 3+1 = 4
        for n=2: 6+4 = 10
        for n=3: 9+9 = 18 It works!
2) What is the next term in this sequence:
    1, 2, 2, 4, 8, 32, 256, __
Each term is the product of the previous 2 terms,
so the next term is 256 x 32 = 8192
3) What is the 8th term in the following sequence?
    0th     SOHCAHTOA
    1st     AOTHACHOS
    2nd     AOHACHOS
    3rd     SOHCAHOA
    4th     SOCAHOA
  1. The input sequence with some spaces to clarify:
        0th     SOHCAHTOA
        1st     AOTHACHOS
        2nd     AO   HACHOS
        3rd     SOHCAHOA
        4th     SO   CAHOA
  2. The sequence is
    1st term: reverse term 0
    2nd term: remove 3rd letter
    3rd term: reverse term 2
    4th term: remove 3rd letter
  3. Continuing this pattern:
    5th term: reverse term 4         AOHACOS
    6th term: remove 3rd letter    AOACOS
    7th term: reverse term 6         SOCAOA
    8th term: remove 3rd letter:   SOAOA

Problem
Solution
4)A shape is formed by the lines x = 3, y = 2x + 6 and y = 0. Use the coordinate grid to draw the shape. What is the area of the shape?
  1. Plot a couple of points for the equation y = 2x + 6:
        x = 0 y = 6
        x = -3 y = 0
    This is enough to plot the line.
  2. The base of the triangle is 3 - (-3) = 6
  3. Plug x=3 into y = 2x+6 to get the height:
    y = 2(3) + 6 = 12
  4. The area is 6x12/2 = 36 sq. units
5) Mary realizes that she can buy 5 tops and 2 pairs of jeans for $96 because of store deals. She also figures out that she can buy 3 tops and 4 pairs of jeans for $115. If the tops all cost the same and the jeans all cost the same how much is just one pair of jeans? This is 2 equations and 2 unknowns.
Let t = the cost of a top and j = the cost of a single pair of jeans.
  1. Equation 1: 5t + 2j = $96
  2. Equation 2: 3t + 4j = $115
  3. Turn the first equation around into an expressin for t:
    t = (96 - 2j)/5
  4. Plug this into the second equation and solve for j:
    3(96 - 2j)/5 + 4j = 115
    57.6 - 1.2j + 4j = 115
    2.8j = 57.4
    j = $20.50