Worked examples
Three worked MOEMS-style examples
Our own problems, one per strategy, at the level of Division E and Division M. They are not MOEMS problems and are not taken from any MOEMS sample, book or contest.
Example 1: look for a pattern (Division E level). Dots are arranged in triangles: 1 dot, then 3 dots, then 6, then 10. How many dots are in the 12th triangle? List and look: 1, 3, 6, 10 add 2, 3, 4 each time, so the nth triangle is 1 + 2 + ... + n. The 12th is 1 + 2 + ... + 12. Pair the ends: 1 + 12, 2 + 11 and so on, six pairs of 13, which is 78. The pairing trick turns a long sum into one multiplication.
Example 2: make an organised list (Division E level). In how many ways can you make 25 cents using pennies, nickels and dimes, if you must use at least one dime? Fix the dimes first. Two dimes: 5 cents left, made as one nickel or five pennies, 2 ways. One dime: 15 cents left, made with 3, 2, 1 or 0 nickels and pennies for the rest, 4 ways. Total 6 ways; rereading the condition confirms two dimes are allowed, since it says at least one. Listing in order of the biggest coin is what makes the count trustworthy.
Example 3: work backward (Division M level). Maya thinks of a number, doubles it, subtracts 5, then divides by 3 and gets 7. What was her number? Undo from the end: 7 x 3 = 21, then 21 + 5 = 26, then 26 / 2 = 13. Check forward: 13 doubled is 26, minus 5 is 21, divided by 3 is 7. Correct, the number is 13. Undoing each step in reverse order, with the opposite operation, is the whole strategy.
What the checks show
In example 2 the student pauses to reread the condition, and in example 3 they check forward. Both habits catch the kind of slip that loses a point on a five-problem contest, where one point is a fifth of the score.
We ask for a written check on every practice problem until it becomes automatic.
Changing the problem
After each example we change it: the 20th triangle, 30 cents, a different final number. If the student can redo it without help, the strategy has stuck.
Then we ask which strategy fits a problem before anyone solves it. Choosing is the real contest skill.
