The first question uses only two skills, but demonstrates them in a powerful way...
How many of the ways that 27 cents can be made using quarters, dimes, nickels and pennies use and odd number of coins?
I rate this question about a 5 out of 10 because it has a few things going on at once and it's poorly worded so that the child has to read it a few times and think. This is warm up questions for a nervous middle school team, but a great question for 5th grader. On a 40 minute, 8 question test, this question should take no more than a few minutes. We spent about 15 minutes on it.
There are two ways to do this problem. You can spend 15 minutes on the question and 1 minute getting the answer right, or one minute on the question and 15 minutes getting it wrong. We don't use solutions. If the child does the problem correctly, the answer will be obvious to parent and child or easily provable. If it's not, I consider the question wrong. The 'right way' is the payoff, the solutions never show more than a single number, and I'm interested in the learning process and not the final number.
When I'm not helping as the missing team member, I usually look at the answer and declare it wrong. My trainee does it again, usually getting a better answer or learns to verbalize his defense.
There is only one correct way to do this problem. That would be the easiest cheatiest way or organize the work so that the least amount of effort is expended finding the qualifying combination of coins. Solving this problem is like coming up with 'alphabetizing' the qualifying combinations.
We started brainstorming combinations. This technique is a way to understand the problem and works in other cases. In this case, it led to an unverifiable mess of combinations.
The only correct answer - the answer that a 5th grader is unlikely to come up with on the first try, is to start with quarters (one), them move to dimes (2 and then 1, in that order), and on to nickles. Any other approach is too much work for me to grade.
The problem seems deceptively easy if you've never seen one of these before (which is the whole point). I asked 'How do you know you have the correct answer' and with a bit of staring, we found some missing combinations. The faster this problem goes, the worse the result.
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