A problem to get started
Making change
What is the fewest number of coins that it will take to make 43 cents if you have available pennies, nickels, dimes, and quarters? After you have solved this problem, provide an explanation that proves that your answer is correct?

How does the answer (and the justification) change if you only have pennies, dimes, and quarters available?

Note: We first encountered this problem in a conversation with Deborah Ball.