Ran across a nice example problem in Art of Problem Solving’s *Introduction to Counting and Probability* this morning – problem 5.5 in section 5.3:

In how many ways can a dog breeder separate his 10 puppies into a group of 4 and a group of 6 if he has to keep Biter and Nipper, to of the puppies, in separate groups?

The boys worked on the problem together for about 10 minutes and found two different ways to approach the problem. Unfortunately those two ways produced different answers. In the first movie I had them talk through the two approaches:

While explaining their initial thoughts about the problem my older son noticed a mistake in their approach which used complimentary counting. The boys worked through that mistake below:

Next we looked at the approach using case by case counting – could we find the mistake here?

I thought their work on this problem was a really nice example of what learning math can sometimes look like. They did a nice job working through the problem two different ways the first time around. The fact that they found two different answers with those two approaches meant that at least one of those approaches must have had a small error. Reviewing the two approaches helped us find the mistakes and hopefully helped the boys learn a little bit more about counting.