How to get from Boston to New York?

3/12/2010

You can reach Hartford from Boston by bus and by train. You can reach New York City from Hartford by bus, train, and plane. In how many different ways you can reach New York City from Boston if you know that you shall stop in Hartford? Please note that you can change transportation in Hartford (e.g. switching from bus to train).

I asked kids to draw 3 points on the white board representing Boston, Hartford, and New York City. They connected Boston -Hartford and Hartford - New York. Then to help them visualizing they drew bus and train next to Boston - Hartford line and bus, train, and plane next to Hartford - New York line. Then they started counting all combinations: bus - bus, bus - train, etc. One girl wrote 'bus' and 'train' on the left side of her white board and 'bus', 'train', and 'plane' on the right side. Then she connected all possible combinations.

The answer is 6.

Afterward I showed to them that instead of just counting all combinations one by one they could just multiple 2 by 3, where 2 is how many different ways to get from Boston to Hartford and 3 is how many ways to get from Hartford to New York.