Practicing fractions while learning some introductory geometry

Saw a nice introductory geometry problem in our Prealgebra book today. It is problem #7 on the 2005 AMC 8

2005 AMC 8 problem number 7

Here’s the problem:

Bill walks 1/2 mile south, then 3/4 mile east, and finally 1/2 mile south. How many miles is he, in a direct line, from his starting point?

I like a couple of different things about this problem. First, there are multiple ways to solve it, and each provides a little different insight into the geometric situation. Second, it provides a nice opportunity for a little fraction review, since you’ll likely encounter adding, multiplying, and maybe even dividing fractions in the solution.

Here’s my son’s approach to the problem:

His initial solution involved using two different triangles to find the length. I was interested to see if he could do it with just one triangle. This new solution involves drawing in two new lines, or maybe just rearranging the initial picture. I wanted to walk through this second solution because I think it is instructuve, but a little harder for a kid learning geometry to see.

It took a while, but we got there. One of the stumbling blocks was understanding what happened to the distances as we moved some of the triangles around.

I really love problems like this one. It gives a great opportunity to cover a new topic from a few different angles and also gives you an opportunity to sneak in a little review of an old topic. Definitely a fun morning.