Going back to Similar Triangles

My own history of learning math is a little strange. When I was in 5th grade we lived in Europe and I taught myself Algebra out of of an Italian algebra book. My elementary school didn’t really know what to do with me in 6th grade, so I ended up learning algebra again from a tutor from the University of Nebraska at Omaha. In 7th grade me Junior High also didn’t know what to do with me, and I was place in – you guessed it – algebra.

I don’t remember much from that class in jr. high. The teacher’s name was Mr. Stribley. The class was out in one of the portable classrooms that were constructed after the school was hit by a tornado in 1975. My girlfriend in high school was also in the class, though I doubt she knew my name then.

Anyway, I’ve always wondered whether or not that three year tour of algebra was useful or not. If nothing else, though, it left me with a reasonably good base in basic algebra!

I bring this little bit of history up because last week I noticed that my son was struggling a little with similar triangles. We covered the similar triangles chapter in our book a few months ago and maybe we just haven’t returned to the topic enough in the interim so the topic wasn’t fresh in his mind. Likely a much better explanation is that the ideas didn’t really sink in the first time around and I failed to notice. But, thinking about how to best learn the topic and also thinking about my own history of learning math, I decided that it would be worth spending a little time going back over that chapter again. Not as a review, but just starting from scratch. Hopefully this idea isn’t a huge mistake, but we have already had some good discussions about similar triangles, so I think the re-do is going to be useful.

Today we worked through a really challenging example in the section about “Side / Angle / Side” similarity. For me anyway, this type of similarity is the most difficult to see in problems. The picture for this particular problem makes the similarity especially hard to see because the two similar triangles are not oriented the same way. Our first time through the problem this morning probably took 30 minutes. The videos below are the 2nd time through. I thought that talking through the problem a second time would help him digest a few of the ideas in his mind. It takes most of the first video to get our arms around the picture:

 

Once we have a good picture, the problem becomes much easier to solve:

Definitely a hard problem, but definitely a good lesson in similar triangles. Hopefully this second time through the section on similar triangles will help him build up a good base in geometry.

A challenging number pattern problem

My younger son struggled with a number pattern problem from an old MOEMS test today. I enjoyed talking through it with him tonight because it was interesting to see how he approached the pattern in the numbers once he saw it – his approach was quite a bit different that what I was expecting.

Here’s an introduction to the problem and our initial talk that gets us on the path that surprised me:

 

So, my surprise in the last video is that he wanted to go to the end of a row and subtract a certain amount to get back to the beginning. I thought it would be interesting to see if he could see that you could also add 1 to the square at the end of the last row. This idea was hard for him to see, but eventually we got there.

 

At the end of the last video we talked about how the odd numbers relate to the perfect squares. The sequence of rows in the original problem hints at the relationship, though for me, at least, the connection doesn’t jump off the page. To get a better sense of that relationship we went to our kitchen table and looked at the relationship using snap cubes:

 

So, a fun little project starting from an old math contest problem. Ultimately the lesson I’m hoping to convey with my son here is about looking for patterns. The connection between arithmetic and geometry in the last part is also something that I hope he finds interesting. I always find it fun when geometry helps us understand arithmetic a little better.