From arithmetic to algebra to geometry

We found a really neat question about triangles in our Introduction to Geometry book. The question gives you two sides of a triangle and asks about the possible values of the third side that would make the triangle obtuse.

It is pretty interesting question on several levels. One of those levels is that you can solve the problem with just basic arithmetic. In the case we study below, I restrict the values of the third side to integers.

 

At the end of the discussion we have an answer to the problem, but no pictures. Pretty surprising for a geometry problem! We went to Wolfram Alpha to take a look at some of the triangles.

 

So, a fun little project which serves as a nice illustration of the connection between arithmetic, algebra, and geometry. Pretty amazing what you can learn about a triangle just from some simple observations about the side lengths.

Integers, fractions, and decimals are totally different things to kids.

My younger son and I are in the middle of talking about percents. This morning the topic at hand was questions like – what number is 20% more than 50?

It was interesting to me to see the difference in approach that my son had to problems involving integers, fractions, and decimals. It is easy for me to forget how different these representations of numbers are to kids.

Integers first – what is 20% more than 70?

Now fractions – what is 30% less than 1/5?

Finally decimals – what is 10% more than 3.4

I don’t know at what level of mathematical development integers, fractions, and decimals all meld into numbers. It sure is easy for me to forget that they don’t appear to be the same things to a kid.